Jean-Louis Verger-Gaugry
email: jean-louis.verger-gaugry@univ-smb.fr

A Panorama on the Minoration of the Mahler Measure: from the Problem of Lehmer to its Reformulations in Topology and Geometry

Keywords: Mathematics Subject Classification (2010 ). 11R06; 11G50, 11R04, 11S05, 14J28, 14J50, 20F55, 22E40, 30B10, 30C15, 37B40, 51F15, 57M10, 57M15, 57M25, 57M27, 57M50 Lehmer conjecture, Lehmer problem, Lehmer's number, Schinzel-Zassenhaus conjecture, mapping class, stretch factor, coxeter system, Salem number, Perron number, Pisot number, Mahler measure, minoration, algebraic dynamical system, torsion growth, heights, knots, links, Alexander polynomial, Jones polynomial, Coxeter systems . . . . small stretch factors . . .

This text is addressed to readers who are already engaged in their own research in the problem of Lehmer in at least one domain concerned, like graduate students, post-docs and active researchers. Initially the problem of Lehmer appeared in the original paper of D. Lehmer in Number Theory in 1933. The great richness of the problem of Lehmer is reflected today by the fact that the problem of minoration of the Mahler measure can been reformulated in several domains. Many interesting directions of research have appeared in doing so, and are developping in parallel. These domains are evoked in the present article. The reader not accustomed to the problem of Lehmer may find an interest to discover the different reformulations of the Mahler measure and its realizations in topology, geometry,... The aims of this text is to provide an extremely sketchy panorama of a large area of this part of mathematics, now spread over Number Theory, Arithmetic Geometry, and much more. It can be used as a guide to those who want to embark on a more serious study of one of these domains, or develop new analogues in existing theories.

Introduction

The question (called Problem of Lehmer) asked by Lehmer in [START_REF] Lehmer | Factorization of Certain Cyclotomic Functions[END_REF] (1933) about the existence of integer univariate polynomials of Mahler measure arbitrarily close to one became a conjecture. Let us recall it: Problem of Lehmer. If is a positive quantity, to find a polynomial of the form f (x) = x r + a 1 x r-1 + . . . + a r where the a i s are integers, such that the absolute value of the product of those roots of f which lie outside the unit circle, lies between 1 and 1 + ... Whether or not the problem has a solution for < 0.176 we do not know.

This problem takes its origin in the search for large prime numbers. In [START_REF] Lehmer | Factorization of Certain Cyclotomic Functions[END_REF] Lehmer introduces an arithmetic process to obtain explicitely large prime numbers from integer polynomials of very small Mahler measure > 1, where "very small" Mahler measures would correspond to "very large" prime numbers. Since then, the strategy of his method has been revisited (see §2.1). Lehmer's Conjecture is addressed to minimal polynomials of algebraic integers. It is stated as follows:

Conjecture 1.1 (Lehmer's Conjecture). There exists an universal constant c > 0 such that the Mahler measure M(α) satisfies M(α) ≥ 1 + c for all nonzero algebraic numbers α, not being a root of unity.

If α is a nonzero algebraic integer, M(α) = 1 if and only if α = 1 or is a root of unity by Kronecker's Theorem (1857) [START_REF] Kronecker | Zwei Sätze über Gleichungen mit Ganzzahligen Coefficienten[END_REF]. Lehmer's Conjecture asserts a discontinuity of the value of M(α), α ∈ O Q , at 1. Then solving Lehmer's Conjecture amounts to a problem of minoration of the Mahler measure M(α) when the absolute value > 1 of α ∈ O Q tends to 1 + .

Lehmer's Conjecture has been extensively studied in Number Theory, e.g. by Amoroso [START_REF] Amoroso | Algebraic Numbers Close to 1: Results and Methods, in Number Theory[END_REF], [START_REF] Amoroso | Algebraic Numbers Close to 1 and Variants of Mahler's Measure[END_REF], Bertin, Decomps-Guilloux, Grandet-Hugot, Pathiaux-Delefosse and Schreiber [START_REF] Bertin | Pisot and Salem Numbers[END_REF], Blansky and Montgomery [START_REF] Blansky | Algebraic Integers Near the Unit Circle[END_REF], Boyd [START_REF] Boyd | Pisot and Salem Numbers in Intervals of the Real Line[END_REF], [START_REF] Boyd | Variations on a Theme of Kronecker[END_REF], Cantor and Strauss [START_REF] Cantor | On a conjecture of D.H Lehmer[END_REF], Dobrowolski [START_REF] Dobrowolski | On a Question of Lehmer and the Number of Irreducible Factors of a Polynomial[END_REF], Dubickas [START_REF] Dubickas | On Algebraic Numbers of Small Measure[END_REF], Langevin [START_REF] Langevin | Méthode de Fekete-Szegő et Problème de Lehmer[END_REF], Louboutin [START_REF] Louboutin | Sur la Mesure de Mahler d'un Nombre Algébrique[END_REF], Mossinghoff, Rhin and Wu [START_REF] Mossinghoff | Minimal Mahler Measures[END_REF], Schinzel [START_REF] Schinzel | On the Product of the Conjugates Outside the Unit Circle of an Algebraic Number[END_REF], Smyth [START_REF] Smyth | The Mahler Measure of Algebraic Numbers: A Survey[END_REF], [START_REF] Smyth | Seventy Years of Salem Numbers[END_REF], Stewart [START_REF] Stewart | On a Theorem of Kronecker and a Related Question of Lehmer[END_REF], Waldschmidt [START_REF] Waldschmidt | Sur le Produit des Conjugués Extérieurs au Cercle[END_REF], [START_REF] Waldschmidt | Auxiliary Functions in Transcendental Number Theory, SASTRA Ramanujan lectures[END_REF]). It has been extended to Arithmetic Geometry by replacing the Mahler measure M by a suitable notion of "height", and by reformulating the minoration problem to elliptic curves, to Abelian varieties, ..., e.g. by Amoroso and David [START_REF] Amoroso | Minoration de la Hauteur Normalisée dans un Tore[END_REF], David and Hindry [START_REF] David | Minorations de Hauteurs sur les Variétés Abéliennes[END_REF], [START_REF] David | Minoration de la hauteur de Néron-Tate sur les variétés de type C.M[END_REF], Hindry and Silverman [START_REF] Hindry | On Lehmer's Conjecture for Elliptic Curves[END_REF], Laurent [START_REF] Laurent | Minoration de la Hauteur de Néron-Tate[END_REF], Masser [START_REF] Masser | Counting Points of Small Height on Elliptic Curves[END_REF], Silverman [START_REF] Silverman | Lower Bound for the Canonical Height on Elliptic Curves[END_REF].

In his attempt to solve Lehmer's Conjecture in 1965, Schinzel has refined the minoration problem of the Mahler measure M(α) by replacing the condition "when the absolute value > 1 of α ∈ O Q tends to 1 + " by the condition "when the house α > 1 of α ∈ O Q tends to 1 + ". The following Conjecture has been formulated in [START_REF] Schinzel | A Refinement of Two Theorems of Kronecker[END_REF].

Conjecture 1.2 (Schinzel -Zassenhaus's Conjecture). Denote by m h (n) the minimum of the houses α of the algebraic integers α of degree n which are not a root of unity. There exists a (universal) constant C > 0 such that

m h (n) ≥ 1 + C n , n ≥ 2. (1.1)
The objective of this Survey is first to review a certain number of results in Number Theory ( § 2 and § 3) and in higher dimension in Arithmetic Geometry ( § 4), then to go beyond to various extensions where different reformulations of the minoration problem of Lehmer exist ( § 5). In other domains ( § 6) the minoration problem of Lehmer emphasizes the role played by algebraic integers like Pisot numbers, Salem numbers, Perron numbers and calls for the problem of the realization when a nontrivial minimum is reached. It occurs in some cases that the minimum is realized by Lehmer's number 1.17628 . . . (cf (2.7) and (2.8)).

This Panorama/Survey tries to take stock of the problem of minoration of the Mahler measure in all its forms.

Standard notations are reported in the Appendix.

Number theory

2.1. Prime numbers, asymptotic expansions, minorations. The search for very large prime numbers has a long history. The method of linear recurrence sequences of numbers (∆ m ), typically satisfying

∆ m+n+1 = A 1 ∆ m+1 + A 2 ∆ m+2 + . . . + A n ∆ m+n , (2.1)
in which prime numbers can be found, has been investigated from several viewpoints, by many authors [START_REF] Bilu | Existence of Prime Divisors of Lucas and Lehmer Numbers[END_REF], [START_REF] Everest | Primes Generated by Recurrence Sequences[END_REF], [START_REF] Everest | Recurrent Sequences[END_REF]: in 1933 Lehmer [START_REF] Lehmer | Factorization of Certain Cyclotomic Functions[END_REF] developped an exhaustive approach from the Pierce numbers [START_REF] Pierce | The Numerical Factors of the Arithmetic Forms n i=1 (1 ± α m i )[END_REF] ∆ n = ∆ n (P ) =

d i=1 (α n i -1) (2.2)
of a monic integer polynomial P where α i are the roots of P . The sequence (A i ) in (2.1) is then the coefficient vector of the integer monic polynomial which is the least common multiple of the d + 1 polynomials: P (0) (x) = x -1,

P (1) (x) = d i=1 (x -α i ), P (2) (x) = d-1 i>j=1
(x -α i α j ), . . . , P (d) (x) = x -α 1 α 2 . . . α d (Theorem 13 in [START_REF] Lehmer | Factorization of Certain Cyclotomic Functions[END_REF]). Large prime numbers, possibly at a certain power, can be found in the factorizations of |∆ n | that have large absolute values (Dubickas [START_REF] Dubickas | Divisibility of the Resultant of a Polynomial and a Cyclotomic Polynomial[END_REF], Ji and Qin [START_REF] Ji | The Numerical Factors of ∆n(f, g)[END_REF] in connection with Iwasawa theory). This can be done fairly quickly if the absolute values |∆ n | do not increase too rapidly (slow growth rate).

If P has no root on the unit circle, Lehmer proves lim n→∞ ∆ n+1 ∆ n = M(P ).

(2.3)

Einsiedler, Everest and Ward [START_REF] Einsiedler | Primes in Sequences Associated to Polynomials (after Lehmer)[END_REF] revisited and extended the results of Lehmer in terms of the dynamics of toral automorphisms ( [START_REF] Everest | Heights of Polynomials and Entropy in Algebraic Dynamics[END_REF], Lind [START_REF] Lind | Dynamical Properties of Quasihyperbolic Toral Automorphisms[END_REF]). They considered expansive (no root on |z| = 1), ergodic (no α i is a root of unity) and quasihyperbolic (if P is ergodic but not expansive) polynomials P and number theoretic heuristic arguments for estimating densities of primes in (∆ n ). In the quasihyperbolic case (for instance for irreducible Salem polynomials P ), more general than the expansive case considered by Lehmer, (2.3) does not extend but the following more robust convergence law holds [START_REF] Lind | Dynamical Properties of Quasihyperbolic Toral Automorphisms[END_REF]:

lim n→∞ ∆ 1/n n = M(P ).
(2.4)

If P has a small Mahler measure, < Θ, it is reciprocal by [START_REF] Smyth | On the Product of the Conjugates Outside the Unit Circle of an Algebraic Integer[END_REF] and the quotients ∆ n /∆ 1 are perfect squares for all n ≥ 1 odd. With Γ n (P ) := ∆ n /∆ 1 in such cases, they obtain the existence of the limit

lim j→∞ j Log Log Γ nj , (2.5)
(n j ) being a sequence of integers for which Γ nj is prime, as a consequence of Merten's Theorem. This limit, say E P , is likely to satisfy the inequality: E P ≥ 2e γ /Log M(P ), where γ = 0.577 . . . is the Euler constant. Moreover, by numberfields analogues of the heuristics for Mersenne numbers (Wagstaff, Caldwell), they suggest that the number of prime values of Γ nj (P ) with n j ≤ x is approximately 2e γ Log M(P ) Log x. (2.6) This result shows the interest of having a polynomial P of small Mahler measure to obtain a sequence (∆ n ) associated with P very rich in primes. These authors consider many examples which fit coherently the heuristics. However, the discrepancy function is still obscure and reflects the deep arithmetics of the factorization of the integers |∆ n | and of the quantities Γ n . More generally, than in the case of (2.1), the Conjecture of Lehmer means that the growth rate of an integer linear recurrence sequence is uniformly bounded from below ( § 7.6 in [START_REF] Everest | Recurrent Sequences[END_REF], [START_REF] Everest | Primes in Divisibility Sequences[END_REF]).

In view of understanding the size of the primes p ≥ 3 found in (∆ n ) generated by the exhaustive method of the Pierce's numbers, Lehmer, in [START_REF] Lehmer | Prime Factors of Cyclotomic Class Numbers[END_REF] (1977), established correlations between the Pierce's numbers |∆ n | and the prime factors of the first factor of the class number of the cyclotomic fields Q(ξ p ) (ξ p is a primitive pth root of unity), using Kummer's formula, the prime factors being sorted out into arithmetic progressions: let h(p) the class number of Q(ξ p ) and let h + (p) be the class number of of the real subfield Q(ξ p + ξ -1 p ). Kummer (1851) established that the ratio h -(p) = h(p)/h + (p) is an integer, called relative class number or first factor of the class number, and that p divides h(p) if and only if p divides h -(p). The factorization and the arithmetics of the large values of h -(p) is a deep problem [START_REF] Agoh | On the Relative Class Number of Special Cyclotomic Fields[END_REF], [START_REF] Fung | Computation of the First Factor of the Class Number of Cyclotomic Fields[END_REF], [START_REF] Granville | On the Size of the First Factor of the Class Number of a Cyclotomic Field[END_REF], [START_REF] Lehmer | Table of the Cyclotomic Class Numbers h * (p) and their Factors for 200[END_REF], related to class field theory in [START_REF] Masley | On the First Factor of the Class Number of Prime Cyclotomic Fields[END_REF], where the validity of Kummer's conjectured asymptotic formula for h -(p) was reconsidered by Granville [START_REF] Debaenne | The First Factor of the Class Number of the p-th Cyclotomic Field[END_REF] [START_REF] Granville | On the Size of the First Factor of the Class Number of a Cyclotomic Field[END_REF].

The smallest Mahler measure M(α)(> 1) known, where α is a nonzero algebraic number which is not a root of unity, is Lehmer's number = 1.17628 . . . , (2.7) the smallest Salem number discovered by Lehmer [START_REF] Lehmer | Factorization of Certain Cyclotomic Functions[END_REF] in 1933 as dominant root of Lehmer's polynomial (2.8).

X 10 + X 9 -X 7 -X 6 -X 5 -X 4 -X 3 + X + 1.

(2.8)

It is the smallest Salem number known [389] [391]. Lehmer discovered other small Salem numbers (cf §3). Small Salem numbers were reinvestigated by Boyd in [START_REF] Boyd | Small Salem Numbers[END_REF], [START_REF] Boyd | Pisot and Salem Numbers in Intervals of the Real Line[END_REF], [START_REF] Boyd | Reciprocal Polynomials Having Small Mahler Measure[END_REF], by Flammang, Grandcolas and Rhin [START_REF] Flammang | Small Salem numbers[END_REF]. The search of small Mahler measures was reconsidered by Mossinghoff [START_REF] Mossinghoff | Algorithms for the Determination of Polynomials with Small Mahler Measure[END_REF], [START_REF] Mossinghoff | Polynomials with Small Mahler Measure[END_REF], then, using auxiliary functions, by Mossinghoff, Rhin and Wu [START_REF] Mossinghoff | Minimal Mahler Measures[END_REF]. For degrees up to 180, the list of Mossinghoff [START_REF] Mossinghoff | Known Polynomials with Small Mahler Measure Through Degree 180[END_REF] (2001), with contributions of Boyd, Flammang, Grandcolas, Lisonek, Poulet, Rhin and Sac-Epée [START_REF] Rhin | New Methods Providing High Degree Polynomials with Small Mahler Measure[END_REF], Smyth, gives primitive, irreducible, noncyclotomic integer polynomials of degree at most 180 and of Mahler measure less than 1.3; this list is complete for degrees less than 40 [START_REF] Mossinghoff | Minimal Mahler Measures[END_REF], and, for Salem numbers, contains the list of the 47 known smallest Salem numbers, all of degree ≤ 44 [START_REF] Flammang | Small Salem numbers[END_REF].

Lehmer's Conjecture is true (solved) in the following particular cases.

(1) for the closed set S of Pisot numbers (Salem [451], Siegel [START_REF] Siegel | Algebraic Integers whose Conjugates Lie in the Unit Circle[END_REF], Bertin et al [START_REF] Bertin | Pisot and Salem Numbers[END_REF]), [START_REF] Adler | Topological Entropy and Equivalence of Dynamical Systems[END_REF] for the set of algebraic numbers α for which the minimal polynomial P α is nonreciprocal by Smyth's Theorem (1971) [START_REF] Smyth | On the Product of the Conjugates Outside the Unit Circle of an Algebraic Integer[END_REF], [START_REF] Smyth | Topics in the Theory of Numbers[END_REF], which asserts:

M(α) = M(P α ) ≥ Θ, (2.9)
proved to be an isolated infimum by Smyth [START_REF] Smyth | Topics in the Theory of Numbers[END_REF] (Θ is defined in (7.2)), (3) for every nonzero algebraic integer α ∈ L, of degree d, assuming that L is a totally real algebraic number field, or a CM field (a totally complex quadratic extension of a totally real number field); then Schinzel [START_REF] Schinzel | On the Product of the Conjugates Outside the Unit Circle of an Algebraic Number[END_REF] obtained the minoration

M(α) ≥ 1 + √ 5 2 d/2
, from which : M(α) ≥ ((1 + √ 5)/2) 1/2 = 1.2720 . . .

(2.10) Improvments of this lower bound, by Bertin, Rhin, Zaimi and Garza are given in § 3, (4) for α an algebraic number of degree d such that there exists a prime number p ≤ d Log d that is not ramified in the field Q(α); then Mignotte [378] [379] showed: M(α) ≥ 1.2; by extension, Silverman [START_REF] Silverman | Lower Bounds for Height Functions[END_REF] proved that the Conjecture of Lehmer is true if there exist primes p 1 , p 2 , . . . ,

p d in Q(α) satisfying N p i ≤ √ d Log d, (5)
for any noncyclotomic irreducible polynomial P with all odd coefficients; Borwein, Dobrowolski and Mossinghoff [START_REF] Borwein | Lehmer's Problem for Polynomials With Odd Coefficients[END_REF] [236] proved (cf Theorem 2.3 and Silverman's Theorem 2.4 for details) M(P ) ≥ 5 1/4 = 1.4953 . . . , (2.11) [START_REF] Allcock | A Banach Space Determined by the Weil Height[END_REF] in terms of the Weil height, Amoroso and David [START_REF] Amoroso | Le Problème de Lehmer en Dimension Supérieure[END_REF] proved that there exists a constant c > 0 such that, for all nonzero algebraic number α, of degree d, not being a root of unity, under the assumption that the extension Q(α)/Q is Galois, then h(α) ≥ c d .

(2.12) Some minorations are known for some classes of polynomials (Panaitopol [408]). Bazylewicz [START_REF] Bazylewicz | On the Product of the Conjugates Outside the Unit Circle of an Algebraic Integer[END_REF] extended Smyth's Theorem, i.e. the lower bound given by (2.9), to polynomials over Kroneckerian fields K (i.e. for which K/Q is totally real or is a totally complex nonreal quadratic extension of such fields). Notari [START_REF] Notari | Sur le Produit des Conjugués à l'Extérieur du Cercle Unité d'un Nombre Algébrique[END_REF] and Lloyd-Smith [START_REF] Lloyd-Smith | Algebraic Numbers Near the Unit Circle[END_REF] extended such results to algebraic numbers. Lehmer's problem is related to the minoration problem of the discriminant (Bertrand [50]). Perron numbers, topology and limit points of their subclasses: Mahler measures, Salem numbers. Mahler measures {M(α) | α ∈ Q} are Perron numbers, by the work of Adler and Marcus [START_REF] Adler | Topological Entropy and Equivalence of Dynamical Systems[END_REF] in topological dynamics, as a consequence of the Perron-Frobenius theory. The set P of Perron numbers is everywhere dense in [1, +∞) and is important since it contains subcollections which have particular topological properties for which conjectures exist. Let us recall them.

The set P admits a nonfactorial multiplicative arithmetics [START_REF] Brunotte | Algebraic Properties of Weak Perron Numbers[END_REF], [START_REF] Lind | The Entropies of Topological Markov Shifts and a Related Class of Algebraic Integers[END_REF], [START_REF] Verger-Gaugry | Uniform Distribution of the Galois Conjugates and Beta-Conjugates of a Parry Number Near the Unit Circle and Dichotomy of Perron Numbers[END_REF], for which the restriction of the usual addition + to a given subcollection is not necessarily internal [START_REF] Dubickas | Sumsets of Pisot and Salem Numbers[END_REF]. Salem [START_REF] Salem | Power Series with Integral Coefficients[END_REF] proved that S ⊂ P is closed, and that S ⊂ T . Boyd [START_REF] Boyd | Pisot Sequences Which Satisfy No Linear Recurrence[END_REF] conjectured that S ∪ T is closed (2.13) and that the first derived set of S ∪ T (cf Appendix for a definition) satifies S = (S ∪ T) (1) (2.14) ( [START_REF] Boyd | Small Salem Numbers[END_REF], p. 237). This second Conjecture would imply that all Salem numbers < Θ would also be isolated Salem numbers, not only Lehmer's number. The set of Mahler measures {M(α) | α ∈ Q} and the semi-group {M(P ) | P (X) ∈ Z[X]} are strict subsets of P and are distinct (Boyd [82], Dubickas [177] [178]). The probabilistic distribution and values of Mahler measures were studied by Boyd [START_REF] Boyd | The Maximal Modulus of an Algebraic Integer[END_REF], [START_REF] Boyd | Inverse Problems for Mahler's Measure[END_REF], [START_REF] Boyd | Perron Units Which Are Not Mahler Measures[END_REF], Chern and Vaaler [START_REF] Chern | The distribution of values of Mahler's measure[END_REF], Dixon and Dubickas [START_REF] Dixon | The Values of Mahler Measures[END_REF], Dubickas [START_REF] Dubickas | Nonreciprocal Algebraic Numbers of Small Measure[END_REF], Schinzel [START_REF] Schinzel | On Values of the Mahler Measure in a Quadratic Field (Solution of a Problem of Dixon and Dubickas)[END_REF], Sinclair [START_REF] Sinclair | The Distribution of Mahler's Measures of Reciprocal Polynomials[END_REF]. Boyd [START_REF] Boyd | Perron Units Which Are Not Mahler Measures[END_REF] has shown that the Perron numbers γ n which are the dominant roots of the height one (irreducible) trinomials -1-z +z n , n ≥ 4, are not Mahler measures. The inverse problem for the Mahler measure consists in determining whether, or not, a Perron number γ is the Mahler measure M(P ) of an integer polynomial P , and to give formulas for the number

#{P ∈ Z[X] | M(P ) = γ} (2.15)
of such polynomials with measure γ and given degree (Boyd [81], Dixon and Dubickas [START_REF] Dixon | The Values of Mahler Measures[END_REF], Staines [START_REF] Staines | On the Inverse Problem for Mahler Measure[END_REF]). Drungilas and Dubickas [START_REF] Drungilas | Every Real Algebraic Number is the Difference of Two Mahler Measures[END_REF] and Dubickas [START_REF] Dubickas | Nonreciprocal Algebraic Numbers of Small Measure[END_REF], [START_REF] Dubickas | Mahler Measures Generate the Largest Possible Groups[END_REF], proved that the subset of Mahler measures is very rich: namely, for any Perron number β, there exists an integer n ∈ N such that nβ is a Mahler measure, and any real algebraic integer is the difference of two Mahler measures.

Minorations and extremality. Algebraic numbers close to 1 ask many questions [START_REF] Dobrowolski | On a Question of Lehmer[END_REF] and require new methods of investigation, as reported by Amoroso [START_REF] Amoroso | Algebraic Numbers Close to 1: Results and Methods, in Number Theory[END_REF]. For α an algebraic integer of degree d > 1, not a root of unity, Blansky and Montgomery [START_REF] Blansky | Algebraic Integers Near the Unit Circle[END_REF] showed, with multivariate Fourier series,

M(α) > 1 + 1 52 1 dLog (6d) . (2.16)
By a different approach, using an auxiliary function and a proof of transcendence (Thue's method), Stewart [START_REF] Stewart | On a Theorem of Kronecker and a Related Question of Lehmer[END_REF] obtained the same minoration but with a constant c = 1/52 instead of 1/52 [START_REF] Bugeaud | Nombres Algébriques de Petite Mesure et Formes Linéaires en un Logarithme[END_REF], [START_REF] Laurent | Sur Quelques Résultats Récents de Transcendance[END_REF], [START_REF] Mignotte | On Algebraic Numbers of Small Height: Linear Forms in One Logarithm[END_REF], [START_REF] Waldschmidt | Sur le Produit des Conjugués Extérieurs au Cercle[END_REF]. In 1979 a remarkable minoration has been obtained by Dobrowolski [START_REF] Dobrowolski | On a Question of Lehmer and the Number of Irreducible Factors of a Polynomial[END_REF] who showed

M(α) > 1 + (1 -) Log Log d Log d 3 , d > d 1 ( ).
(2.17)

for any nonzero algebraic number α of degree d, with 1 -replaced by 1/1200 for an effective version (then with d ≥ 2), in particular for |α| > 1 arbitrarily close to 1. The minoration (2.17) was also obtained by Mignotte in [START_REF] Mignotte | e année: 1977/78[END_REF] [379] but with a constant smaller than 1 -. Cantor and Strauss [START_REF] Cantor | On a conjecture of D.H Lehmer[END_REF], [START_REF] Pathiaux-Delefosse | Résultat de Cantor et Strauss sur la Conjecture de Lehmer[END_REF], then Louboutin [START_REF] Louboutin | Sur la Mesure de Mahler d'un Nombre Algébrique[END_REF], improved the constant 1 -: they obtained 2(1 + o(1)), resp. 9/4 (cf also Rausch [START_REF] Rausch | On a Theorem of Dobrowolski about the Product of Conjugate Numbers[END_REF] and Lloyd-Smith [START_REF] Lloyd-Smith | Algebraic Numbers Near the Unit Circle[END_REF]). If α is a nonzero algebraic number of degree d ≥ 2, Voutier [START_REF] Voutier | An Effective Lower Bound for the Height of Algebraic Numbers[END_REF] obtained the better effective minorations:

M(α) > 1 + 1 4 Log Log d Log d 3 and M(α) > 1 + 2 (Log (3d)) 3 .
(2.18)

For sufficiently large degree d, Waldschmidt ([527], Theorem 3.17) showed that the constant 1-could be replaced in (2.17) by 1/250 with a transcendence proof which uses an interpolation determinant. It is remarkable that these minorations only depend upon the degree of α and not of the size of the coefficients, i.e. of the (naïve) height of their minimal polynomial. Dobrowolski's proof is a transcendence proof (using Siegel's lemma, extrapolation at finite places) which has been extended to the various Lehmer problems ( § 4). An algebraic integer α, of degree n, is said extremal if α = m h (n). An extremal algebraic integer is not necessarily a Perron number [START_REF] Bertrand-Mathis | Nombres de Pisot, Matrices Primitives et Beta-Conjugués[END_REF].

In 1965 Schinzel and Zassenhaus [START_REF] Schinzel | A Refinement of Two Theorems of Kronecker[END_REF] formulated Conjecture 1.2 and obtained the first result: for α = 0 being an algebraic integer of degree n ≥ 2 which is not a root of unity, then

α > 1 + 4 -(s+2) , (2.19)
where 2s is the number of nonreal conjugates of α. For a nonreciprocal algebraic integer α of degree n, Cassels [START_REF] Cassels | On a problem of Schinzel and Zassenhaus[END_REF] obtained:

α > 1 + c 2 n , with c 2 = 0.1; (2.20)
Breusch [START_REF] Breusch | On the Distribution of the Roots of a Polynomial with Integral Coefficients[END_REF] independently showed that c 2 = Log (1.179 . . .) = 0.165 . . . could be taken; Schinzel [START_REF] Schinzel | On the Product of the Conjugates Outside the Unit Circle of an Algebraic Number[END_REF] showed that c 2 = 0.2 could also be taken. Finally Smyth [START_REF] Smyth | On the Product of the Conjugates Outside the Unit Circle of an Algebraic Integer[END_REF] improved the minoration (2.20) with c 2 = Log Θ = 0.2811 . . .. On the other hand, Boyd [START_REF] Boyd | The Maximal Modulus of an Algebraic Integer[END_REF] showed that c 2 cannot exceed 3 2 Log Θ = 0.4217 . . .. In 1997 Dubickas [START_REF] Dubickas | The Maximal Conjugate of a Non-Reciprocal Algebraic Integer[END_REF] showed that c 2 = ω -with ω = 0.3096 . . . the smallest root of an equation in the interval (Log Θ, +∞), with > 0, n 0 ( ) an effective constant, and for all n > n 0 ( ). These two bounds seem to be the best known extremities delimiting the domain of existence of the constant c 2 [START_REF] Dobrowolski | On a Problem of Schinzel[END_REF].

The expression of the minorant in (1.1), "in 1/n", as a function of n, is not far from being optimal, being "in 1/n 2 " at worse in (2.22). Indeed, for nonzero algebraic integers α, Kronecker's Theorem [START_REF] Kronecker | Zwei Sätze über Gleichungen mit Ganzzahligen Coefficienten[END_REF] implies that α = 1 if and only if α is a root of unity. The sufficient condition in Kronecker's Theorem was weakened by Blansky and Montgomery [START_REF] Blansky | Algebraic Integers Near the Unit Circle[END_REF] who showed that α, with deg α = n, is a root of unity provided

α ≤ 1 + 1 30n 2 Log (6n)
.

(2.21) Dobrowolsky [START_REF] Dobrowolski | On the Maximal Modulus of Conjugates of an Algebraic Integer[END_REF] sharpened this condition by: if

α < 1 + Log n 6n 2 , (2.22)
then α is a root of unity. Matveev [START_REF] Matveev | On the Cardinality of Algebraic Integers[END_REF] proved, for α, with deg α = n, not being a root of unity,

α ≥ exp Log (n + 1 2 ) n 2 .
Rhin and Wu [START_REF] Rhin | On the Absolute Mahler Measure of Polynomials Having all Zeros in a Sector II[END_REF] verified Schinzel Zassenhaus's Conjecture up to n = 28 and improved Matveev's minoration as:

α ≥ exp 3Log ( n 3 ) n 2 4 ≤ n ≤ 12, (2.23)
and, for n ≥ 13,

α ≥ exp 3Log ( n 2 ) n 2 . (2.24)
Matveev's minoration is better than Voutier's lower bound [START_REF] Voutier | An Effective Lower Bound for the Height of Algebraic Numbers[END_REF]

m h (n) ≥ 1 + 1 4 Log Log n Log n 3 1/n (2.25)
for n ≤ 1434, and Rhin Wu's minoration is better than Voutier's bound for 13 ≤ n ≤ 6380. For reciprocal nonzero algebraic integers α, deg(α) = n ≥ 2, not being a root of unity, Dobrowolski's lower bound is

α > 1 + (2 -) Log Log n Log n 3 1 n , n ≥ n 0 ( ), (2.26)
where the constant 2 -could be replaced by 9 2 -(Louboutin [START_REF] Louboutin | Sur la Mesure de Mahler d'un Nombre Algébrique[END_REF]), or, better, by 64 π 2 -(Dubickas [START_REF] Dubickas | On a Conjecture of A[END_REF]). Callahan, Newman and Sheingorn [START_REF] Callahan | Fields with Large Kronecker Constants[END_REF] introduce a weaker version of Schinzel Zassenhaus's Conjecture: given a number field K, they define the Kronecker constant of K as the least η K > 0 such that α ≥ 1 + η K for all α ∈ K. Under certain assumptions on K, they prove that there exists c > 0 such that

η K ≥ c/[K : Q].
The sets of extremal algebraic integers are still unknown. In Boyd [START_REF] Boyd | Kronecker's Theorem and Lehmer's Problem for Polynomials in Several Variables[END_REF] [533] the following conjectures on extremality are formulated: Conjecture 2.1 (Lind -Boyd). The smallest Perron number of degree n ≥ 2 has minimal polynomial

X n -X -1 if n ≡ 3, 5 mod 6, (X n+2 -X 4 -1)/(X 2 -X + 1) if n ≡ 3 mod 6, (X n+2 -X 2 -1)/(X 2 -X + 1) if n ≡ 5 mod 6. Conjecture 2.2 (Boyd). (i) If α is extremal, then it is always nonreciprocal, (ii) if n = 3k
, then the extremal α has minimal polynomial

X 3k + X 2k -1, or X 3k -X 2k -1, (2.27)
(iii) the extremal α of degree n has asymptotically a number of conjugates α (i) outside the closed unit disc equal to

∼ = 2 3 n, n → ∞. (2.28)
This asymptotic proportion of 2 3 n would correspond to a fairly regular angular distribution of the complete set of conjugates in a small annulus containing the unit circle, in the sense of the Bilu-Erdős-Turán-Amoroso-Mignotte equidistribution theory [START_REF] Amoroso | On the Distribution on the Roots of Polynomials[END_REF], [START_REF] Belotserkovski | Uniform Distribution of Algebraic Numbers Near the Unit Circle, Vests[END_REF], [START_REF] Bilu | Limit Distribution of Small Points on Algebraic Tori[END_REF], [START_REF] Erdös | On the Distribution of Roots of Polynomials[END_REF].

Structure of coefficients vectors and minorations. The nature of the coefficient vector of an integer polynomial P is linked to the Mahler measure M(P ) and to extremal properties [START_REF] Mahler | On Two Extremal Properties of Polynomials[END_REF]. If some inequalities between coefficients occur, then Brauer [START_REF] Brauer | On Algebraic Equations with All But One Root in the Interior of the Unit Circle[END_REF] proved that P is a Pisot polynomial; in this case Lehmer's problem is solved for this class {P }. Stankov [START_REF] Stankov | The Necessary and Sufficient Condition for an Algebraic Integer to Be a Salem Number[END_REF] proved that a real algebraic integer τ > 1 is a Salem number if and only if its minimal polynomial is reciprocal of even degree ≥ 4 and if there is an integer n ≥ 2 such that τ n has minimal polynomial P n (x) = a 0,n + a 1,n x + . . . + a d,n x n which is also reciprocal of degree d and satisfies the condition

|a d-1,n | > 1 2 d d -2 (2 + d-2 k=2 |a k,n |). (2.29)
Related to Kronecker's Theorem [START_REF] Kronecker | Zwei Sätze über Gleichungen mit Ganzzahligen Coefficienten[END_REF] is the problem of finding necessary and sufficient conditions on the coefficient vector of reciprocal, self-inversive, resp. selfreciprocal polynomials to have all their roots on the unit circle (unimodularity): Lakatos [START_REF] Lakatos | On Zeros of Reciprocal Polynomials[END_REF] proved that a polynomial

P (x) = m j=0 A j x j ∈ R[x] satisfying the conditions A m-j = A j for j ≤ m and |A m | ≥ m j=0 |A j -A m | (2.30)
has all zeroes on the unit circle. Schinzel [START_REF] Schinzel | Self-Inversive Polynomials With All Zeroes on the Unit Circle[END_REF], Kim and Park [START_REF] Kim | On the Zeros of Certain Self-Reciprocal Polynomials[END_REF], Kim and Lee [START_REF] Kim | On the Zeros of Self-Reciprocal Polynomials Satisfying Certain Coefficient Conditions[END_REF], Lalin and Smyth [START_REF] Lalín | Unimodularity of Zeros of Self-Inversive Polynomials[END_REF] obtained generalizations of this result. Suzuki [START_REF] Suzuki | On Zeros of Self-Reciprocal Polynomials[END_REF] established correlations between this problem and the theory of canonical systems of ordinary linear differential equations. Lakatos and Losonczi [START_REF] Lakatos | Self-Inversive Polynomials Whose Zeros are on the Unit Circle[END_REF] [319] proved that, for a self-inversive polynomial

P m (z) = m j=0 A k z k ∈ C[z], m ≥ 1, the roots of P m are all on the unit circle if |A m | ≥ m-1 k=1 |A k |; moreover if
this inequality is strict then the zeroes e iϕ l , l = 1, . . . , m, are simple and can be arranged such that, with

β m = arg(A m A 0 /A m 1/2 ) , 2((l -1)π -β m ) m < ϕ l < 2(lπ -β m ) m . (2.31)
In the direction of Salem polynomials, ν-Salem polynomials and more [START_REF] Kerada | Une Caractérisation de Certaines Classes d'Entiers Algébriques Généralisant les Nombres de Salem[END_REF] [454], a generalization was obtained by Vieira [START_REF] Vieira | On the Number of Roots of Self-Inversive Polynomials on the Complex Unit Circle[END_REF]: if a sufficient condition is satisfied then a self-inversive polynomial has a fixed number of roots on the unit circle. Namely, let P (z) = a n z n + a n-1 z n-1 + . . . + a 1 z + a 0 ∈ C[z], a n = 0, be such that

P (z) = ω z n P (1/z) with |ω| = 1. If the inequality |a n-l | > 1 2 n n -2l n k=0,k =l,k =n-l |a k |, l < n/2 (2.32)
is satisfied, then P (z) has exactly n -2l roots on the unit circle and these roots are simple; moreover, if n is even and l = n/2, then P (z) has no root on |z| = 1 if the inequality

|a n/2 | > n k=0,k =n/2 |a k | is satisfied.
Questions of irreducibility of P as a function of the coefficient vector were studied in [START_REF] Dubickas | Polynomials Irreducible by Eisenstein's Criterion[END_REF]. Flammang [START_REF] Flammang | Inégalités sur la Mesure de Mahler d'un Polynôme[END_REF] obtained new inequalities for the Mahler measure M(P ) [START_REF] Waldschmidt | Diophantine Approximation on Linear Algebraic Group: Transcendence Properties of the Exponential Function in Several Variables[END_REF], and Flammang, Rhin and Sac-Epée [START_REF] Flammang | Integer Transfinite Diameter and Polynomials with Small Mahler Measure[END_REF] proved relations between the integer transfinite diameter and polynomials having a small Mahler measure.

The lacunarity of P and the minoration of M(P ) are correlated: when P is a noncyclotomic (sparse) integer polynomial, Dobrowolski, Lawton and Schinzel [START_REF] Dobrowolski | On a Problem of Schinzel[END_REF], then Dobrowolski [START_REF] Dobrowolski | Mahler's Measure of a Polynomial in Function of the Number of its Coefficients[END_REF], [START_REF] Dobrowolski | Mahler's Measure of a Polynomial in Function of the Number of its Monomials[END_REF], obtained lower bounds of M(P ) as a function of the number k of its nonzero coefficients: e.g. in [START_REF] Dobrowolski | Mahler's Measure of a Polynomial in Function of the Number of its Monomials[END_REF], with a < 0.785,

M(P ) ≥ 1 + 1 exp(a3 (k-2)/4 k 2 Log k) , (2.33)
and, if P is irreducible, [START_REF] Dubickas | Counting Degenerate Polynomials of Fixed Degree and Bounded Height[END_REF] with the counting problem, studied the number of integer polynomials as a function of their (naïve) height and resp. their Mahler measure. The next two theorems show that Lehmer's Conjecture is true for the set of the algebraic integers which are the roots of polynomials in particular families of monic integer polynomials. Theorem 2.3 (Borwein, Dobrowolski, Mossinghoff [START_REF] Borwein | Lehmer's Problem for Polynomials With Odd Coefficients[END_REF]). Let m ≥ 2, and let

M(P ) ≥ 1 + 0.17 2 k/2 k/2 ! . ( 2
f (X) ∈ Z[X] be a monic polynomial of degree D with no cyclotomic factors that satisfies f (X) ≡ X D + X D-1 + . . . + X 2 + X + 1 mod m. Then f (α)=0 h(α) ≥ D D + 1 C m , (2.35)
where we may take

C 2 = 1 4 Log 5 and C m = Log √ m 2 + 1 2 for m ≥ 3.
Theorem 2.4 (Silverman [START_REF] Silverman | Lehmer's Conjecture for Polynomials Satisfying a Congruence Divisibility Condition and an Analogue for Elliptic Curves[END_REF]). For all > 0 there exists a constant C > 0 with the following property: let f (X) ∈ Z[X] be a monic polynomial of degree D such that

f (X) is divisible by X n-1 + X n-2 + . . . + X + 1 in (Z/mZ)[X].
for some integers m ≥ 2 and n ≥ max{ D, 2}. Suppose further that no root of f (X) is a root of unity. Then

f (α)=0 h(α) ≥ C Log m. (2.36)
2.2. Limit points. The set of limit points of {M(P ) | P (X) ∈ Z[X]} is obtained by the following useful Theorem of Boyd and Lawton [START_REF] Boyd | Kronecker's Theorem and Lehmer's Problem for Polynomials in Several Variables[END_REF], [START_REF] Boyd | Speculations Concerning the Range of Mahler's Measure[END_REF], [START_REF] Lawton | A Problem of Boyd Concerning Geometric Means of Polynomials[END_REF], which correlates Mahler measures of univariate polynomials to Mahler measures of multivariate polynomials:

Theorem 2.5. let P (x 1 , x 2 , . . . , x n ) ∈ C[x 1 , . . . , x n ]) and r = (r 1 , r 2 , . . . , r n ), r i ∈ N >0 . Let P r (x) := P (x r1 , x r2 , . . . , x rn ). Let

q(r) := min{H(t) | t = (t 1 , t 2 , . . . , t n ) ∈ Z n , t = (0, . . . 0), n j=1 t j r j = 0},
where

H(t) = max{|t i | | 1 ≤ j ≤ n}. Then lim q(r)→∞ M(P r ) = M(P ). (2.37)
This theorem allows the search of small limit points of (univariate) Mahler's measures, by several methods [START_REF] Boyd | Small Limit Points of Mahler's Measure[END_REF]; a more recent method relies upon the (EM) Expectation-Maximization algorithm [START_REF] Mclachlan | The EM Algorithm and Extensions[END_REF] [START_REF] El Otmani | The EM Algorithm Applied to Determining New Limit Points of Mahler Measures[END_REF]. The set of limit points of the Salem numbers was investigated either by the "Construction of Salem" [START_REF] Boyd | Small Salem Numbers[END_REF], [START_REF] Boyd | Pisot Numbers and the Width of Meromorphic Functions[END_REF], or sets of solutions of some equations by Boyd and Parry [START_REF] Boyd | Limit Points of Salem Numbers[END_REF]. Everest [START_REF] Everest | Estimating Mahler Measure[END_REF], then Condon [START_REF] Condon | Mahler Measure Evaluations in Terms of Polylogarithms[END_REF], [START_REF] Condon | Asymptotic Expansion of the Difference of Two Mahler Measures[END_REF], established asymptotic expansions of the ratio M(P r )/M(P ). For bivariate polynomials P (x, y) ∈ C[x, y] such that P and ∂P/∂y do not have a common zero on T × C, then Condon (Theorem 1 in [START_REF] Condon | Asymptotic Expansion of the Difference of Two Mahler Measures[END_REF]) establishes the expansion, for k large enough,

Log M(P r ) M(P ) = Log M(P (x, x n )) M(P (x, y)) = k-1 j=2 c j n j + O P,k 1 n k , (2.38)
(n is not the degree of the univariate polynomial P (x, x n )) where the coefficients c j are values of a quasiperiodic function of n, as finite sums of real and imaginary parts of values of Li a polylogarithms, 2 ≤ a ≤ j, weighted by some rational functions deduced from the derivatives of P , where the sums are taken over algebraic numbers deduced from the intersection of T 2 and the hypersurface of C 2 defined by P (affine zero locus). In particular, if P is an integer polynomial, the coefficients c j are Q-linear combinations of polylogarithms evaluated at algebraic arguments.

For instance, for P (x, y) = -1+x+y, G n (x) = -1+x+x n , the coefficient c 2 (n) in the expansion of Log (M(G n )/M(P )), though a priori quasiperiodic, is a periodic function of n modulo 6 which can be directly computed (Theorem 1.3 in [START_REF] Verger-Gaugry | On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions[END_REF]), as: for n odd:

c 2 (n) = √ 3π/18 = +0.3023 . . . if n ≡ 1 or 3 (mod 6), - √ 3π/6 = -0.9069 . . . if n ≡ 5 (mod 6),
for n even: For the height one trinomial 1 + x + y, Corollary 2 in [START_REF] Condon | Asymptotic Expansion of the Difference of Two Mahler Measures[END_REF] gives the coefficients c j , j ≥ 2, as linear combinations of polylogaritms evaluated at third roots of unity, with coefficients coming from the Stirling numbers of the first and second kind, i.e. in 1 2π Z[

c 2 (n) = - √ 3π
√ 3]
. The method of Condon also provides the other coefficients c j , j ≥ 3, for the trinomial -1 + x + y in the same way.

Doche in [START_REF] Doche | On the Spectrum of the Zhang-Zagier Height[END_REF] obtains an alternate method to Boyd-Lawton's Theorem, in the objective of obtaining estimates of the Mahler measures of bivariate polynomials: let P (y, z) ∈ C[y, z] be a polynomial such that deg z (P ) > 0, let ξ n := e 2iπ n and assume that P (ξ k n , z) ≡ 0 for all n, k. Then M(P (y, z)) Limit points of Mahler measures of univariate polynomials are algebraic numbers or transcendental numbers: by (2.39) and Theorem 2.5, they are Mahler measures of multivariate polynomials. The problem of finding a positive lower bound of the set of such limit points of Mahler measures is intimately correlated to the problem of Lehmer [START_REF] Schinzel | Reducibility of Lacunary Polynomials[END_REF]. [START_REF] Smyth | On the Product of the Conjugates Outside the Unit Circle of an Algebraic Integer[END_REF] [START_REF] Smyth | On Measures of Polynomials in Several Variables[END_REF] found the remarkable identity: the Dirichlet L-series for the character χ 3 , with χ 3 the uniquely specified odd character of conductor 3 (χ 3 (m) = 0, 1 or -1 according to whether m ≡ 0, 1 or 2 (mod 3), equivalently χ 3 (m) = m 3 the Jacobi symbol). The values of logarithmic Mahler measures of multivariate polynomials are sums of special values of different L-functions, often conjecturally [START_REF] Boyd | Mahler's Measure and Special Values of L-Functions[END_REF]; the remarkable conjectural identities discovered by Boyd in [START_REF] Boyd | Mahler's Measure and Special Values of L-Functions[END_REF] (1998), also by Smyth [START_REF] Smyth | On Measures of Polynomials in Several Variables[END_REF] and Ray [START_REF] Ray | Relations between Mahler's Measure and Values of L-Series[END_REF], serve as starting points for further studies, some of them being now proved, e.g. [START_REF] Lalín | An Algebraic Integration for Mahler Measures[END_REF], [START_REF] Lalín | Mahler Measures and Computations with Regulators[END_REF], [START_REF] Rogers | Hypergeometric Formulas for Lattice Sums and Mahler Measures[END_REF], [START_REF] Shinder | Linear Mahler Measures and Double L-Values of Modular Forms[END_REF], [START_REF] Zudilin | Regulator of Modular Units and Mahler Measures[END_REF].

Log M(1 + x + y) = Λ (2.
Indeed, after the publication of [START_REF] Boyd | Mahler's Measure and Special Values of L-Functions[END_REF], Deninger [START_REF] Deninger | Deligne Periods of Mixed Motives, K-Theory and the Entropy of Certain Z n -Actions[END_REF] reinterpreted the logarithmic Mahler measures Log M(P ) of Laurent polynomials

P ∈ Z[Z n ] = Z[x ±
1 , . . . , x ± n ] as topological entropies in the theory of dynamical systems of algebraic origin, with Z n -actions (Schmidt [START_REF] Schmidt | Dynamical Systems of Algebraic Origin[END_REF], Chap. V, Theorem 18.1; Lind, Schmidt and Ward [START_REF] Lind | Mahler Measures and Entropy for Commuting Automorphisms of Compact Groups[END_REF]). This new approach makes a link with higher K-theory, mixed motives (Deninger [START_REF] Deninger | On Extensions of Mixed Motives, Journées Arithmétiques[END_REF]), real Deligne cohomology, the Bloch-Beilinson conjectures on special values of L-functions, and Mahler measures. There are two cases: either P does not vanish on T n , in which case Log M(P ) is a Deligne period of the mixed motive over Q which corresponds to the nonzero symbol {P, x 1 , . . . , x n } (Theorem 2.2 in [START_REF] Deninger | Deligne Periods of Mixed Motives, K-Theory and the Entropy of Certain Z n -Actions[END_REF]), or, if P vanishes on T n , under some assumptions, it is a difference of two Deligne periods of certain mixed motives, equivalently, the difference of two symbols evaluated against topological cycles ("integral K-theory cycles") (Theorem 3.4 in [START_REF] Deninger | Deligne Periods of Mixed Motives, K-Theory and the Entropy of Certain Z n -Actions[END_REF], with a motivic reinterpretation in Theorem 4.1 in [START_REF] Deninger | Deligne Periods of Mixed Motives, K-Theory and the Entropy of Certain Z n -Actions[END_REF]).

Let G n m,A := Spec(A[Z n ]) be the split n-torus defined over the commutative ring A = Z, Q or R. The polynomial P ≡ 0 defines the irreducible closed subscheme

Z := Spec(Z[Z n ]/(P )) ⊂ G n m,Z , Z = G n m,Z
, For any coherent sheaf F on G n m,A , the group Γ(G n m,A , F) of global sections, equipped with the discrete topology, admits a Pontryagin dual Γ(G n m,A , F) * which is a compact group. This compact group endowed with the canonical Z n -action constitute an arithmetic dynamical system for which the entropy can be defined according to [START_REF] Schmidt | Dynamical Systems of Algebraic Origin[END_REF], and correlated to the Mahler measure (Theorem 18.1 in [START_REF] Schmidt | Dynamical Systems of Algebraic Origin[END_REF]); the application to P , A = Z and F = O Z provides the identity with the entropy: h(O Z ) = Log M(P ). The definition

Log M(P ) := 1 (2iπ) n T n Log |P (x 1 , . . . , x n )| dx 1 x 1 . . . dx n x n (2.42)
corresponds to the integration of a differential form in connection with the cup-

product Log |P | ∪ Log |x 1 | ∪ . . . ∪ Log |x n | in the real Deligne cohomology of G n m,R \ Z R .
The link between the L-series L(M, s) of a motive M , and its derivatives, Deligne periods, and the Beilinson conjectures, comes from the Conjecture of Deligne-Scholl ([149] Conjecture 2.1). Further, Rodriguez-Villegas [START_REF] Rodriguez-Villegas | Modular Mahler Measures I[END_REF] studied the conditions of applicability of the conjectures of Bloch-Beilinson for having logarithmic Mahler measures Log M(P ) expressed as L-series.

The example of Log M(P ( [START_REF] Schmidt | Dynamical Systems of Algebraic Origin[END_REF]. For instance, for k = 3, we have Log M((

x 1 , x 2 )) = Log M((x 1 + x 2 ) 2 + k), with k ∈ N, is computed in Proposition 19.10 in
x 1 + x 2 ) 2 + 3) = 2 3 Log 3 + √ 3 π L(2, χ 3 ). (2.43)
Deninger shows (ex. [START_REF] Deninger | Deligne Periods of Mixed Motives, K-Theory and the Entropy of Certain Z n -Actions[END_REF] p. 275) the cohomological origin of each term:

√ 3 π L(2, χ 3 ) from the first M-cohomology group H 1 M (∂A, Q(2)), 2 3 Log 3 from the second M- cohomology group H 2 M (Z reg , Q (2)
). Bornhorn [START_REF] Bornhorn | K-Theory and Values of L-Functions[END_REF], and later Standfest [START_REF] Standfest | [END_REF], reinvestigated further the conjectural identities of Boyd [START_REF] Boyd | Mahler's Measure and Special Values of L-Functions[END_REF] in particular the formulas of mixed type, containing several types of L-series. The logarithmic Mahler measure Log M(P ) is then written

= * L (s 1 , χ) + * L (E, s 2 ), (2.44)
where χ is a Dirichlet character, L(s 1 , χ) the corresponding Dirichlet series, L(E, s 2 ) the Hasse-Weil L-function of an elliptic curve E/Q deduced from P , and s 1 , s 2 algebraic numbers. Following Deninger and Rodriguez-Villegas, Lalin [START_REF] Lalín | An Algebraic Integration for Mahler Measures[END_REF], [START_REF] Lalín | Mahler Measures and Computations with Regulators[END_REF], introduces techniques for applying Goncharov's constructions of the regulator on polylogarithmic motivic complexes in the objective of computing Mahler measures of multivariate Laurent polynomials. With some three-variable polynomials, whose zero loci define singular K3 surfaces, Bertin et al [START_REF] Bertin | Mahler Measures of Some Singular K3-surfaces[END_REF] prove that the logarithmic Mahler measure is of the form * L (g, O) + * L (χ, -1)

(2.45)
where g is the weight 4 newform associated with the K3 surface and χ is a quadratic character. Other three-variable Mahler measures are associated with special values of modular and Dirichlet L-series [START_REF] Samart | Three-Variable Mahler Measures and Special Values of Modular and Dirichlet L-Series[END_REF]. Some four-variables polynomials define a Calabi-Yau threefold and the logarithmic Mahler measure is of the form * L (f, O)

+ * ζ (-2) (2.46)
where f is a normalized newform deduced from the Dedekind eta function [START_REF] Papanikolas | The Mahler Measure of a Calabi-Yau Threefold and Special L-Values[END_REF]. Multivariable Mahler measures are also related to mirror symmetry and Picard -Fuchs equations in Zhou [START_REF] Zhou | Integrality Properties of Variations of Mahler Measures[END_REF].

In comparison, the limit points of the set S of Pisot numbers were studied by analytical methods by Amara [START_REF] Amara | Ensembles Fermés de Nombres Algébriques[END_REF]. The set of values

{Log M(P ) | P ∈ Z[Z n ], n ≥ 1} (2.47)
is conjecturally (Boyd [80]) a closed subset of R for the usual topology.

Salem numbers, interlacing, association equations, dynamics and dichotomy, partially and totally real algebraic numbers

The set of (positive) Salem numbers is a subcollection of Mahler measures of algebraic integers.

A negative Salem number is by definition the opposite of a Salem number, a negative Pisot number is by definition the opposite of a Pisot number. Negative and (positive) Salem numbers occur in number theory, e.g. for graphs or integer symmetric matrices in [START_REF] Mckee | Salem Numbers, Pisot Numbers, Mahler Measures and Graphs[END_REF], [START_REF] Mckee | Integer Symmetric Matrices of Small Spectral Radius and Small Mahler Measure[END_REF], [START_REF] Mckee | Salem Numbers and Pisot Numbers via Interlacing[END_REF], and in other domains (cf § 6.4), like Alexander polynomials of links of the variable "-x", e.g. in Theorem 6.11 (Hironaka [263]).

The set of Pisot numbers admits the minorant Θ by a result of Siegel [START_REF] Siegel | Algebraic Integers whose Conjugates Lie in the Unit Circle[END_REF]. The set S of Pisot numbers is closed (Salem [452]). Its successive derived sets S (i) , were extensively studied by Dufresnoy and Pisot [START_REF] Dufresnoy | Sur les éléments d'accumulation d'un ensemble fermé d'entiers algébriques[END_REF], and their students (Amara [START_REF] Amara | Ensembles Fermés de Nombres Algébriques[END_REF], . . . ), by means of compact families of meromorphic functions, following ideas of Schur. This analytic approach is reported extensively in the book [START_REF] Bertin | Pisot and Salem Numbers[END_REF]. [START_REF] Salem | A Remarkable Class of Algebraic Integers. Proof of a Conjecture of Vijayaraghavan[END_REF] (Samet [454], [START_REF] Salem | A Remarkable Class of Algebraic Integers. Proof of a Conjecture of Vijayaraghavan[END_REF]) proved that every Salem number is the quotient of two Pisot numbers. Apart from this result, few relations were known between Salem numbers and Pisot numbers. The set of Pisot numbers is better known than the set of Salem numbers.

To study Salem numbers association equations between Pisot numbers and Salem numbers have been introduced by Boyd ([74] Theorem 4.1), Bertin and Pathiaux-Delefosse ( [START_REF] Bertin | A characterization of two related classes of Salem numbers[END_REF], [START_REF] Bertin | Conjecture de Lehmer et Petits Nombres de Salem[END_REF] pp 37-46, [48] chapter 6). Association equations are generically written (X 2 + 1)P Salem = XP P isot (X) + P * P isot (X), (3.1) to investigate the links between infinite collections of Pisot numbers and a given Salem number, of respective minimal polynomials P P isot and P Salem .

The theory of interlacing of roots on the unit circle is somehow a powerful tool for studying classes of polynomials having special geometry of zeroes of modulus one [START_REF] Lakatos | Salem Numbers, PV Numbers and Spectral Radii of Coxeter Transformations[END_REF], [START_REF] Lakatos | A New Construction of Salem Polynomials[END_REF], [START_REF] Lakatos | Salem Numbers Defined by Coxeter Transformations[END_REF], in particular Salem polynomials. In [START_REF] Bertin | A characterization of two related classes of Salem numbers[END_REF] ( [START_REF] Bertin | Conjecture de Lehmer et Petits Nombres de Salem[END_REF]) Bertin and Boyd obtained two interlacing theorems, namely Theorem A and Theorem B, turned out to be fruitful with their limit-interlacing versions. McKee and Smyth in [START_REF] Mckee | Salem Numbers and Pisot Numbers via Interlacing[END_REF] obtained new interlacing theorems. Theorem 5.3 in [START_REF] Mckee | Salem Numbers and Pisot Numbers via Interlacing[END_REF] shows that all Pisot numbers are produced by a suitable interlacing condition, supporting the second Conjecture of Boyd, i.e. (2.14). Similarly Theorem 7.3 in [START_REF] Mckee | Salem Numbers and Pisot Numbers via Interlacing[END_REF], using Boyd's association Theorems, shows that all Salem numbers are produced by interlacing and that a classification of Salem numbers can be made.

In [START_REF] Guichard | On Salem Numbers, Expansive Polynomials and Stieltjes Continued Fractions[END_REF] Guichard and Verger-Gaugry reconsider the interest of the interlacing Theorems of [START_REF] Bertin | A characterization of two related classes of Salem numbers[END_REF], as potential tools for the study of the limit points of sequences of algebraic integers in neighbourhoods of Salem numbers, as analogues of those of McKee and Smyth. Focusing on Theorem A of [START_REF] Bertin | A characterization of two related classes of Salem numbers[END_REF] they obtain association equations between Salem polynomials (and/or cyclotomic polynomials) and expansive polynomials, generically

(z -1)P Salem (z) = zP expansive (z) -P * expansive (z), (3.2)
and deduce rational n-dimensional representations of the neighbourhoods of a Salem number of degree n, using the formalism of Stieltjès continued fractions. These representations are tools to study the limit points of sequences of algebraic numbers in the neighbourhood of a given Salem number, for instance by dynamical methods.

In the same direction association equations between Salem numbers and (generalized) Garsia numbers are obtained by Hare and Panju [START_REF] Hare | Some Comments on Garsia Numbers[END_REF] using the theory of interlacing on the unit circle.

As counterpart, Salem numbers are linked to units: they are given by closed formulas from Stark units in Chinburg [START_REF] Chinburg | On the Arithmetic of Two Constructions of Salem Numbers[END_REF] [START_REF] Chinburg | Salem Numbers and L-functions[END_REF], exceptional units in Silverman [START_REF] Silverman | Small Salem Numbers, Exceptional Units, and Lehmer's Conjecture[END_REF]. From [START_REF] Chinburg | Salem Numbers and L-functions[END_REF] they are related to relative regulators of number fields [START_REF] Amoroso | Une Minoration pour l'Exposant du Groupe des Classes d'un Corps Engendré par un Nombre de Salem[END_REF], [START_REF] Amoroso | Small Points on a Multiplicative Group and Class Number Problem[END_REF], [START_REF] Christopoulos | Galois Theory of Salem Polynomials[END_REF], [START_REF] Costa | Ratios of Regulators in Totally Real Extensions of Number Fields[END_REF], [START_REF] Ghate | The Arithmetic and Geometry of Salem numbers[END_REF].

Dynamics of Salem numbers, Parry Salem numbers. Let β > 1 and assume

β ∈ N. Let T β : [0, 1] → [0, 1], x → {βx} the β-transformation.
T 0 β (1) = 1, T j β (1) = β j -t 1 β j-1 -t 2 β j-2 -. . . -t j ∈ Z[β] ∩ [0, 1] (3.5)
for all j ≥ 1. The digits t i belong to the finite alphabet A β = {0, 1, . . . , β }. We say that d β (1) is finite if it ends in infinitely many zeros.

Definition 3.1. If d β (1)
is finite or ultimately periodic (i.e. eventually periodic), then the real number β > 1 is said to be a Parry number. In particular, a Parry number β is said to be simple if d β (1) is finite.

Let P denote the set of Perron numbers. The set P of Perron numbers contains the subset P P of all (simple and nonsimple) Parry numbers by a result of Lind [START_REF] Lind | Dynamical Properties of Quasihyperbolic Toral Automorphisms[END_REF] (Blanchard [START_REF] Blanchard | β-Expansions and Symbolic Dynamics[END_REF], Boyle [START_REF] Boyle | Ergodic Theory and Topological Dynamics[END_REF], Denker, Grillenberger and Sigmund [START_REF] Denker | Ergodic Theory on Compact Spaces[END_REF], Frougny in [START_REF] Lothaire | Algebraic Combinatorics on Words[END_REF]

(1) = 0.t 1 t 2 . . . t m , t m = 0, the polynomial P β,P (X) := X m -t 1 X m-1 -t 2 X m-2 -. . . t m (3.7)
is called the Parry polynomial of β. If β is a Parry number which is not simple, with d β (1) = 0.t 1 t 2 . . . t m (t m+1 t m+2 . . . t m+p+1 ) ω and not purely periodic (m is = 0), then

P β,P (X) := X m+p+1 -t 1 X m+p -t 2 X m+p-1 -. . . -t m+p X -t m+p+1 -X m + t 1 X m-1 + t 2 X m-2 + . . . + t m-1 X + t m (3.8) is the Parry polynomial of β. If β is a nonsimple Parry number such that d β (1) = 0.(t 1 t 2 . . . t p+1 ) ω is purely periodic (i.e. m = 0), then P β,P (X) := X p+1 -t 1 X p -t 2 X p-1 -. . . -t p X -(1 + t p+1 ) (3.9)
is the Parry polynomial of β. By definition the degree d P of P β,P (X) is respectively m and p + 1 in the first and third case, and m + p + 1 in the second case (m is taken minimal).

If β is a Parry number, the Parry polynomial P β,P (X), belonging to the ideal P β (X)Z[X], admits β as simple root and is often not irreducible. For instance Parry polynomials have applications in the geometric and topological properties of Thurston's Master Teapot and the Thurston set (defined in Thurston [START_REF] Thurston | Entropy in Dimension One[END_REF]) for superattracting unimodal continuous self-maps of intervals [START_REF] Bray | The Shape of Thurston's Master Teapot[END_REF].

The small Salem numbers found by Lehmer in [START_REF] Lehmer | Factorization of Certain Cyclotomic Functions[END_REF], reported below, either given by their minimal polynomial or equivalently by their β-expansion ("dynamization"), are Parry Salem numbers: The smallest Salem numbers of degree ≤ 44 are all known from the complete list of Mahler measures ≤ 1.3 of Mossinghoff [START_REF] Mossinghoff | Known Polynomials with Small Mahler Measure Through Degree 180[END_REF] of irreducible monic integer polynomials of degree ≤ 180. Recall that, for n ≥ 5, θ n denotes the real root > 0 of the polynomial -1 + X + X n . Table 1 gives the subcollection of those Salem numbers β which are Parry numbers within the intervals of extremities the Perron numbers θ -1 n , n = 5, 6, . . . , 12. In each interval dyg(β) is constant while the increasing order of the βs corresponds to a certain disparity of the degrees deg(β). The remaining Salem numbers in [START_REF] Mossinghoff | Known Polynomials with Small Mahler Measure Through Degree 180[END_REF] are very probably nonParry numbers though proofs are not available yet; they are not included in Table 1. Apart from them, the other Salem numbers which exist in the intervals (θ -1 n , θ -1 n-1 ), n ≥ 6, if any, should be of (usual) degrees deg > 180.

deg(β) β = M(β) minimal pol. of β d β (1) 4 1.722 . . . X 4 -X 3 -X 2 -X + 1 0.1(100) ω 6 1.401 . . . X 6 -X 4 -X 3 -X 2 + 1 0.1(0 2 10 4 ) ω 8 1.2806 . . . X 8 -X 5 -X 4 -X 3 + 1 0.1(0 5 10 5 10 7 ) ω 10 1.17628 . . . X 10 + X 9 -X 7 -X 6 -X 5 0.1(0 10 10 18 10 12 10 18 10 12 ) ω -X 4 -X 3 + X + 1 Let Θ = 1.3247 . . . be the dominant root of X 3 -X -1. If β ∈ (1, Θ]
For some families of algebraic integers, the "dynamization" of the minimal polynomial is known explicitly, the digits being algebraic functions of the coefficients of the minimal polynomials: e.g. for Salem numbers of degree 4 and 6 (Boyd [START_REF] Boyd | Perron Units Which Are Not Mahler Measures[END_REF] [84]), for Salem numbers of degree 8 (Hichri [START_REF] Hichri | On the beta-expansion of Salem Numbers of Degree 8[END_REF] [258] [START_REF] Hichri | Beta Expansion of Salem Numbers Approaching Pisot Numbers with the Finiteness Property[END_REF]), for Pisot numbers (Boyd [87], Frougny and Solomyak [START_REF] Frougny | Finite Beta-Expansions[END_REF], Bassino [START_REF] Bassino | Beta Expansions for Cubic Pisot Numbers[END_REF] in the cubic case, Hare [START_REF] Hare | Beta-expansions of Pisot and Salem Numbers[END_REF], Panju [START_REF] Panju | Beta Expansion for Regular Pisot Numbers[END_REF] for regular Pisot numbers). Schmidt [START_REF] Schmidt | On Periodic Expansions of Pisot Numbers and Salem Numbers[END_REF], independently Bertrand-Mathis [START_REF] Bertrand-Mathis | Nombres de Perron et Questions de Rationalité[END_REF], proved that Pisot numbers are Parry numbers. Many Salem numbers are known to be Parry numbers. For Salem numbers of degree 4 it is the case [START_REF] Boyd | Salem Numbers of Degree Four Have Periodic Expansions[END_REF]. For Salem numbers of degree ≥ 6, Boyd [START_REF] Boyd | On the Beta Expansion for Salem Numbers of Degree 6[END_REF] gave an heuristic argument and a probabilistic model, for the existence of nonParry Salem numbers as a metric approach of the dichotomy of Salem numbers. This approach, coherent with Thurston's one ( [START_REF] Thurston | Entropy in Dimension One[END_REF], p. 11), is in contradiction with the conjecture of Schmidt. Hichri [START_REF] Hichri | On the beta-expansion of Salem Numbers of Degree 8[END_REF] [258] [START_REF] Hichri | Beta Expansion of Salem Numbers Approaching Pisot Numbers with the Finiteness Property[END_REF] further developped the heuristic approach of Boyd for Salem numbers of degree 8. The Salem numbers of degree ≤ 8 are all greater than 1.280638 . . . from [START_REF] Mossinghoff | Known Polynomials with Small Mahler Measure Through Degree 180[END_REF].

Using the "Construction of Salem", Hare and Tweedle [START_REF] Hare | Beta-expansions for Infinite Families of Pisot and Salem Numbers[END_REF] obtain convergent families of Salem numbers, all Parry numbers, having as limit points the limit points of the set S of Pisot numbers in the interval (1, 2) (characterized by Amara [START_REF] Amara | Ensembles Fermés de Nombres Algébriques[END_REF]). These families of Parry Salem numbers do not contain Salem numbers smaller than Lehmer's number.

Parry numbers are studied from the negative β-shift. The negative β-shift was introduced by Ito and Sadahiro [START_REF] Ito | Beta-expansions with Negative Bases[END_REF] (Liao and Steiner [START_REF] Liao | Dynamical Properties of the Negative beta-transformation[END_REF], Masakova and Pelantova [START_REF] Masáková | Ito-Sadahiro Numbers vs. Parry Numbers[END_REF], Nguéma Ndong [START_REF] Ndong | On the Lyndon dynamical system[END_REF] [START_REF] Ndong | Zeta function and negative beta-shifts[END_REF]) and the generalized β-shift by Gora [START_REF] Gora | Invariant Densities for Generalized β-maps[END_REF] [243] and Thompson [START_REF] Thompson | Irregular Sets, the β-Transformation and the Almost Specification Property[END_REF]), in the general context of iterated interval maps and post-critical finite (PCF) interval maps [START_REF] Milnor | On Iterated Maps of the Interval[END_REF] [START_REF] Thurston | Entropy in Dimension One[END_REF]. Indeed, Kalle [START_REF] Kalle | Isomorphisms Between Positive and Negative β-transformations[END_REF] showed that nonisomorphisms exist between the β-shift and the negative β-shift, possibly leading to new Parry numbers arising from "negative" Parry numbers (called Ito-Sadahiro numbers in [START_REF] Masáková | Ito-Sadahiro Numbers vs. Parry Numbers[END_REF], Irrap numbers in [START_REF] Liao | Dynamical Properties of the Negative beta-transformation[END_REF] reading "Parry" from the right to the left). More generally negative Parry numbers and generalized Parry numbers are defined as poles of the corresponding dynamical zeta functions [START_REF] Ndong | On the Lyndon dynamical system[END_REF] [396] [START_REF] Thompson | Generalized β-Transformation and the Entropy of Unimodal Maps[END_REF]. Negative Pisot and Salem numbers appear naturally in several domains: as roots of Newman polynomials [START_REF] Hare | Negative Pisot and Salem Numbers as Roots of Newman Polynomials[END_REF], in association equations with negative Salem polynomials [START_REF] Guichard | On Salem Numbers, Expansive Polynomials and Stieltjes Continued Fractions[END_REF], in topology with the Alexander polynomials of pretzel links ( § 6.4), as Coxeter polynomials for Coxeter elements (Hironaka [263]; § 6.2), in studies of numeration with negative bases (Frougny and Lai [START_REF] Frougny | On Negative Bases, in Developments in Language Theory[END_REF]). Generalizing Solomyak's constructions to the generalized β-shift, Thompson [START_REF] Thompson | Generalized β-Transformation and the Entropy of Unimodal Maps[END_REF] investigates the fractal domains of existence of the conjugates.

Partially and totally real algebraic numbers. For totally real algebraic integers, the basic result is the minoration (3.10) given by Schinzel. Let us recall it. Let L be a totally real algebraic number field, or a CM field (a totally complex quadratic extension of a totally real number field). Then, for any nonzero algebraic integer α ∈ L, of degree d, not being a root of unity, Schinzel [START_REF] Schinzel | On the Product of the Conjugates Outside the Unit Circle of an Algebraic Number[END_REF] obtained the minoration

M(α) ≥ θ -d/2 2 = 1 + √ 5 2 d/2 . (3.10) More precisely, if H(X) ∈ Z[X]
is monic with degree d, H(0) = ±1 and H(-1)H(1) = 0, and if the zeroes of H are all real, then

M(H) ≥ 1 + √ 5 2 d/2 (3.11)
with equality if and only if H(X) is a power of X 2 -X -1. Bertin [START_REF] Bertin | Quelques Résultats Nouveaux sur les Nombres de Pisot et de Salem[END_REF] improved Schinzel's minoration (3.11) for the algebraic integers α, of degree d, of norm N (α), which are totally real, as

M(α) ≥ max θ -d/2 2 , N (α) θ - d 2|N (α)| 1/d 2 .
(3.12)

The totally real algebraic numbers form a subfield, denoted by

Q tr , in Q ∩ R.
Following [START_REF] Bertin | Quelques Résultats Nouveaux sur les Nombres de Pisot et de Salem[END_REF], the natural extension of a Salem number is a ν-Salem number, intermediate between Salem numbers and totally real algebraic numbers. Let us define a ν-Salem as an algebraic integer α having ν conjugates outside {|z| ≥ 1} and at least one conjugate α (q) satisfying |α (q) | = 1; denote by 2ν + 2k its degree. Such an algebraic integer is totally real in the sense that its conjugates of modulus > 1 are all real, and then

M(α) ≥ θ -ν 2 k/ν 2 . (3.13)
Further, extending Pisot numbers, lower bounds of M(α) were obtained by Zaimi [START_REF] Zaïmi | Sur les Nombres de Pisot Relatifs[END_REF] [540] when α is a K-Pisot number. Rhin [START_REF] Rhin | A Generalization of a Theorem of Schinzel[END_REF], following Zaimi (cf references in [START_REF] Rhin | A Generalization of a Theorem of Schinzel[END_REF]), obtained minorations of M(α) for totally positive algebraic integers α as functions of the discriminant disc(α). Let K be an algebraic number field and α an algebraic integer of minimal polynomial R over K; by definition [START_REF] Bergé | Notions Relatives de Régulateurs et de Hauteurs[END_REF] α is K-Pisot number if, for any embedding σ : K → C, σ(R K ) admits only one root of modulus > 1 and no root of modulus 1. Denote by ∆ the discriminant of K. Lehmer's problem and small discriminants were studied by [START_REF] Mahler | An Inequality for the Discriminant of a Polynomial[END_REF], Bertrand [START_REF] Bertrand | Problème de Lehmer et Petits Discriminants[END_REF], Matveev [START_REF] Matveev | On a Connection Between the Mahler Measure and the Discriminant of Algebraic Numbers[END_REF], Rhin [START_REF] Rhin | A Generalization of a Theorem of Schinzel[END_REF]. For any K-Pisot number α, Zaimi [START_REF] Zaïmi | Sur les Nombres de Pisot Relatifs[END_REF], [START_REF] Zaïmi | Sur les K-Nombres de Pisot de Petite Mesure[END_REF], showed

M(α) ≥ √ ∆ 2 K quadratic, (3.14)
M(α) ≥ ∆ 1/4 √ 6 K cubic and totally real. (3.15)
Other minorations of totally positive algebraic integers were obtained by Mu and Wu [START_REF] Mu | The Measure of Totally Positive Algebraic Integers[END_REF]. Denote Z tr := Q tr ∩ O Q . Because the degree d of the algebraic number commonly appears in the exponent of the lower bounds of the Mahler measure, the (absolute logarithmic) Weil height h is more adapted than the Mahler measure. Schinzel's bound, originally concerned with the algebraic integers in Z tr , reads:

α ∈ Z tr , α = 0, = ±1 ⇒ h(α) ≥ h θ -1 2 = 1 2 Log ( 1 + √ 5 2 ) = 0.2406059 . . .
Smyth [START_REF] Smyth | On the Measure of Totally Real Algebraic Integers[END_REF], [START_REF] Smyth | On Measures of Polynomials in Several Variables[END_REF], proved that the set with exactly 6 isolated points in the interval (0, 0.271327 . . .), the smallest one being Schinzel's bound 0.2406059 . . . In fact, passing from algebraic integers to algebraic numbers lead to various smaller minorants of h(α): for instance (Log 5)/12 = 0.134119 . . . by Amoroso and Dvornicich [START_REF] Amoroso | A Lower Bound for the Height in Abelian Extensions[END_REF] for any nonzero α ∈ L which is not a root of unity, where L/Q is an abelian extension of number fields, or, by Ishak, Mossinghoff, Pinner and Wiles [START_REF] Ishak | Lower Bounds for Heights in Cyclotomic Extensions[END_REF], for nonzero α ∈ Q(ξ m ), not being a root of unity, (i) h(α) ≥ 0.155097 . . ., for 3 not dividing m,

{exp(h(α)) | α totally real algebraic integer, α = 0, = ±1} (3.
(ii) h(α) ≥ 0.166968 . . ., for 5 not dividing m, unless α = α ±1 0 ζ, with ζ a root of unity, whence h(α) ≥ (Log 7)/12 = 0.162159 . . ., α 0 being a root of 7X 12 -13X 6 + 7, (iii) h(α) ≥ 0.162368 . . ., for 7 not dividing m.
(cf also [START_REF] Amoroso | A Lower Bound for the Height in Abelian Extensions[END_REF], [START_REF] Amoroso | A Uniform Relative Dobrowolski's Lower Bound over Abelian Extensions[END_REF], [START_REF] Garza | On the Height of Algebraic Numbers With Real Conjugates[END_REF], [START_REF] Garza | The Mahler Measure of Dihedral Extensions[END_REF], [START_REF] Ishak | Lower Bounds for Heights in Cyclotomic Extensions[END_REF], [START_REF] Pottmeyer | Heights and Totally Real Numbers[END_REF], for other results). For totally real numbers α, Fili and Miner [START_REF] Fili | Norms Extremal with Respect to the Mahler Measure[END_REF], using results of Favre and Rivera-Letelier [START_REF] Favre | Equidistribution Quantitative des Points de Petite Hauteur sur la Droite Projective[END_REF] on the equidistribution of points of small Weil height, obtained the limit infimum of the height lim inf α∈Q tr h(α) ≥ Bombieri and Zannier [START_REF] Bombieri | A Note on Heights in Certain Infinite Extensions of Q[END_REF] have recently introduced the concept of "Bogomolov property", by analogy with the "Bogomolov Conjecture". Let us recall it. Assuming a fixed choice of embedding Q → C, a field K ⊂ Q is said to possess the Bogomolov property relative to h is and only if h(α) is zero or bounded from below by a positive constant for all α ∈ K. The search of small Weil's heights is important [START_REF] Amoroso | A Lower Bound for the Height in Abelian Extensions[END_REF], [START_REF] Amoroso | Algebraic Numbers of Small Weil's Height in CM-Fields: On a Theorem of Schinzel[END_REF], Choi [START_REF] Choi | On the Distribution of Points in Projective Space of Bounded Height[END_REF]. Every number field has the Bogomolov property relative to h by Northcott's theorem [START_REF] Schmidt | Asymptotic Formulae for Point Lattices of Bounded Determinant and Subspaces of Bounded Height[END_REF] [START_REF] Schmidt | Northcott's Theorem on Heights II. The Quadratic Case[END_REF]. Other fields are known to possess the Bogomolov property: (i) Q tr [START_REF] Schinzel | On the Product of the Conjugates Outside the Unit Circle of an Algebraic Number[END_REF], (ii) finite extensions of the maximal abelian extensions of number fields [START_REF] Amoroso | A Relative Dobrowolski Lower Bound over Abelian Extensions[END_REF] [29], (iii) totally p-adic fields [START_REF] Bombieri | A Note on Heights in Certain Infinite Extensions of Q[END_REF], i.e. for algebraic numbers all of whose conjugates lie in Q p , (iv) Q(E tors ) for E/Q an elliptic curve [START_REF] Habegger | Small Height and Infinite Non-Abelian Extensions[END_REF].

For partially real algebraic integers, let us recall Garza's lower bound. Garza [START_REF] Garza | On the Height of Algebraic Numbers With Real Conjugates[END_REF] established the following minoration of the Mahler measure M(α) for α an algebraic number, different from 0 and ±1, having a certain proportion of real Galois conjugates: if deg(α) = d ≥ 1 and 1 ≤ r ≤ d be the number of real Galois conjugates of α, then

M(α) ≥ 2 1-1/R + √ 4 1-1/R + 4 2 r 2
, where R := r/d. An elementary proof of this minoration was given by Höhn [START_REF] Höhn | On a Theorem of Garza Regarding Algebraic Numbers With Real Conjugates[END_REF].

If r = d, Garza's bound is Schinzel's bound (3.11) for totally real algebraic integers [START_REF] Höhn | Un résultat de Schinzel[END_REF]. Garza's minorant satisfies lim d→∞ 2 -r/2 2 1-d/r + √ 4 1-d/r + 4 r/2 = 1, for any r fixed, where the limit 1 is reached "without any discontinuity". In some sense, a better minorant is expected since Garza's lower bound does not take into account the discontinuity at 1 claimed by the Conjecture of Lehmer.

Small points and Lehmer problems in higher dimension

The theory of heights [START_REF] Bombieri | Heights in Diophantine Geometry[END_REF], [START_REF] Schanuel | Heights in Number Fields[END_REF], [START_REF] Waldschmidt | Diophantine Approximation on Linear Algebraic Group: Transcendence Properties of the Exponential Function in Several Variables[END_REF], is a powerful tool for studying distributions of algebraic numbers, algebraic points on algebraic varieties, and of subvarieties in projective spaces by extension. Points having a small height, or "small points", resp. "small" projective varieties, together with their distribution, have a particular interest in the problem of Lehmer in higher dimension.

In the classical Lehmer problem, the "height" is the Weil height, and Lehmer's Conjecture is expressed by a Lehmer inequality where the minorant is "a function of the degree", i.e. it states that there exists a universal constant c > 0 such that

h(α) ≥ c deg(α) (4.1)
unless α = 0 or is a root of unity. The generalizations of Lehmer's problem are still formulated by a minoration as in (4.1), but in which "α" is replaced by a rational point "P " of a (abelian) variety, or replaced by a variety "V ", where "h" is replaced by another height, more suitable, where the degree "deg(α)" may be replaced by the more convenient "obstruction index" ("degree of a variety"), where the minorant function of the "degree" may be more sophisticated than the inverse "deg(α) (i) The classical Lehmer problem. On G m , Dobrowolski's and Voutier's minorations, given by (2.17) and (2.18), with "(Log deg(α)) 3 " at the denominator, were up till now considered as the best general lower bounds, as functions of the degree deg(α). Generalizations to higher dimension (below) have been largely studied: e.g. Amoroso and David [START_REF] Amoroso | Le Problème de Lehmer en Dimension Supérieure[END_REF], [START_REF] Amoroso | Densité des Points à Coordonnées Multiplicativement Indépendantes[END_REF], [START_REF] Amoroso | Minoration de la Hauteur Normalisée dans un Tore[END_REF], Pontreau [START_REF] Pontreau | Minoration Effective de la Hauteur des Points d'une Courbe de G 2 m[END_REF] [425], W. Schmidt [START_REF] Schmidt | Heights of Points on Subvarieties of G n m , in Number Theory[END_REF] for points on G n m , Anderson and Masser [START_REF] Anderson | Lower Bounds for Heights on Elliptic Curves[END_REF], David [START_REF] David | Points de Petite Hauteur sur les Courbes Elliptiques[END_REF], Galateau and Mahé [START_REF] Galateau | Some Consequences of Masser's Counting Theorem on Elliptic Curves[END_REF], Hindry and Silverman [START_REF] Hindry | On Lehmer's Conjecture for Elliptic Curves[END_REF], Laurent [START_REF] Laurent | Minoration de la Hauteur de Néron-Tate[END_REF], Silverman [START_REF] Silverman | Lower Bound for the Canonical Height on Elliptic Curves[END_REF], [START_REF] Silverman | Lower Bounds for Height Functions[END_REF], [START_REF] Silverman | A Lower Bound for the Canonical Height on Elliptic Curves over Abelian Extensions[END_REF], for elliptic curves, David and Hindry [START_REF] David | Minorations de Hauteurs sur les Variétés Abéliennes[END_REF], [START_REF] David | Minoration de la hauteur de Néron-Tate sur les variétés de type C.M[END_REF], Masser [START_REF] Masser | Counting Points of Small Height on Elliptic Curves[END_REF] for abelian varieties.

Conjecture 4.1. (Elliptic Lehmer problem) Let E/K be an elliptic curve over a number field K. There is a positive constant C(E) > 0 such that, if P ∈ E(K) has infinite order,

h(P ) ≥ c(E) [K(P ) : K] . (4.2)
Theorem 4.2 (Laurent [START_REF] Laurent | Minoration de la Hauteur de Néron-Tate[END_REF]). Let E/K be an elliptic curve with complex multiplication over a number field K. There is a positive constant c(E/K) such that

h(P ) ≥ c(E/K) D Log Log 3D Log 2D 3 for all P ∈ E(K) \ E tors (4.3)
where D = [K(P ) : K].

Masser [START_REF] Masser | Small Values of the Quadratic Part of the Néron-Tate Height on an Abelian Variety[END_REF], [START_REF] Masser | Letter to D. Bertrand[END_REF], [START_REF] Masser | Counting Points of Small Height on Elliptic Curves[END_REF], and David [START_REF] David | Points de Petite Hauteur sur les Courbes Elliptiques[END_REF], gave estimates of lower bounds of h(P ) for elliptic curves and abelian varieties, on families of abelian varieties [START_REF] Masser | Small Values of Heights on Families of Abelian Varieties[END_REF], for P of infinite order. Galateau and Mahé [START_REF] Galateau | Some Consequences of Masser's Counting Theorem on Elliptic Curves[END_REF] solved the elliptic Lehmer problem in the Galois case, extending Amoroso David's Theorem ( [START_REF] Amoroso | Le Problème de Lehmer en Dimension Supérieure[END_REF], and [START_REF] Amoroso | Small Points on Rational Subvarieties of Tori, Commentarii Math[END_REF] for sharper estimates): [START_REF] Galateau | Some Consequences of Masser's Counting Theorem on Elliptic Curves[END_REF]). Let E/K be an elliptic curve over a number field K. There is a positive constant C(E) > 0 such that, if P ∈ E(K) has infinite order and the field extension K(P )/K is Galois,

Theorem 4.3 (Galateau -Mahé
h(P ) ≥ c(E) [K(P ) : K] . (4.4) Let α = (α 1 , . . . , α n ) ∈ G n m (Q) ⊂ P n (Q). The height of α in G n m (Q) is defined by h(α) = h(1 : α) the absolute logarithmic height. Let F 0 ∈ Q[x 1 , . . . , x n ] be a nonzero polynomial vanishing at α. The obstruction index of α is by definition deg(F 0 ), denoted by δ Q (α).
(n) > 0 such that h(α) ≥ c(n) δ Q (α) (4.5) for all α = (α 1 , . . . , α n ) ∈ G n m (Q) such that α 1 , . . . , α n are multiplicatively inde- pendent.
Small points of subvarieties of algebraic tori were studied by Amoroso [START_REF] Amoroso | Small Points on Subvarieties of Algebraic Tori: Results and Methods, Summer School in Analytic Number Theory and Diophantine Approximation[END_REF].

Theorem 4.5 (Amoroso -David). There exist a positive constant c(n) > 0 such that, for all α = (α 1 , . . . , α n ) ∈ G n m (Q) such that α 1 , . . . , α n are multiplicatively independent,

h(α) ≥ c(n) δ Q (α) Log (3δ Q (α)) -η(n) (4.6) with η(n) = (n + 1)(n + 1)! n -n.
As a consequence of the main Theorem in [START_REF] Amoroso | Small Points on Rational Subvarieties of Tori, Commentarii Math[END_REF] Amoroso and Viada improved the preceding Theorem and proved: Theorem 4.6 (Amoroso -Viada). Let α 1 , . . . , α n be multiplicatively independent algebraic numbers in a number field

K of degree D = [K : Q]. Then h(α 1 ) . . . h(α n ) ≥ 1 D 1 (1050 n 5 Log (3D)) n 2 (n+1) 2 . (4.7)
The assumption of being multiplicatively independent was reconsidered in the multiplicative group Q × /Tor(Q × ) by Vaaler in [START_REF] Vaaler | Heights on Groups and Small Multiplicative Dependencies[END_REF].

Let A/K be an abelian variety over K a number field. Let L be a line bundle over A. Let V be a subvariety of A defined over K. The degree deg L (V ) of V relatively to the Cartier divisor D associated with L is defined by the theory of intersection [START_REF] Ratazzi | Problème de Lehmer pour les Hypersurfaces de Variétés Abéliennes de Type C.M[END_REF]. In particular, if P ∈ A(K), and

V = {P }, then deg L (V ) = [K(P ) : K].
For any P ∈ A(K) the obstruction index δ K,L (P ) of P is now extended as

:= min{deg L (V ) 1 codim(V ) | V /K subvariety of A, for which P ∈ V (K)}. (4.8)
Conjecture 4.7 (David -Hindry, 2000 -Abelian Lehmer problem). Let A/K be an abelian variety over a number field K and L an ample symmetric line bundle over A. Then there exists a real number c(A, K, L) > 0 such that the canonical height h L (P ) of P satisfies

h L (P ) ≥ c(A, K, L) δ K,L (P ) (4.9)
for every point P ∈ A(K) of infinite order modulo every proper abelian subvariety

V /K of A. Moreover, if D = [K(P ) : K],
for any P ∈ A(K) not being in the torsion,

h L (P ) ≥ c(A, K, L) D 1/g0 (4.10)
where g 0 is the dimension of the smallest algebraic subgroup containing P .

For any abelian variety A defined over a number field K [START_REF] Hindry | The Analog of Lehmer Problem on Abelian Varieties with Complex Multiplication[END_REF], let us denote, for any integer n ≥ 1, K n := K(A[n]) the extension generated by the group of the torsion points

A[n], so that K tors = ∪ n≥1 K(A[n]).
Theorem 4.8 (Ratazzi [START_REF] Ratazzi | Minoration de la Hauteur sur les Variétés Abéliennes de type C.M. et Applications[END_REF]). Let A/K be a CM abelian variety of dimension g over a number field K and L an symmetric ample line bundle over A. Then there exists a real number c(A, K, L) > 0 such that, for every point P ∈ A(K), the canonical height h L (P ) satisfies either

(i) h L (P ) ≥ c(A/K, L) δ Kn,L (P ) Log Log 3 [K n : K] δ Kn,L (P ) Log 2 [K n : K] δ Kn,L (P ) η(g) (4.11)
with η(g) = (2g + 5)(g + 2)(g + 1)!(2g.g!) g ; or (ii) the point P belongs to a proper torsion subvariety, B ⊂ A Kn , defined over K n , having a degree bounded by

(deg L B) 1/codimB ≤ 1 c(A/K, L) δ Kn,L (P ) (Log 2[K n : K]δ Kn,L (P )) 2g+2η(g) .
(4.12)

(ii) The relative Lehmer problem. The generalization of the classical Lehmer problem for subfields K ⊂ Q is decomposed into two steps:

(ii-i) does there exist a real number c(K) > 0 such that h(α) ≥ c(K) for all α ∈ G m (K)/G m (K) tors ? (ii-ii) if (i) is satisfied, does there exist a real number c (K) > 0 such that, for all α ∈ G m (K)/G m (K) tors , h(α) ≥ c (K) [K(α):K] ? If K is a number field, (ii-i
) is satisfied by Northcott's Theorem and (ii-ii) amounts to the classical Lehmer problem. If K is an infinite extension of Q the problem is more difficult. In (ii-i), when the field K is Q ab , or the abelian closure of a number field, it is usual to speak of the abelian Lehmer problem. The abelian Lehmer problem was solved by Amoroso and Dvornicich [START_REF] Amoroso | A Lower Bound for the Height in Abelian Extensions[END_REF]: they proved that, if L/Q is an abelian extension of number fields,

h(α) ≥ Log 5 12 (4.13)
for any nonzero α ∈ L which is not a root of unity. As for (ii-ii), it is usual to speak of relative Lehmer problem. The abelian and the relative Lehmer problems are naturally extended in higher dimension. If G denotes either an abelian variety A/K over a number field K or the n-torus G n m , and K tors = K(G tors ), the minorant function of the height is expected to depend upon the "nonabelian part of the degree D", where D = [K(P ) : K]. This "nonabelian part : D tors = [K tors (P ) :

K tors ] of D" is equal to [K ab (P ) : K ab ], where K ab is the abelian closure of K (if G = A, A is assumed CM).
Given an abelian extension L/K of number fields and a nonzero algebraic number α which is not a root of unity, with D := [L(α) : L], Amoroso and Zannier [START_REF] Amoroso | A Relative Dobrowolski Lower Bound over Abelian Extensions[END_REF] proved the following result, which makes use of Dobrowolski's minoration and the previous minoration:

h(α) ≥ c(K) D Log Log 5D Log 2D 13 , (4.14)
where c(K) > 0, in the direction of the relative problem. Amoroso and Delsinne [START_REF] Amoroso | Une Minoration Relative Explicite pour la Hauteur dans une Extension d'une Extension Abélienne, Diophantine geometry[END_REF] computed a lower bound, depending upon the degree and the discriminant of the number field K, for the constant c(K). In 2010, given K/Q an extension of algebraic number fields, of degree d, Amoroso and Zannier [START_REF] Amoroso | A Uniform Relative Dobrowolski's Lower Bound over Abelian Extensions[END_REF] showed

h(α) ≥ 3 -d 2 -2d-6 (4.15)
for any nonzero algebraic number α which is not a root of unity such that K(α)/K is abelian. As a corollary they obtained

h(α) ≥ 3 -14 (4.16)
for any dihedral extension L/Q and any nonzero α ∈ L which is not a root of unity.

For cyclotomic extensions, they obtained sharper results: (i) if K is a number field of degree d, there exists an absolute constant c 2 > 0 such that, with L denoting the number field generated by K and any given root of unity, then

h(α) ≥ c 2 d (Log Log 5d) 3 (Log 2d) 4 , (4.17)
for any nonzero α ∈ L which is not a root of unity; (ii) if K is a number field of degree d, and α any nonzero algebraic number, not a root of unity, such that α n ∈ K for some integer n under the assumption that K(α)/K is an abelian extension, then

h(α) ≥ c 3 d (Log Log 5d) 2 (Log 2d) 4 , (4.18)
for some constant c 3 > 0. In higher dimension [START_REF] David | Minorations de Hauteurs sur les Variétés Abéliennes[END_REF] [432], with G = A an abelian variety over a number field K, the torsion obstruction index δ tors K,L (P ) of a point P is now defined by := min{deg L tors (V )

1 codim(V ) | V /Ktors subvariety of A Ktors , for which P ∈ V (K)}. (4.19)
Conjecture 4.9 (David). Let A/K be an abelian variety over a number field K and L an ample symmetric line bundle over A. Then there exists a real number c(A, K, L) > 0 such that the canonical height h L (P ) satisfies

h L (P ) ≥ c(A, K, L) δ tors K,L (P ) (4.20)
for every point P ∈ A(K) of infinite order modulo every proper abelian subvariety

V /K of A.
The analogue of Amoroso and Dvornicich's theorem [START_REF] Amoroso | A Lower Bound for the Height in Abelian Extensions[END_REF] (abelian Lehmer problem) was obtained by Baker and Silverman for abelian varieties [START_REF] Baker | Lower Bounds for the Canonical Height on Elliptic Curves over Abelian Extensions[END_REF], [START_REF] Baker | A Lower Bound for the Canonical Height on Abelian Varieties Over Abelian Extensions[END_REF], and for elliptic curves by Baker [START_REF] Baker | Lower Bounds for the Canonical Height on Elliptic Curves over Abelian Extensions[END_REF], then by Silverman [START_REF] Silverman | A Lower Bound for the Canonical Height on Elliptic Curves over Abelian Extensions[END_REF]: Theorem 4.10 (Baker -Silverman). Let A/K be an abelian variety over a number field K and L an symmetric ample line bundle over A. Let h(P ) : A(K) → R the associated canonical height. Then there exists a real number c(A, K, L) > 0 such that

h(P ) ≥ c(A, K, L)
for all nontorsion points P ∈ A(K ab ). (4.21)

The proof relies upon Zahrin's theorem on torsion points of abelian varieties deduced from the proof of Faltings's theorem [START_REF] Faltings | Endlichkeitssätze für Abelsche Varietäten über Zahlkörpern[END_REF] of the Mordell Conjecture. Theorem 4.11 (Silverman). Let K/Q be a number field, let E/K be an elliptic curve, and h : E(K) → R be the canonical height on E. There is a constant C(E/K) > 0 such that every nontorsion point P ∈ E(K ab ) satisfies

h(P ) > C(E/K). (4.22)
Small points were studied by Carrizosa [START_REF] Carrizosa | Petits Points et Multiplication Complexe[END_REF]. Ratazzi [START_REF] Ratazzi | Théorème de Dobrowolski-Laurent pour les Extensions Abéliennes sur une Courbe Elliptique à Multiplications Complexes[END_REF] obtained the relative version of Amoroso and Zannier's minoration [START_REF] Amoroso | A Relative Dobrowolski Lower Bound over Abelian Extensions[END_REF]: Theorem 4.12 (Ratazzi). Let E/K be an elliptic curve with complex multiplication over a number field K. Then there exists a constant c(E, K) > 0 such that

h(P ) ≥ c(E, K) D Log Log 5D Log 2D 13
for all nontorsion points P ∈ E(K), In the direction of the relative problem, a better lower bound of the canonical height of a point P in a CM abelian variety A/K in terms of the degree of the field generated by P over K(A tors ) was obtained by Carrizosa [START_REF] Carrizosa | Problème de Lehmer et Variétés Abéliennes CM[END_REF]. For tori Delsinne [START_REF] Delsinne | Le Problème de Lehmer Relatif en Dimension Supérieure[END_REF] obtained the following (the obstruction index ω K (α) is defined below): Theorem 4.13 (Delsinne). Let n ≥ 1 be an integer. There exist constants c 1 (n),

κ 1 (n), µ(n), η 1 (n) > 0 such that, for any α ∈ G n m (Q) satisfying h(α) ≤ c 1 (n)ω Q ab (α) Log (3ω Q ab (α)) κ1(n) -1 , (4.24)
there exists a torsion subvariety B containing α, the degree of B being bounded by

(deg B) 1/codimB ≤ c 1 (n)ω Q ab (α) η1(n) Log (3ω Q ab (α)) µ(n) ; (4.25)
the constants are effective and one can take the following values:

c 1 (n) = exp 64nn!(2(n + 1) 2 (n + 1)!) 2n , (4.26
)

κ 1 (n) = 3(2(n + 1) 2 (n + 1)!) n , µ(n) = 8n!(2(n + 1) 2 (n + 1)!) n , (4.27
)

η 1 (n) = (n -1)! n-3 i=0 1 i! + 1 (4.28)
(iii) Lehmer's problem for subvarieties. The extension from points to subvarieties has been formulated for nontorsion subvarieties V of the multiplicative group G n m or of an abelian variety A/K over a number field K by David and Philippon [START_REF] David | Minorations des Hauteurs Normalisées des Sous-Variétés de Variétés Abéliennes[END_REF], [START_REF] David | Minorations des Hauteurs Normalisées des Sous-Variétés des Tores[END_REF], and Ratazzi [START_REF] Ratazzi | Densité de Points et Minoration de Hauteur[END_REF], [START_REF] Ratazzi | Problème de Lehmer pour les Hypersurfaces de Variétés Abéliennes de Type C.M[END_REF]. The natural extension of the minoration problem for the height consists in obtaining the best minoration of the height h L (V ), resp. of the essential minimum, as a function of the degree of V or of the obstruction index of V . The obstruction index δ K,L (V ) of V , resp. ω K (V ), extends the obstruction index δ K,L (P ) of a point P [START_REF] David | Minorations des Hauteurs Normalisées des Sous-Variétés des Tores[END_REF]. As for the definition of the height of V relatively to an symmetric ample line bundle L, two approaches were followed [START_REF] Ratazzi | Problème de Lehmer pour les Hypersurfaces de Variétés Abéliennes de Type C.M[END_REF]: one by Philippon [START_REF] Philippon | Sur des Hauteurs Alternatives I[END_REF], another one by Bost, Gillet and Soulé [START_REF] Bost | Heights of Projective Varieties and Positive Green Forms[END_REF], using theorems of Soulé [START_REF] Soulé | Géométrie d'Arakelov et Théorie des Nombres Transcendants[END_REF] and Zhang [START_REF] Zhang | Small Points and Adelic Metrics[END_REF]. In the second construction Zhang [START_REF] Zhang | Small Points and Adelic Metrics[END_REF] showed how to consider the canonical height (or Néron-Tate height, or normalized height) h L (V ) as a limit of Arakelov heights.

Define the canonical height (say) h on G n m (Q) by h(α 1 , . . . , α n ) = h(α 1 ) + . . . + h(α n ). For θ > 0, let V be a subvariety of G n m defined over Q. For θ > 0, let:

V θ := {P ∈ V (Q) |≤ θ}, (4.29)
and the essential minimum

µ ess (V ) := inf{θ > 0 | V θ is Zariski dense in V }. (4.30)
The generalized Bogomolov conjecture for subvarieties of tori asserts that µ ess (V ) = 0 is and only if V is a torsion subvariety. In the case where V is a point, V = {P }, µ ess (V ) = h(P ). Zhang [START_REF] Zhang | Positive Line Bundles on Arithmetic Surfaces[END_REF], [START_REF] Zhang | Small Points and Adelic Metrics[END_REF], [START_REF] Zhang | Equidistribution of Small Points on Abelian Varieties[END_REF], showed that the minoration problem of µ ess (V ) is essentially the same problem as finding lower bounds for the canonical height h(V ) of V , in the sense of Arakelov theory. Indeed, from his Theorem of the Successive Minima, Zhang proved:

µ ess (V ) ≤ h(V ) deg(V ) ≤ (dim(V ) + 1) µ ess (V ) (4.31)
for V any subvariety of G n m over Q. Zhang obtained similar results for subvarieties of abelian varieties. The canonical height h(V ) of V is related to the problem of minoration of multivariate Mahler measures by the following: for V being a hypersurface defined by a polynomial F (x 1 , . . . , x n ) ∈ Z[x 1 , . . . , x n ] (having relatively prime integer coefficients), then is the logarithmic Mahler measure Log M(F ) of F . Let K be a field of characteristic zero, and let V be a subvariety of G n m defined over Q. Define the index of obstruction ω K (V ) to be the minimum degree of a nonzero polynomial F ∈ K[x 1 , . . . , x n ] vanishing identically on V . Equivalently, it is the minimum degree of a hypersurface defined over K and containing V . The higher-dimensional Lehmer Conjecture takes the following form (i.e. the two following conjectures): Conjecture 4.14 [START_REF] Amoroso | Le Problème de Lehmer en Dimension Supérieure[END_REF]. Let V be a subvariety of G n m , and assume that V is not contained in any torsion subvariety (i.e., a translate of a proper subgroup by a torsion point). Then there exists a constant C(n) > 0 such that

h(V ) = 1 0 . . .
µ ess (V ) ≥ C(n) ω Q (V ) . (4.33)
A 0-dimensional subvariety V = (α 1 , . . . , α n ) of G n m is contained in a torsion subvariety if and only if α 1 , . . . , α n are multiplicatively dependent.

In a similar way, for θ > 0, V a subvariety of an abelian variety A defined over a number field K, and L a symmetric ample line bundle on A, we define:

V (θ, L) := {x ∈ V (K) | h L (K) ≤ θ}. The essential minimum of V is µ ess L (V ) := {θ > 0 | V (θ, L) = V } (4.34)
where V (θ, L) is the adherence of Zariski of V (θ, L) in A.

Conjecture 4.15 [START_REF] David | Minorations des Hauteurs Normalisées des Sous-Variétés de Variétés Abéliennes[END_REF]. Let A be an abelian variety defined over a number field K, and L a symmetric ample line bundle on A. Let V /K be a proper subvariety of A, K-irreducible and such that V K is not the union of torsion subvarieties, then

h L (V ) deg L (V ) ≥ c(A/K, L) (deg L (V )) 1/(s-dim(V )) (4.35)
for some constant c(A/K, L) > 0 depending on A/K and L, where s is the dimension of the smallest algebraic subgroup containing V .

Generalizing (4.6) the higher dimensional Dobrowolski bound takes the following form, proved in [START_REF] Amoroso | Le Problème de Lehmer en Dimension Supérieure[END_REF] for dim(V ) = 0, in [START_REF] Amoroso | Minoration de la Hauteur Normalisée des Hypersurfaces[END_REF] for codim(V ) = 1 and in [START_REF] Amoroso | Densité des Points à Coordonnées Multiplicativement Indépendantes[END_REF] for varieties of arbitrary dimension. Theorem 4.16 (Amoroso -David). Let V be a subvariety of G n m defined over Q of codimension k. Let us assume that V is not contained in any union of proper torsion varieties. Then, there exist two constants c(n) and κ(n

) = (k + 1)(k + 1)! k -k such that µ ess (V ) ≥ C(n) ω Q (V ) 1 (Log 3ω Q (V )) κ(k) .
(4.36) Amoroso and Viada [START_REF] Amoroso | Small Points on Subvarieties of a Torus[END_REF] introduced relevant invariants of a proper projective subvariety V ⊂ P n : e.g. δ(V ) defined as the minimal degree δ such that V is, as a set, the intersection of hypersurfaces of degree ≤ δ. Theorem 4.17 (Amoroso -Viada [START_REF] Amoroso | Small Points on Rational Subvarieties of Tori, Commentarii Math[END_REF]).

Let V ⊂ G n m be a Q-irreducible variety of dimension d. Then, for any α ∈ V * (Q), h(α) ≥ 1 δ(V ) 1 (935 n 5 Log (n 2 δ(V )) (d+1)(n+1) 2 .
(4.37)

Following the main Theorem1.3 in [START_REF] Amoroso | Small Points on Rational Subvarieties of Tori, Commentarii Math[END_REF] the essential minimum admits the following lower bound: Theorem 4.18 (Amoroso -Viada). Let V ⊂ G n m be a Q-irreducible variety of dimension k which is not contained in any union of proper torsion varieties. Then,

µ ess (V ) ≥ 1 ω Q (V ) 1 (935 n 5 Log (n 2 ω Q (V )) k(k+1)(n+1) .
(4.38)

Theorem 4.19 (Ratazzi [432]). Let A be a CM abelian variety defined over a number field K, and L a symmetric ample line bundle on A. Let V /K be a proper subvariety of A, K-irreducible and such that V K is not the union of torsion subvarieties. Then

h L (V ) deg L (V ) ≥ µ ess L (V ) ≥ c(A/K, L) (deg L (V )) 1/(n-dim(V ) 1 (Log (2 deg L (V )) κ(n) (4.39)
with κ(n) = (2n(n + 1)!) n+2 , for some constant c(A/K, L) > 0 depending only on A/K and L.

Ratazzi in [START_REF] Ratazzi | Problème de Lehmer pour les Hypersurfaces de Variétés Abéliennes de Type C.M[END_REF] obtained more precise minorations of h L (V ) in the case where V is an hypersurface. In [START_REF] Ratazzi | Densité de Points et Minoration de Hauteur[END_REF] Ratazzi proves that the optimal lower bound given by David and Philippon [START_REF] David | Minorations des Hauteurs Normalisées des Sous-Variétés de Variétés Abéliennes[END_REF] in Conjecture 4.15 is a consequence of a Conjecture of David and Hindry on the abelian Lehmer problem.

On the way of proving the relative abelian Lehmer Conjecture, Carrizosa [START_REF] Carrizosa | Problème de Lehmer et Variétés Abéliennes CM[END_REF] [118] obtained a lower bound of the canonical height of a point P in a CM abelian variety A/K defined over a number field K in terms of the degree of the field generated by P over K(A tors ). As Corollary of Theorem 4.13, with the same constants, Delsinne obtained the relative result: Theorem 4.20 (Delsinne). Let V be a subvariety of G n m which is not contained in any proper algebraic subgroup of G n m . Then

µ ess (V ) ≥ c 3 (n)ω Q ab (V )(Log (3ω Q ab (V ))) κ1(n) -1 (4.40) with c 3 (n) = c 1 (n)(dim(V ) + 1).
Concomitantly to the Lehmer problems, the geometry of the distribution of the small points, their Galois orbits, the limit equidistribution of conjugates on some subvarieties, the theorems of finiteness, were investigated, e.g. in Amoroso and David [START_REF] Amoroso | Distribution des Points de Petite Hauteur Normalisée dans les Groupes Multiplicatifs[END_REF] [20] Bilu [START_REF] Bilu | Limit Distribution of Small Points on Algebraic Tori[END_REF], Bombieri [START_REF] Bombieri | Problems and Results on the Distribution of Algebraic Points on Algebraic Varieties[END_REF], Burgos Gil, Philippon, Rivera-Letelier and Sombra [START_REF] Burgos Gil | The Distribution of Galois Orbits of Points of Small Height in Toric Varieties[END_REF], Chambert-Loir [START_REF] Chambert-Loir | Théorèmes d'équidistribution pour les Systèmes Dynamiques d'Origine Arithmétique[END_REF], D'Andrea, Galligo, Narváez-Clauss and Sombra [START_REF] D'andrea | Quantitative Equidistribution for the Solutions of Systems of Sparse Polynomial Equations[END_REF] [136], Favre and Rivera-Letelier [START_REF] Favre | Equidistribution Quantitative des Points de Petite Hauteur sur la Droite Projective[END_REF], Habegger [START_REF] Habegger | Small Height and Infinite Non-Abelian Extensions[END_REF], Hughes and Nikeghbali [START_REF] Hughes | The zeros of random polynomials cluster uniformly near the unit circle[END_REF], Litcanu [START_REF] Litcanu | Petits Points et Conjecture de Bogomolov[END_REF], Petsche [START_REF] Petsche | The Distribution of Galois Orbits of Low Height[END_REF] [416], Pritsker [START_REF] Pritsker | Distribution of Algebraic Numbers[END_REF], Ratazzi and Ullmo [START_REF] Ratazzi | Galois + Equidistribution = Manin-Mumford[END_REF], Rémond [START_REF] Rémond | Intersection de Sous-Groupes et de Sous-Variétés I[END_REF], Rumely [START_REF] Rumely | On Bilu's Equidistribution Theorem[END_REF], Szpiro, Ullmo and Zhang [START_REF] Szpiro | Equirépartition des Petits Points[END_REF], Zhang [START_REF] Zhang | Small Points and Adelic Metrics[END_REF], [START_REF] Zhang | Equidistribution of Small Points on Abelian Varieties[END_REF].

The proof of Dobrowolski in [START_REF] Dobrowolski | On a Question of Lehmer and the Number of Irreducible Factors of a Polynomial[END_REF] has been revisited and generalized, e.g. by Amoroso and David [START_REF] Amoroso | Le Théorème de Dobrowolski en Dimension Supérieure[END_REF], Carrizosa [START_REF] Carrizosa | Survey on Lehmer Problems[END_REF], Laurent [START_REF] Laurent | Minoration de la Hauteur de Néron-Tate[END_REF], Meyer [START_REF] Meyer | Le Problème de Lehmer, Méthode de Dobrowolski et Lemme de Siegel "à la Bombieri-Vaaler[END_REF], Ratazzi [START_REF] Ratazzi | Problème de Lehmer sur Gm et Méthode des Pentes[END_REF]. It is a keystone to the above-mentionned minoration problems.

Analogues of the Mahler measure and Lehmer's problem

Several generalizations and analogues of the Mahler measure were introduced, for which the analogue of the problem of Lehmer holds, or not.

The Zhang-Zagier height H(α) of an algebraic number α is defined as H(α) = M(α)M(1 -α). After Zhang [START_REF] Zhang | Positive Line Bundles on Arithmetic Surfaces[END_REF] and Zagier [START_REF] Zagier | Algebraic Numbers Close to 0 and 1[END_REF] [537], if α is an algebraic number different from the roots of (z 2 -z)(z 2 -z + 1), then

H(α) ≥ 1 + √ 5 2 = 1.2720196 . . . (5.1)
Doche [START_REF] Doche | On the Spectrum of the Zhang-Zagier Height[END_REF], [START_REF] Doche | Zhang-Zagier Heights of Perturbed Polynomials[END_REF], using (2.39), obtains the following better minorant: if α is an algebraic number different from the roots of (z

2 -z)(z 2 -z + 1)Φ 10 (z)Φ 10 (1 -z), then H(α) ≥ 1.2817770214 =: η, (5.2)
and the smallest limit point of

{H(α) | α ∈ Q} lies in [1.2817770214, 1.289735].
Dresden [START_REF] Dresden | Orbits of Algebraic Numbers with Low Heights[END_REF] introduced a generalization of the Zhang-Zagier height:

given G a subgroup of P SL(2, Q), the G-orbit height of α ∈ P 1 (Q) is h G (α) := g∈G h(gα).
(5.3)

For G the cyclic group generated by

1 0 0 1 , 0 1 -1 1 , 1 -1 1 0
Dresden finds, for α = 0, = 1 not being a primitive sixth root of unity,

h(α) + h( 1 1 -α ) + h( 1 α ) ≥ 0.42179 . . . (5.4)
with equality for α any root of (X 2 -X + 1) 3 -X 2 (X -1) 2 ; otherwise, h G (α) = 0. The G-invariant Lehmer problem is stated as follows in van Ittersum ( [START_REF] Van Ittersum | A Group-Invariant Version of Lehmer's Conjecture on Heights[END_REF] p. 146): given G a finite subgroup of P SL(2, Q), does there exist a positive constant

D = D G > 0 such that h G (α) = 0 or h G (α) ≥ D,
for all α ∈ P 1 (Q)? (

If G is trivial this constant D does not exist [START_REF] Zagier | Algebraic Numbers Close to 0 and 1[END_REF]. Denote by Orb G the set of all orbits of the action of G on C = C ∪ {∞} and Orb G,unit := {Y ∈ Orb G | for all α ∈ Y, α = 0 or |α| = 1}. Dresden's result [START_REF] Dresden | Orbits of Algebraic Numbers with Low Heights[END_REF] was generalized in [START_REF] Van Ittersum | A Group-Invariant Version of Lehmer's Conjecture on Heights[END_REF]: van Ittersum [START_REF] Van Ittersum | A Group-Invariant Version of Lehmer's Conjecture on Heights[END_REF] proved the G-invariant Lehmer problem under the assumption on G that Orb G,unit is finite. The (logarithmic) metric Mahler measure m : G → [0, ∞) was introduced by Dubickas and Smyth in [START_REF] Dubickas | On the Metric Mahler Measure[END_REF], [START_REF] Dubickas | On Metric Heights[END_REF], where

G := Q × /Tor(Q × ) (5.6)
is the Q-vector space of algebraic numbers modulo torsion, written multiplicatively.

For α ∈ G it is defined by

m(α) := inf N n=1 Log M(α n ) | N ∈ N, α n ∈ Q × , α = N n=1 α n (5.7)
where the infimum is taken over all possible ways of writing any representative α of α as a product of other algebraic numbers. The construction may be applied to any height function [START_REF] Dubickas | On Metric Heights[END_REF] and is extremal in the sense that any other function

g : G → [0, ∞) satisfying (i) g(α) ≤ m(α) for any α ∈ G, (ii) g(α β -1 ) ≤ g(α) + g(β) for any α, β ∈ G (triangle inequality),
is smaller than m.

The structure of the completion of G, as a Banach space over the field R of real numbers, endowed with the norm deduced from the Weil height has been studied by Allcock and Vaaler [START_REF] Allcock | A Banach Space Determined by the Weil Height[END_REF]. Indeed, by construction, the Weil height satisfies: for any α ∈ Q × and any root of unity ζ, h(α) = h(ζα), so that h extends to h : G → ∞ with the properties:

(i) h(α) = 0 if and only if α is the identity element 1 in G, (ii) h(α) = h(α -1 ) for all α ∈ G, (iii) h(α β) ≤ h(α) + h(β) for all α, β ∈ G.
These conditions imply that the map (α, β) → h(α β -1 ) is a metric on the quotient group G, on which the Q-action is defined by (r/s, α) → α r/s by the roots of the polynomials z s -( Let λ be the unique regular measure on B, positive on open sets, finite on compact sets, which satisfies:

(i) λ(Y (k, v)) = [k v : Q v ] [k : Q] for any Galois k/Q, any place v of k, (5.9)
(ii) λ(τ E) = λ(E) for all τ ∈ Aut(Q/k) and E ∈ B. Allcock and Vaaler [START_REF] Allcock | A Banach Space Determined by the Weil Height[END_REF] proved that the (not surjective) map

f : G → L 1 (Y, B, λ), α → f α given by f α (y) := Log α y (5.10)
is a linear isometry of norm 2h, i.e.

f αβ (y) = f α (y) + f β (y), f α r/s (y) = (r/s)f α (y), Y |f α (y)|dλ(y) = 2h(α),
with the property:

Y f α (y)dλ(y) = 0.
Denote by

F := f (G) the image of G in L 1 (Y, B, λ) and χ := {F ∈ L 1 (Y, B, λ) | Y F (y)dλ(y) = 0}
the co-dimension one linear subspace of L 1 (Y, B, λ). They proved that F is dense in χ ([6] Theorem 1), i.e. that χ is the completion of (G, h), up to isometry. They also proved that, for any real 1 < p < ∞, F is dense in L p (Y, B, λ) ([6] Theorem 2), and F is dense in the Banach space C 0 (Y ) of continuous real valued functions on Y which vanish at infinity, equipped with the sup-norm ([6] Theorem 3). Fili and Miner [START_REF] Fili | Orthogonal Decomposition of the Space of Algebraic Numbers and Lehmer's Problem[END_REF] proved that the space F admits linear operators canonically associated to the Mahler measure and to the L p norms on Y . They introduced norms, called Mahler p-norms, from orthogonal decompositions of F, and, in this context, obtained extended formulations, called L p Lehmer Conjectures, of the Lehmer Conjecture and the Conjecture of Schinzel-Zassenhaus. Namely, let K be the set of finite extensions of Q and

K G := {K ∈ K | σ(K) = K for all σ ∈ Gal(Q/Q)}.
For each K ∈ K, denote by V K := {f α | α ∈ K × /Tor(K × )} the Q-vector subspace of F constituted by the nonzero elements of K modulo torsion, and, for n ≥ 0, V (n) := K∈K,[K:Q]≤n V K . Denote by f, g = Y f (y)g(y)dλ(y) the inner product on F.

Theorem 5.1 (Fili -Miner). (i) There exist projection operators T

K : F → F for each K ∈ K G such that T K (F) ⊂ V K , T K (F)⊥ T L (F) for all K, L ∈ K G , K = L,
with respect to the inner product on F, and

F = K∈K G T K (F), (5.11)
(ii) for all n ≥ 1, there exist projections

T (n) : F → F such that T (n) (F) ⊂ V (n) , T (m) (F)⊥ T (n) (F) for all m = n, and
F = K∈K G T (n) (F), (5.12)
(iii) for every K ∈ K G and n ≥ 0, the projections T K and T (n) commute.

Now, for any α ∈ Q × and any real number 1 ≤ p ≤ ∞, let h p (α) := f α p (recalling that h 1 (α) = 2h(α)). Conjecture 5.2 (Fili -Miner (L p Lehmer Conjectures)). For any real number 1 ≤ p ≤ ∞, there exists a real constant c p > 0 such that The operator M :

( * p ) m p (α) := deg Q (α) h p (α) ≥ c p for all α ∈ Q \ Tor(Q). ( 5
F → F, f → ∞ n=1 nT (n)
f is well-defined, unbounded, invertible, and is always a finite sum. The norm f → M f p is called the Mahler p-norm on F. For any f ∈ F, let d(f

) := min{deg Q (α) | α ∈ Q × , f α = f } be
the smallest degree possible in the class of f . For any f ∈ F, the minimal field, denoted by K f , is defined to be the minimal element of the set

{K ∈ K | f ∈ V K }. Let δ(f ) = [K f : Q].
The P K operators on F are defined from the T K operators as:

P K := F ∈K G ,F ⊂K T F . An element f ∈ F is said to be Lehmer irreducible (or representable) if δ(f ) = d(f ).
The set of Lehmer irreducible elements of F is denoted by L. An element f ∈ F is said to be projection irreducible if P H (f ) = 0 for all propers subfields H of K f . The set of projection irreducible elements of F is denoted by

P. Let U = {f ∈ F | supp Y (f ) ⊂ Y (Q, ∞)}
( * * p ) ∞ n=1 nT (n) f α p ≥ c p for all f α ∈ L ∩ P ∩ U, f α = 0. (5.14)
Further, for 1 ≤ p ≤ q ≤ ∞, if ( * * p ) holds, then ( * * q ) also holds.

f ∈ F such that δ(f ) = 1, i.e.
for which K f = Q and M f p = f p = h p (f ); a surd is projection irreducible. The t-metric Mahler measure, was introduced by Samuels [START_REF] Jankauskas | The t-Metric Mahler Measures of Surds and Rational Numbers[END_REF], [START_REF] Samuels | A Collection of Metric Mahler Measures[END_REF], [START_REF] Samuels | The Parametrized Family of Metric Mahler Measures[END_REF]. For t ≥ 1, the t-metric Mahler measure is defined by

M t (α) := inf N n=1 (Log M(α n )) t 1/t | N ∈ N, α n ∈ Q × , α = N n=1 α n (5.15)
and, by extension, for t = ∞, by

M ∞ (α) := inf max 1≤n≤N Log M(α n ) | N ∈ N, α n ∈ Q × , α = N n=1 α n . (5.16)
For t = 1, M 1 is the metric Mahler measure introduced in [START_REF] Dubickas | On the Metric Mahler Measure[END_REF]. These functions satisfy an analogue of the triangle inequality [START_REF] Jankauskas | The t-Metric Mahler Measures of Surds and Rational Numbers[END_REF], and the map (α, β) → M t (αβ -1 ) defines a metric on G := Q × /Tor(Q × ) which induces the discrete topology if and only if Lehmer's Conjecture is true. For t ∈ [1, ∞] and α ∈ Q we say that the infimum in M t (α) is attained by α 1 , . . . , α n if the equality case holds: i.e.,

• for 1

≤ t < ∞, if M t (α) = N n=1 (Log M(α n )) t 1/t and, • for t = ∞, if M ∞ (α) = max 1≤n≤N {Log M(α n )}.
For α ∈ Q, denote by K α the Galois closure of Q(α)/Q, and let

Rad(K α ) := {β ∈ Q | β m ∈ K α for some m ∈ N}.
Following a conjecture of Dubickas and Smyth [START_REF] Dubickas | On the Metric Mahler Measure[END_REF], Samuels [START_REF] Samuels | A Collection of Metric Mahler Measures[END_REF] [START_REF] Samuels | The Infimum in the Metric Mahler Measure[END_REF] proved that the infimum of M t (α) is attained in Rad(K α ). Whether this infimum is attained in proper subsets of Q leads to many open questions ([282], Question 1.5), though Jankauskas and Samuels proved some results for certain cases of decompositions of rational numbers in prime numbers ( [START_REF] Jankauskas | The t-Metric Mahler Measures of Surds and Rational Numbers[END_REF], Theorem 1.3, Theorem 1.4). In particular for α ∈ Q, they proved that the infimum of M t (α) may be attained using only rational points.

The p-metric, resp. the t-metric, constructions of Fili and Miner [START_REF] Fili | Norms Extremal with Respect to the Mahler Measure[END_REF] and Jankauskas and Samuels [START_REF] Jankauskas | The t-Metric Mahler Measures of Surds and Rational Numbers[END_REF] are of different nature, though they are esentially the same for p = 1. Fili and Miner [START_REF] Fili | Norms Extremal with Respect to the Mahler Measure[END_REF] studied the minimality of the Mahler measure by several norms, related to the metric Mahler measure introduced in [START_REF] Dubickas | On the Metric Mahler Measure[END_REF], using results of de la Masa and Friedman [START_REF] De La Masa | Heights of Algebraic Numbers Modulo Multiplicative Group Actions[END_REF] on heights of algebraic numbers modulo multiplicative group actions. Fili and Miner [START_REF] Fili | Norms Extremal with Respect to the Mahler Measure[END_REF] introduced an infinite collection (h t ) t of vector space norms on G, called L t Weil heights, t ∈ [1, ∞], which satisfy extremality properties, and minimal logarithmic L t Mahler measures (m t ) t from (h t ) t . By definition, for K a number field, Σ K its set of places and ||.|| ν the absolute value on K extending the usual p-adic absolute value on Q if ν is finite or the usual archimedean absolute value if ν is infinite, for 1 ≤ t < ∞ real,

h t (α) := ν∈Σ K [K ν : Q ν ] [K : Q] . |Log ||α|| ν | t 1/t , α ∈ K × (5.17)
for which 2h = h 1 [START_REF] Allcock | A Banach Space Determined by the Weil Height[END_REF], and

h ∞ (α) := sup ν∈Σ K |Log ||α|| ν | , α ∈ K × . (5.18)
They reformulated Lehmer's Conjecture in this context (with 1 ≤ t < ∞). Because h ∞ serves as a generalization of the (logarithmic) house of an algebraic integer, they also reformulated the Conjecture of Schinzel and Zassenhaus. In [START_REF] Jankauskas | The t-Metric Mahler Measures of Surds and Rational Numbers[END_REF] Jankauskas and Samuels investigate the t-metric Mahler measures of surds and rational numbers. The ultrametric Mahler measure was introduced by Fili and Samuels [START_REF] Fili | On the Non-Archimedean Metric Mahler Measure[END_REF], [START_REF] Samuels | The Infimum in the Metric Mahler Measure[END_REF], to give a projective height of G, which satisfies the strong triangle inequality.

The ultrametric Mahler measure induces the discrete topology on G if and only if Lehmer's Conjecture is true.

Two p-adic Mahler measures are introduced by Besser and Deninger in [START_REF] Besser | p-adic[END_REF] in view of developping natural analogues of the classical logarithmic Mahler measures of Laurent polynomials, following Deninger [START_REF] Deninger | Deligne Periods of Mixed Motives, K-Theory and the Entropy of Certain Z n -Actions[END_REF]. The p-adic analogue of Deligne cohomology is now Besser's modified syntomic cohomology, but with the same symbols in the algebraic K-theory groups. For one p-adic Mahler measure the authors show that there is no analogue of Lehmer's problem.

Generalized Mahler measures, higher Mahler measures and multiple k-higher Mahler measures were introduced by Gon and Oyanagi [START_REF] Gon | Generalized Mahler Measures and Multiple Sine Functions[END_REF], resp. Kurokawa, Lalín and Ochiai [START_REF] Kurokawa | Higher Mahler Measures and Zeta Functions[END_REF] and reveal deep connections between zeta functions, polylogarithms, multiple L-functions (Sasaki [START_REF] Sasaki | On Multiple Higher Mahler Measures and Multiple L values[END_REF]) and multiple sine functions. For any n ≥ 1, given P 1 , . . . , P s ∈ C[x 1 , . . . , x n ] (not necessarily distinct) nonzero polynomials, the generalized Mahler measure is defined by m max (P 1 , . . . , P s ) :=

1 (2πi) n T n max{Log |P 1 (x 1 , . . . , x n )|, . . . ,
) := 1 (2πi) n T n Log |P 1 (x 1 , . . . , x n )| . . . Log |P s (x 1 , . . . , x n )| dx 1 x 1 . . . dx n x n , (5.20)
the k-higher Mahler measure of P by m k (P ) := m(P, . . . , P ) =

1 (2πi) n T n Log k |P 1 (x 1 , . . . , x n )| dx 1 x 1 . . . dx n x n , (5.21)
The k-higher Mahler measures are deeply related to the zeta Mahler measures, and their derivatives, introduced by Akatsuka [5]. The problem of Lehmer for k-higher Mahler measures is considered by Lalín and Sinha in [START_REF] Lalín | Higher Mahler Measure for Cyclotomic Polynomials and Lehmer's Question[END_REF]. Asymptotic formulas of m k (P ), with k, are given in [START_REF] Biswas | Asymptotic Nature of Higher Mahler Measure[END_REF] and [START_REF] Lalín | Higher Mahler Measure for Cyclotomic Polynomials and Lehmer's Question[END_REF], for some families of polynomials P . Analogues of Boyd-Lawton's Theorem are studied in Issa and Lalín [START_REF] Issa | A Generalization of a Theorem of Boyd and Lawton[END_REF]. By analogy with Deninger's approach, the motivic reinterpretation of the values of k-higher Mahler measures in terms of Deligne cohomology is given by Lalín in [START_REF] Lalín | Higher Mahler Measure as a Massey Product in Deligne Cohomology[END_REF].

The logarithmic Mahler measure m G over a compact abelian group G is introduced by Lind [START_REF] Lind | Lehmer's Problem for Compact Abelian Groups[END_REF]. The group is equipped with the normalized Haar measure µ. By Pontryagin's duality the dual group G (characters) is discrete and the class of functions f to be considered is Z

[ G]. For f ∈ Z[ G] m G (f ) = G Log |f |dµ (5.22) generalizes m(f ) = 1 0 Log |f (e 2iπt )|dt for f ∈ Z[x ±1 ].
(5.23)

The Lehmer constant of G is then defined by

λ(G) := inf{m G (f ) | f ∈ Z[ G], m G (f ) > 0}.
(5.24)

The author considers several groups G (connected, finite) and the problem of Lehmer in each case. The classical Lehmer's problem asks whether λ(T) = 0, where T = R/Z. Let n ≥ 2, denote by ρ(n) the smallest prime number that does not divide n. Lind proves that λ(G) = λ(T) for any nontrivial connected compact abelian group, and λ(Z/nZ) ≤ Log ρ(n) n for n ≥ 2. This Lehmer's constant has been named Lind-Lehmer's constant more recently. It is known in some cases [START_REF] Pinner | Polynomials with Lind Mahler Measure One[END_REF]. Kaiblinger [START_REF] Kaiblinger | On the Lehmer Constant of Finite Cyclic Groups[END_REF] obtained results on λ(G) for finite cyclic groups G of cardinality not divisible by 420; Pigno and Pinner [START_REF] Pigno | The Lind-Lehmer Constant for Cyclic Groups of Order Less[END_REF] solved the case |G| = 420. De Silva and Pinner [START_REF] Silva | Lind-Lehmer Constant for Groups of the Form Z n p[END_REF], [START_REF] Silva | The Lind-Lehmer Constant for Z n p[END_REF], made progress on noncyclic finite abelian groups G = Z n p , then Pigno, Pinner and Vipismakul [START_REF] Vipismakul | The Stabilizer of the Group Determinant and Bounds for Lehmer's Conjecture on Finite Abelian Groups[END_REF], [START_REF] Pigno | The Lind-Lehmer Constant for Zm×Z n p[END_REF], on general p-groups G p = Z p l 1 × . . . × Z p ln and G = Z m × G p for m not divisible by p.

An areal analogue of Mahler's measure is reported by Pritsker [START_REF] Pritsker | Small Polynomials with Integer Coefficients[END_REF], linked to Hardy and Bergman normed spaces of functions on the unit disk.

Lehmer's problems in positive characteristic and Drinfeld modules: let k = F q (T ) be the fraction field of the ring F q [T ] of polynomials with coefficients in the finite field F q (p is a prime number and q a power of p). Let k ∞ = F q ((1/T )) be the completion of k for the 1/T -adic valuation v. The valuation, still denoted by v, is extended to the algebraic closure

k of k, resp. k ∞ of k ∞ . The degree deg(x) of x ∈ k ∞ is equal to the integer-valued -v(x), with the convention deg(0) = -∞.
Let t denote a formal variable. By definition a t-module of dimension N and rank d on k is given by the additive group (G a ) N and an injective ring homomorphism Φ : F q [T ] → End(G a ) N which satisfies:

Φ(t) = a 0 F 0 + . . . + a d F d , (5.25)
where F is the Frobenius endomorphism on (G a ) N and a 0 , a 1 , . . . , a d are N × N matrices with coefficients in k. In [START_REF] Denis | Hauteurs Canoniques et Modules de Drinfeld[END_REF] Denis constructed a canonical height h = h Φ on t-modules for which a d is invertible, from the Weil height. Denis formulated Lehmer's problem for t-modules as follows, in two steps: (i) for α ∈ (k) n , defined over a field of degree ≤ δ, not in the torsion of the t-module, does there exist c(δ) = c a0,...,a d ,N,d,q,F (δ) > 0 such that h(α) ≥ c(δ)?; (ii) if (i) is satisfied, on a Drinfeld module of rank d, does there exist c > 0 such that, for any α not belonging to the torsion,

h(α) ≥ c δ ? (5.26)
The second problem is the extension of the classical Lehmer problem [START_REF] Pacheco | Analogues of Lehmer's Problem in Positive Characteristic[END_REF]. Denis partially solved Lehmer's problem ( [START_REF] Denis | Hauteurs Canoniques et Modules de Drinfeld[END_REF] Theorem 2) in the case of Carlitz modules, i.e. with N = 1 and d = 1 for which Φ(T )(x) = T x+x q . He obtained the following minoration which is an analogue of Laurent's Theorem 4.2 for CM elliptic curves (elliptic Lehmer problem) and Dobrowolski's inequality (2.17):

Theorem 5.4 (Denis). There exists a real number η > 0 such that, for any α belonging to the regular separable closure of k, not to the torsion, of degree ≤ δ, the minoration holds:

h(α) ≥ η 1 δ Log Log δ Log δ 3 (5.27)
(the real number η is effective and computable from q).

Grandet-Hugot in [START_REF] Grandet-Hugot | Eléments Algébriques Remarquables dans un Corps de Séries Formelles[END_REF] studied analogues of Pisot and Salem numbers in fields of formal series: x ∈ k ∞ is a Salem number if it is algebraic on k, deg(x) > 0, and all its conjugates satisfy: deg(x i ) ≤ 0. In this context Denis ([152] Theorem 1) proved the fact that there is no Salem number too close to 1, namely: Theorem 5.5 (Denis). Let α ∈ k ∞ having at least one conjugate in k ∞ . If α does not belong to the torsion, is of degree D on k, then h(α) ≥ 1 q D

(5.28)

Extending the previous results, Denis ([152] Theorem 3) solved Lehmer's problem for the following infinite family of t-modules:

Theorem 5.6 (Denis). Let Φ(t) = a 0 F 0 + a 1 F + . . . a d-1 F d-1 + F d be a t-module of dimension 1 such that a i ∈ k ∞ ∩ k, 0 ≤ i ≤ d -1, is integral over F q [T ].
Then there exists a real number c Φ > 0 depending only upon Φ, such that, if α is an algebraic element of k ∞ , not in the torsion, of degree D on k, then

h Φ (α) ≥ c Φ D (5.29)
The abelian Lehmer problem for Drinfeld modules was solved by David and Pacheco [START_REF] David | Le Problème de Lehmer Abélien pour un Module de Drinfeld[END_REF] using Denis's construction of the canonical height (with A = F q [T ]): Theorem 5.7 (David -Pacheco). Let K/k be a finite extension, K an algebraic closure of K, and K ab the largest abelian extension of K in K. Let φ : A → K{τ } be a Drinfeld module of rank ≥ 1. Then there exists c = c(φ, K) > 0 which depends only upon φ and K such that, for any α ∈ K ab , not being in the torsion, h Φ (α) ≥ c.

(5.30)

In [START_REF] Ghioca | The Local Lehmer Inequality for Drinfeld Modules[END_REF] Ghioca investigates statements, for Drinfeld modules of generic characteristic, which would imply that the classical Lehmer problem for Drinfeld modules is true. In [START_REF] Ghioca | The Lehmer Inequality and the Mordell-Weil Theorem for Drinfeld Modules[END_REF] Ghioca obtained several Lehmer type inequalities for the height of nontorsion points of Drinfeld modules. Using them, as consequence of Theorem 5.8 below, Ghioca proved several Mordell-Weil type structure theorems for Drinfeld modules over certain infinitely generated fields (the definitions of the terms can be found in [START_REF] Ghioca | The Lehmer Inequality and the Mordell-Weil Theorem for Drinfeld Modules[END_REF]): Theorem 5.8 (Ghioca). Let K/F q be a field extension, and φ : A → K{τ } be a Drinfeld module. Let L/K be a finite field extension. Let t be a non-constant element of A and assume that φ t = r i=0 a i τ i is monic. Let U be a good set of valuations on L and let C(U ) be the field of constants with respect to U . Let S be the finite set of valuations v ∈ U such that φ has bad reduction at v. The degree of the valuation v is denoted by d(v). Let x ∈ L.

a) If S is empty, then either x ∈ C(U ) or there exists v ∈ U such that

h U,v (x) ≥ d(v), (5.31)
b) If S is not empty, then either x ∈ φ tors , or there exists v ∈ U such that

h U,v (x) > d(v) q 2r+r 2 N φ |S| ≥ d(v) q r 2+(r 2 +r)|S| .
(5.32)

Moreover, if S is not empty and x ∈ φ tors , then there exists a polynomial b(t) ∈ F q (t) of degree at most rN φ |S| such that φ b(t) (x) = 0.

Let K be a finitely generated field extension of F q , and K alg an algebraic closure of K. Ghioca [START_REF] Ghioca | The Local Lehmer Inequality for Drinfeld Modules[END_REF] developped global heights associated to a Drinfeld module φ : A → K{τ } and, for each divisor v, local heights h v : K alg → R + associated to φ. For Drinfeld modules of finite characteristic Ghioca [START_REF] Ghioca | The Local Lehmer Inequality for Drinfeld Modules[END_REF] obtained Lehmer type inequalities with the local heights, extending the classical Lehmer problem: Theorem 5.9 (Ghioca). For φ : A → K{τ } a Drinfeld module of finite characteristic, there exist two positive constants C and r depending only on φ such that if x ∈ K alg and v is a place of K(x) for which h v (x) > 0, then

h v (x) ≥ C d r (5. 33
)
where d = [K(x) : K].
Bauchère [START_REF] Bauchère | Minoration de la Hauteur Canonique pour les Modules de Drinfeld à Multiplications Complexes[END_REF] generalized David Pacheco's Theorem 5.7 to Drinfeld modules having complex multiplications, proving the abelian Lehmer problem in this context: Theorem 5.10 (Bauchère). Let φ be a A-Drinfeld module defined over k having complex multiplications. Let K/k be a finite field extension, L/K a Galois extension (finite or infinite) with Galois group G =Gal(L/K). Let H be a subgroup of the center of G and E ⊂ L the subfield fixed by H. Let d 0 be an integer. We assume that there exists a finite place v of K such that [E w : K v ] ≤ d 0 for every place w of E, v|w. Then there exists a constant c 0 = c 0 (φ) > 0 such that, for any α ∈ L, not belonging to the torsion for φ,

h φ (α) ≥ 1 q c0 d(v) d 2 0 [K:k] .
(5.34)

Theorem 5.10 is the analogue of a result obtained by Amoroso, David and Zannier [START_REF] Amoroso | On Fields with the Property (B)[END_REF] for the multiplicative group. Theorem 5.10 is particularly interesting when L/K is infinite. Bauchère [START_REF] Bauchère | Minoration de la Hauteur Canonique pour les Modules de Drinfeld à Multiplications Complexes[END_REF] deduced special minorations of the heights h φ (α) in two Corollaries, for L = K ab , and in the case where the subgroup H is trivial.

In other domains

The conjectural discontinuity of the Mahler measure M(α), α ∈ Q, at 1 has consequences in different domains of mathematics. It is linked to the notions of "smallest complexity", "smallest growth rate", "smallest geometrical dilatation", "smallest geodesics", "smallest Salem number" or "smallest topological entropy" (Hironaka [START_REF] Hironaka | What is ... Lehmer's number?[END_REF]). We will keep an interdisciplinary viewpoint as in the recent survey [START_REF] Smyth | Seventy Years of Salem Numbers[END_REF] by C. Smyth and refer the reader to [START_REF] Ghate | The Arithmetic and Geometry of Salem numbers[END_REF], [START_REF] Smyth | Seventy Years of Salem Numbers[END_REF]; we only mention below a few more or less new results. The smallest Mahler measures, or smallest Salem numbers, correspond to peculiar geometrical constructions in their respective domains.

6.1. Coxeter polynomials, graphs, Salem trees. Let Γ = (Γ 0 , Γ 1 ) be a simple graph with set of enumerated vertices Γ 0 = {v 1 , . . . , v n }, Γ 1 being the set of edges where (v i , v j ) ∈ Γ 1 if there is an edge connecting the vertices v i and v j . The adjacency matrix of Γ is Ad is Lehmer's polynomial. A tree T is said to be cyclotomic, resp. a Salem tree, if cox T (x) is a product of cyclotomic polynomials, resp. the product of cyclotomic polynomials by an irreducible Salem polynomial. Such objects generalize E n as far as their Coxeter polynomials remains of the same form. Evripidou [START_REF] Evripidou | Coxeter Polynomials of Salem Trees[END_REF], following Lakatos [START_REF] Lakatos | On the Coxeter Polynomials of Wild Stars[END_REF], [START_REF] Lakatos | Salem Numbers, PV Numbers and Spectral Radii of Coxeter Transformations[END_REF], [START_REF] Lakatos | A New Construction of Salem Polynomials[END_REF], [START_REF] Lakatos | Salem Numbers Defined by Coxeter Transformations[END_REF] and [START_REF] Gross | Cyclotomic Factors of Coxeter Polynomials[END_REF], obtained structure theorems and formulations for the Coxeter polynomials of families of Salem trees, for the spectral radii of the respective Coxeter transformations. Lehmer's problem asks whether there exists a Salem tree of minimal Salem number; what would be its decomposition?

Γ := [a ij ] ∈ M n (Z) where a ij = 1, if (v i , v j ) ∈ Γ
The Mahler measure M(G) of a finite graph G, with n vertices, is introduced in McKee and Smyth [START_REF] Mckee | Salem Numbers, Pisot Numbers, Mahler Measures and Graphs[END_REF]. If D G (z) is the characteristic polynomial of G, then the reciprocal integer polynomial associated with G is z n D G (z +1/z). The Mahler measure of this later polynomial is the Mahler measure M(G) of G; explicitely, Assume that G admits a finite generating set S. Define the length of an element g in G = (G, S) to be the least nonnegative integer n such that g can be expressed as a product of n elements from S ∪ S -1 . For every nonnegative integer n let N S (n) be the number of elements in G with length n.

M(G) = D G (χ)=0,|χ|>2 1 2 (|χ| + χ 2 -4). ( 6
Following Milnor [START_REF] Milnor | Problems and Solutions: Advanced Problems 5600-5609[END_REF] the growth series of the group (G, S) is by definition

f (x) = ∞ n=1 N S (n)x n , for which N S (n) ≤ (2|S|) n . (6.4)
The asymptotic growth rate of G = (G, S), finite and ≥ 1, is by definition

lim sup n→∞ (N S (n)) 1/n , (6.5)
its inverse, positive, being the radius of convergence of f (x). A Coxeter group G, with S being a finite generating set for G, is a group where every element of S has order two and all the other defining relators for G are of the form (g h) m(g,h) = 1 G where m(g, h) = m(h, g) and m(g, h) ≥ 2. Steinberg [START_REF] Steinberg | Endomorphisms of Linear Algebraic Groups[END_REF] and Bourbaki [START_REF] Bourbaki | Groupes et Algèbres de Lie[END_REF] showed that the growth series of a Coxeter group is a rational function. Salem numbers, Pisot numbers and Perron numbers occur as roots of the polynomials at the denominator (here the definition of a Salem number is often extended to quadratic Pisot numbers, conveniently and abusively).

Let us consider a hyperbolic cocompact Coxeter group G with generating set of reflections S acting in low dimensions n ≥ 2.

Case n = 2: for the Coxeter reflection groups G p1,...,p d , with p i any positive integer, of presentation G p1,...,p d := (g 1 , . . . , g d | (g i ) 2 = 1, (g i g i+1 ) pi = 1), the denominator ∆ p1,...,p d (x) of the growth series f (x) of G p1,...,p d is explicitely given by the following theorem [START_REF] Cannon | The Growth of the Closed Surface Groups and the Compact Hyperbolic Coxeter Groups[END_REF]. Theorem 6.1 (Cannon -Wagreich [START_REF] Cannon | Growth Functions of Surface Groups[END_REF], Floyd -Plotnick [START_REF] Floyd | Symmetries of Planar Growth Functions of Coxeter Groups[END_REF], Parry [START_REF] Parry | Growth Series of Coxeter Groups and Salem Numbers[END_REF]).

∆ p1,...,p d (x) = [p 1 ][p 2 ] . . . [p d ](x -d + 1) + d i=1 [p 1 ] . . . [p i ] . . . [p d ].
(6.6)

The polynomial ∆ p1,...,p d (x) is either a product of cyclotomic polynomials or a product of cyclotomic polynomials times an irreducible Salem polynomial. The Salem polynomial occurs if and only if G p1,...,p d is hyperbolic, that is,

1 p 1 + . . . + 1 p d < d -2. (6.7)
Then hyperbolic Coxeter reflection groups have Salem numbers as asymptotic growth rates. Such Salem numbers form a subclass of the set of Salem numbers. Lehmer's polynomial is ∆ 2,3,7 (x), denominator of the growth series of the (2,3,7)hyperbolic triangle group (Takeuchi [507]). The Construction of Salem [START_REF] Salem | Power Series with Integral Coefficients[END_REF], [START_REF] Boyd | Small Salem Numbers[END_REF], for establishing the existence of sequences of Salem numbers converging to a given Pisot number, on the left and on the right, admits an analogue in terms of geometric convergence for the fundamental domains of cocompact planar hyperbolic Coxeter groups. Using the Construction of Salem Parry [START_REF] Parry | Growth Series of Coxeter Groups and Salem Numbers[END_REF] gives a new proof of Theorem 6.1. Bartholdi and Ceccherini-Silberstein [START_REF] Bartholdi | Salem Numbers and Growth Series of Some Hyperbolic Graphs[END_REF] studied the Salem numbers which arise from some hyperbolic graphs. Hironaka [START_REF] Ghate | The Arithmetic and Geometry of Salem numbers[END_REF] solves the problem of Lehmer for the subclass of Salem numbers occuring as such asymptotic growth rates: Theorem 6.2 (Hironaka [START_REF] Hironaka | The Lehmer Polynomial and Pretzel Links[END_REF]). Lehmer's number is the smallest Salem number occuring as dominant roots of ∆ p1,...,p d polynomials for any (p 1 , . . . , p d ), p i being positive integers.

Case n = 3: Parry [START_REF] Parry | Growth Series of Coxeter Groups and Salem Numbers[END_REF] extends its 2-dimensional approach to every hyperbolic cocompact reflection Coxeter group on H 3 generated by reflections whose fundamental domain is a bounded polyhedron (not just tetrahedron). Parry's approach is based on the properties of anti-reciprocal rational functions with Salem numbers. Kolpakov [START_REF] Kolpakov | Deformation of Finite-Volume Hyperbolic Coxeter Polyhedra, Limiting Growth rates and Pisot Numbers[END_REF] provides a generalization to the three-dimensional case, by establishing a metric convergence of fundamental domains for cocompact hyperbolic Coxeter groups with finite-volume limiting polyhedron; for instance, the compact polyhedra P(n) ⊂ H 3 of type < 2, 2, n, 2, 2 > converging, as n → ∞, to a polyhedron P ∞ with a single four-valent ideal vertex. In this context, Kolpakov investigates the growth series of Coxeter groups acting on H n , n ≥ 3 and their limit properties. The growth rates of ideal Coxeter polyhedra in H 3 was studied by Nonaka and Kellerhals [START_REF] Nonaka | The Growth Rates of Ideal Coxeter Polyhedra in Hyperbolic 3-Space[END_REF].

Case n ≥ 4: the growth rates of cocompact hyperbolic Coxeter groups are not Salem numbers anymore. Kellerhals and Perren [START_REF] Kellerhals | On the Growth of Cocompact Hyperbolic Coxeter Groups[END_REF], §3 Example 2, show this fact with the example of the compact right-angled 120-cell in H 4 .

Lehmer's problem asks about the geometry of the poles of the growth rates of hyperbolic Coxeter groups acting on H n , and structure theorems about such groups having denominators of growth series of minimal Mahler measure. Conjecture 6.3 (Kellerhals -Perren). Let G be a Coxeter group acting cocompactly on H n with natural generating set S and growth series f S (x). Then, (a) for n even, f S (x) has precisely n 2 poles 0 < x 1 < . . . < x n 2 < 1 in the open unit interval (0, 1) ;

(b) for n odd, f S (x) has precisely the pole 1 and n-1

2 poles 0 < x 1 < . . . < x n-1 2 < 1 in the interval (0, 1).
In both cases, the poles are simple, and the non-real poles of f S (x) are contained in the annulus of radii x m and x -1 m for some m ∈ {1, . . . , n 2 }. Theorem 6.4 (Kellerhals -Perren). Let G be a Lannér group, an Esselmann group or a Kaplinskaya group, respectively, acting with natural generating set S on H 4 . Then, (1) the growth series f S (x) of G is a quotient of relatively prime, monic and reciprocal polynomials of equal degree over the integers,

(2) the growth series f S (x) of G possesses four distinct positive real poles appearing in pairs (x 1 , x -1 1 ) and (x 2 , x -1 2 ) with

x 1 < x 2 < 1 < x -1 2 < x -1 1 ; these poles are simple, (3) the growth rate τ = x -1
1 is a Perron number, (4) the non-real poles of f S (x) are contained in an annulus of radii x 2 , x -1 2 around the unit circle, (5) the growth series f S (x) of the Kaplinskaya group G 66 with graph K 66 has four distinct negative and four distinct positive simple real poles; for G = G 66 , f S (x) has no negative pole.

Kellerhals and Kolpakov [START_REF] Kellerhals | The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3-Space[END_REF] (2014) prove that the simplex group (3, 5, 3) has the smallest growth rate among all cocompact hyperbolic Coxeter groups on H 3 , and that it is, as such, unique. The growth rate is the Salem number τ = 1.35098 . . . of minimal polynomial

X 10 -X 9 -X 6 + X 5 -X 4 -X + 1. (6.8)
Their approach provides a different proof for the analog situation in H 2 where Hironaka [START_REF] Hironaka | The Lehmer Polynomial and Pretzel Links[END_REF] identified Lehmer's number as the minimal growth rate among all cocompact planar hyperbolic Coxeter groups and showed that it is (uniquely) achieved by the Coxeter triangle group [START_REF] Agoh | On the Relative Class Number of Special Cyclotomic Fields[END_REF][START_REF] Amara | Ensembles Fermés de Nombres Algébriques[END_REF].

After Meyerhoff who proved that among all cusped hyperbolic 3-orbifolds the quotient of H 3 by the tetrahedral Coxeter group [START_REF] Agoh | On the Relative Class Number of Special Cyclotomic Fields[END_REF][START_REF] Agoh | On the Relative Class Number of Special Cyclotomic Fields[END_REF][START_REF] Allcock | A Banach Space Determined by the Weil Height[END_REF] has minimal volume, Kellerhals [START_REF] Kellerhals | Cofinite Hyperbolic Coxeter Groups, Minimal Growth Rate and Pisot Numbers[END_REF] (2013) proves that the group [START_REF] Agoh | On the Relative Class Number of Special Cyclotomic Fields[END_REF][START_REF] Agoh | On the Relative Class Number of Special Cyclotomic Fields[END_REF][START_REF] Allcock | A Banach Space Determined by the Weil Height[END_REF] has smallest growth rate among all non-cocompact cofinite hyperbolic Coxeter groups, and that it is as such unique. This result extends to three dimensions some work of Floyd [START_REF] Floyd | Growth of Planar Coxeter Groups, P.V. Numbers, and Salem Numbers[END_REF] who showed that the Coxeter triangle group (3, ∞) has minimal growth rate among all non-cocompact cofinite planar hyperbolic Coxeter groups. In contrast to Floyd's result, the growth rate of the tetrahedral group [START_REF] Agoh | On the Relative Class Number of Special Cyclotomic Fields[END_REF][START_REF] Agoh | On the Relative Class Number of Special Cyclotomic Fields[END_REF][START_REF] Allcock | A Banach Space Determined by the Weil Height[END_REF] is not a Pisot number. The following Theorem completes the picture of growth rate minimality for cofinite hyperbolic Coxeter groups in three dimensions. Theorem 6.5 (Kellerhals). Among all hyperbolic Coxeter groups with non-compact fundamental polyhedron of finite volume in H 3 , the tetrahedral group [START_REF] Agoh | On the Relative Class Number of Special Cyclotomic Fields[END_REF][START_REF] Agoh | On the Relative Class Number of Special Cyclotomic Fields[END_REF][START_REF] Allcock | A Banach Space Determined by the Weil Height[END_REF] has minimal growth rate, and as such the group is unique.

In [START_REF] Komori | On the Growth of Hyperbolic 3-Dimensional Generalized Simplex Reflection Groups[END_REF] Komori and Umemoto, for three-dimensional non-compact hyperbolic Coxeter groups of finite covolume, show that the growth rate of a three-dimensional generalized simplex reflection group is a Perron number. In [START_REF] Komori | On the Growth Rate of Ideal Coxeter Groups in Hyperbolic 3-Space[END_REF] Komori and Yukita show that the growth rates of ideal Coxeter groups in hyperbolic 3-space are also Perron numbers ; a Coxeter polytope P is called ideal if all vertices of P are located on the ideal boundary of hyperbolic space. They prove that the growth rate τ of an ideal Coxeter polytope with n facets in H n satisfies n -3 ≤ τ ≤ n -1. The smallest growth rates occur only if n ≤ 4. They prove that the minimum of the growth rates is 0.492432 -1 ≈ 2.03074, which is uniquely realized by the ideal Coxeter simplex with p = q = s = 2. In [START_REF] Umemoto | The Growth Function of Coxeter Dominoes and 2-Salem Numbers[END_REF] Umemoto shows that infinitely many 2-Salem numbers can be realized as the growth rates of cocompact Coxeter groups in H 4 ; the Coxeter polytopes are here constructed by successive gluing of Coxeter polytopes, which are called Coxeter dominoes [START_REF] Umemoto | Growth Rates of Cocompact Hyperbolic Coxeter Groups and 2 -Salem Numbers[END_REF]. The algebraic integers having a fixed number of conjugates outside the closed unit disk were studied by Bertin [START_REF] Bertin | Quelques Résultats Nouveaux sur les Nombres de Pisot et de Salem[END_REF], Kerada [START_REF] Kerada | Une Caractérisation de Certaines Classes d'Entiers Algébriques Généralisant les Nombres de Salem[END_REF], Samet [START_REF] Samet | Algebraic Integers with two Conjugates Outside the Unit Circle, I[END_REF], Zaimi [START_REF] Zaïmi | Sur les Nombres de Pisot Relatifs[END_REF], [START_REF] Zaïmi | Sur les K-Nombres de Pisot de Petite Mesure[END_REF], in particular 2-Salem numbers in [START_REF] Kerada | Une Caractérisation de Certaines Classes d'Entiers Algébriques Généralisant les Nombres de Salem[END_REF] to which Umemoto refers. In [START_REF] Zehrt | The Growth Function of Coxeter Garlands in H 4[END_REF] Zehrt and Zehrt-Liebendörfer construct infinitely many growth series of cocompact hyperbolic Coxeter groups in H 4 , whose denominator polynomials have the same distribution of roots as 2-Salem polynomials; their Coxeter polytopes are the Coxeter garlands built by the compact truncated Coxeter simplex described by the Coxeter graph on the left side of Figure 1 in [START_REF] Zehrt | The Growth Function of Coxeter Garlands in H 4[END_REF]. Lehmer's problem asks about the minimality of the houses of the 2-Salem numbers involved in these constructions. In [START_REF] Nonaka | The Growth Rates of Ideal Coxeter Polyhedra in Hyperbolic 3-Space[END_REF] Kellerhals and Nonaka prove that the growth rates of three-dimensional Coxeter groups (Γ, S) given by ideal Coxeter polyhedra of finite volume are Perron numbers.

A Coxeter system (W, S) is a Coxeter group W with a finite generating set S; the permuted products s σ(1) s σ(2) . . . s σ(n) , σ ∈ S n , are the Coxeter elements of (W, S). The element w ∈ W is said to be essential if it is not conjugate into any subgroup W I ⊂ W generated by a proper subset I ⊂ S. The Coxeter group (W, S) acts naturally by reflections on V ≡ R S . Let λ(w) be the spectral radius of w|V . When λ(w) > 1 it is also an eigenvalue of w. MacMullen [START_REF] Mcmullen | Coxeter Groups, Salem Numbers and the Hilbert Metric[END_REF] proves the three following results. Theorem 6.6 (MacMullen). Let (W, S) be a Coxeter system and let w ∈ W be essential. Then

λ(w) ≥ inf Sn λ(s σ(1) s σ(2) . . . s σ(n) ). (6.9)
Let α(W, S) be the dominant eigenvalue of the adjacency matrix (A ij ) of (W, S), defined by A ij = 2 cos(π/m ij ) for i = j and A ii = 0. Let β(W, S) be the largest root of the equation β + β -1 + 2 = α(W, S) 2 provided α(W, S) ≥ 2. If α(W, S) < 2 then we put: β(W, S) = 1. Then λ(w) = β(W, S) for all bicolored Coxeter element. Lehmer's problem is solved in this context. The quantity β(W, S) can be viewed as a measure (not in logarithmic terms) of the complexity of a Coxeter system. Denote by Y a,b,c the Coxeter system whose diagram is a tree with 3 branches of lenghts a, b and c, joined by a single node. MacMullen [START_REF] Mcmullen | Coxeter Groups, Salem Numbers and the Hilbert Metric[END_REF] shows that the smallest Salem numbers of respective degrees 6, 8 and 10 coincide with λ(w) for the Coxeter elements of Y 3,3,4 , Y 2,4,5 and Y 2,3,7 respectively; in particular Lehmer's number is λ(w) for the Coxeter elements of Y 2,3,7 . MacMullen shows that the set of irreducible Coxeter systems with β(W, S) < Θ consists exactly of Y 2,4,5 and Y 2,3,n , n ≥ 7. He shows that the infimum of β(W, S) over all high-rank Coxeter systems coincides with Θ. There are 6 Salem numbers < 1.3 that arise as eigenvalues in Coxeter groups, five of them arising from the Coxeter elements of Y 2,3,n , 7 ≤ n ≤ 11.

6.3. Mapping classes: small stretch factors. We refer to Birman [START_REF] Birman | Braids, Links and Mapping Class Groups[END_REF], Farb and Margalit [START_REF] Farb | A primer on Mapping Class Groups[END_REF] and Hironaka [START_REF] Hironaka | Dynamics of Mapping Classes on Surfaces[END_REF]. If S is a surface the mapping class group of S, denoted by Mod(S), is the group of isotopy classes of orientation-preserving diffeomorphisms of S (that restrict to the identity on ∂S if ∂S = ∅). An irreducible mapping class is an isotopy class of homeomorphisms f of a compact oriented surface S to itself so that no power preserves a nontrivial subsurface. The classification of Nielsen-Thuston states that a mapping class [f ] ∈ Mod(S) is either periodic, reducible or pseudo-Anosov [START_REF] Farb | A primer on Mapping Class Groups[END_REF], [START_REF] Fathi | Travaux de Thurston sur les Surfaces[END_REF]. In the periodic case, the situation is "analogous to roots of unity" in Lehmer's problem. The minoration problem of the Mahler measure finds its analogue in the minoration of the dilatation factors of the pseudo-Anosovs. We refer to a mapping class [f ] by one of its representive f . Let S g be a closed, orientable surface of genus g ≥ 2 and Mod(S g ) its mapping class group. For any Pseudo-Anosov element f ∈Mod(S g ), and any integer 0 Thurston [START_REF] Fathi | Travaux de Thurston sur les Surfaces[END_REF], [START_REF] Thurston | On the Geometry and Dynamics of Diffeomorphisms of Surfaces[END_REF], noticed that the set of stretch factors for pseudo-Anosov elements of Mod(S g ) is closed and discrete in R, and proved that any dilatation factor λ(f ) > 1 is a Perron number, with λ(f ) + λ(f ) -1 an algebraic integer of degree ≤ 4g -3. The Perron number λ(f ) is the growth rate of lengths of curves under iteration (of any representant) of f , in any metric on S g . These stretch factors appear as the length spectrum of the moduli space of genus g Riemann surface.

≤ k ≤ 2g, let (i) κ(f ) be the dimension of the subspace of H 1 (S g , R) fixed by f (for which 0 ≤ κ(f ) ≤ 2g), (ii) h(f )
Penner [START_REF] Penner | Bounds on Least Dilatations[END_REF] proved that the asymptotic behaviour L(0, g) 1/g holds. With k = 2g, Farb, Leininger and Margalit [START_REF] Farb | The Lower Central Series and Pseudo-Anosov Dilatations[END_REF] proved L(2g, g) 1.

For the other values of k, since L(0, g) ≤ L(k, g) ≤ L(2g, g), the following inequalities hold, from [START_REF] Aaber | Closed Surface Bundles of Least Volume[END_REF]

[303] [263] [414], Log 2 6 1 2g -2 ≤ L(k, g) ≤ Log (62). (6.12)
For k = 0 and g = 1, L(0, 1) = Log 3+ √ 5 2 for T 2 . For k = 0 and g = 2, Cho and Ham [START_REF] Cho | The Minimal Dilatation of a Genus-2 Surface[END_REF], [START_REF] Lanneau | On the Minimum Dilatation of Pseudo-Anosov Homeomorphisms on Surfaces of Small Genus[END_REF], [START_REF] Zhirov | On the Minimum Dilatation of Pseudo-Anosov Diffeomorphisms of a Double Torus[END_REF], proved L(0, 2) ≈ 0.5435 . . ., as logarithm of the largest root of the Salem polynomial X 4 -X 3 -X 2 -X + 1; these authors showed that this minimum dilatation is given by Zhirov in [START_REF] Zhirov | On the Minimum Dilatation of Pseudo-Anosov Diffeomorphisms of a Double Torus[END_REF], and realized by Pseudo-Anosov 5-braids in [START_REF] Ham | The Minimum Dilatation of Pseudo-Anosov 5-Braids[END_REF]. In [START_REF] Agol | Pseudo-Anosov Stretch Factors and Homology of Mapping Tori[END_REF] Agol, Leininger and Margalit improved the upper bound to: (2g -2)L(0, g) < Log (θ -4

2 ) for all g ≥ 2, where θ -1 2 is the golden mean, and proved the main theorem:

L(k, g) k + 1 g , g ≥ 2, 0 ≤ k ≤ 2g. (6.13)
Arnoux-Yoccoz's Theorem [START_REF] Arnoux | Construction de Difféomorphismes Pseudo-Anosov[END_REF] states that, for g ≥ 2, for any C ≥ 1, there are only finitely many conjugacy classes in Mod(S g ) of pseudo-Anosov mapping classes with stretch factors at most C.

Minimal dilatation problem:

what are the values of L(k, g), except L(0, g) for g = 1, 2 already determined? i.e. what are the minima δ g := exp(L(0, g)), g ≥ 3?

Lower bounds of the entropy are difficult to establish: e.g. Penner [START_REF] Penner | Bounds on Least Dilatations[END_REF], Tsai [START_REF] Tsai | The Asymptotic Behavior of Least Pseudo-Anosov Dilatations[END_REF] on punctured surfaces, Boissy and Lanneau [START_REF] Boissy | Pseudo-Anosov Homeomorphisms on Translation Surfaces in Hyperelliptic Components Have Large Entropy[END_REF] on translation surfaces that belong to a hyperelliptic component, Hironaka and Kin [START_REF] Hironaka | A Family of Pseudo-Anosov Braids With Small Dilatation[END_REF]. Then Kin [START_REF] Kin | Notes on Pseudo-Anosovs with Small Dilatations Coming from the Magic 3-Manifold, in Representation Spaces, Twisted Topological Invariants and Geometric Structures of 3-Manifolds[END_REF], [START_REF] Kin | An Asymptotic Behavior of the Dilatation for a Family of Pseudo-Anosov Braids[END_REF], [START_REF] Kin | Entropy Versus Volume for Pseudo-Anosovs[END_REF], formulated several questions and conjectures on the minimal dilatation problem and its realizations. Bauer [START_REF] Bauer | An Upper Bound for the Least Dilatation[END_REF] studied upper bounds of the least dilatations, and Minakawa [START_REF] Minakawa | Examples of Pseudo-Anosov Homeomorphisms With Small Dilatations[END_REF] gave examples of pseudo-Anosovs with small dilatations. Farb, Leininger and Margalit [START_REF] Farb | Small Dilatation Pseudo-Anosov Homeomorphisms and 3-Manifolds[END_REF] obtained a universal finiteness theorem for the set of all small dilatation pseudo-Anosov homeomorphisms φ : S → S, ranging over all surfaces S. The following questions were posed by in [START_REF] Mcmullen | Polynomial Invariants for Fibered 3-Manifolds and Teichmuller Geodesics for Foliations[END_REF] and [START_REF] Farb | Some Problems on Mapping Class Groups and Moduli Space[END_REF].

Asymptotic behaviour:

(i) Does lim g→∞ g L(0, g) exist? What is its value?

(ii) Is the sequence {δ g } g≥2 (strictly) monotone decreasing?

Kin, Kojima and Takasawa [START_REF] Kin | Entropy Versus Volume for Pseudo-Anosovs[END_REF], for monodromies of fibrations on manifolds obtained from the magic 3-manifold N by Dehn filling three cusps with some restriction, proved lim g→∞ g L(0, g) = Log 3+ √ 5 2

; they also formulated limit conjectures for the asymptotic behaviour relative to compact surfaces of genus g with n boundary components.

A pseudo-Anosov mapping class [f ] is said to be orientable if the invariant (un)-stable foliation of a pseudo-Anosov homeomorphism f ∈ [f ] is orientable. Let λ H (f ) be the spectral radius of the action of f on H 1 (S g , R). It is the homological stretch factor of f . The inequality

λ H (f ) ≤ λ(f ) (6.14)
holds and equality occurs iff the invariant foliations for f are orientable. Stretch factors obey some constraints [START_REF] Shin | Algebraic degrees of Stretch Factors in Mapping Class Groups[END_REF]:

(i) deg(λ(f )) ≥ 2, (ii) deg(λ(f )) ≤ 6g -6, (iii) if deg(λ(f )) > 3g -3, then deg(λ(f )) is even.
Shin [START_REF] Shin | Algebraic degrees of Stretch Factors in Mapping Class Groups[END_REF] formulates the following questions.

Algebraicity of the stretch factors.

(i) Which real numbers can be the stretch factors, the homological stretch factors?

(ii) What degrees of stretch factors can occur on S g ?

Let us define a mapping class f g,k by

f g,k = (T cg ) k T dg . . . T c2 T d2 T c1 T d1 ∈ Mod(S g ), (6.15)
where c i and d i are simple closed curves on S g as in Figure 1 in [START_REF] Shin | Algebraic degrees of Stretch Factors in Mapping Class Groups[END_REF], and T c the Dehn twist about c. Theorem 6.9 (Shin). For each g ≥ 2, k ≥ 3, f g,k is a pseudo-Anosov mapping class which satisfies:

(i) λ(f g,k ) = λ H (f g,k ), (ii) f g,k is a Salem number, (iii) lim g→∞ f g,k = k -1,
where the minimal polynomial of λ(f g,k ) is the irreducible Salem polynomial

t 2g -(k -2) 2g-1 j=1 t j + 1, of degree 2g. (6.16)
Shin [START_REF] Shin | Algebraic degrees of Stretch Factors in Mapping Class Groups[END_REF] deduces that, for each 1 ≤ h ≤ g/2, there exists a pseudo-Anosov mapping class f h ∈ Mod(S g ) such that deg( f h ) = 2h, with λ( f h ) a Salem number. He conjectures that, on S g , there exists a pseudo-Anosov mapping class with a stretch factor of degree d for any even 1 ≤ d ≤ 6g-6. He proves that the conjecture is true for g = 2 to 5. Shin and Strenner [START_REF] Shin | Pseudo-Anosov Mapping Classes Not Arising from Penner's Construction[END_REF] prove that the Perron numbers which are the stretch factors of pseudo-Anosov mapping classes arising from Penner's construction [START_REF] Penner | Bounds on Least Dilatations[END_REF] have conjugates which do not belong to the unit circle. In §3 in [START_REF] Shin | Pseudo-Anosov Mapping Classes Not Arising from Penner's Construction[END_REF] they ask several questions about the geometry of the Galois conjugates of stretch factors, around the unit circle, obtained by several constructions: by Hironaka [START_REF] Hironaka | Small Dilatation Mapping Classes Coming From the Simplest Hyperbolic Braid[END_REF], by Dunfeld and Tiozzo, by Lanneau and Thiffeault [START_REF] Lanneau | On the Minimum Dilatation of Pseudo-Anosov Homeomorphisms on Surfaces of Small Genus[END_REF], [START_REF] Lanneau | On the Minimum Dilatation of Braids on Punctured Discs[END_REF], by Shin [START_REF] Shin | Algebraic degrees of Stretch Factors in Mapping Class Groups[END_REF]. For S g,n being an orientable surface with genus g and n marked points, Tsai [START_REF] Tsai | The Asymptotic Behavior of Least Pseudo-Anosov Dilatations[END_REF] proves that the infimum of stretch factors is of the order of (Log n)/n for g ≥ 2 whereas it is of the order of 1/n for g = 0 and g = 1; Tsai asks the question about the asymptotic behaviour of this infimum of dilatation factors in the (g, n)-plane. For some subcollections of mapping classes, by generalizing Penner's construction and by comparing the smallness of dilatation factors with trivial homological dilatation, Hironaka [START_REF] Hironaka | Penner Sequences and Asymptotics of Minimum Dilatations for Subfamilies of the Mapping Class Group[END_REF] concludes to the existence of subfamilies of pseudoanosovs which have asymptotically small dilatation factors.

In the context of Z n -actions on compact abelian groups (Proposition 17.2 and Theorem 18.1 in Schmidt [START_REF] Schmidt | Dynamical Systems of Algebraic Origin[END_REF]) the topological entropy is equal to the logarithm of the Mahler measure. If we assume that the stretch factors are Mahler measures M(α) of algebraic numbers α (which are Perron numbers by Adler and Marcus [START_REF] Adler | Topological Entropy and Equivalence of Dynamical Systems[END_REF]), then we arrive at a contradiction since Penner [START_REF] Penner | Bounds on Least Dilatations[END_REF] showed that L(0, g) 1 g for surfaces of genus g. Indeed, it suffices to increase the genus g to find pseudo-Anosov elements of Mod(S g ) with dilatation factors arbitrarily close to 1, while Theorem ?? states that a discontinuity should exist. As a consequence of [START_REF] Penner | Bounds on Least Dilatations[END_REF], [START_REF] Tsai | The Asymptotic Behavior of Least Pseudo-Anosov Dilatations[END_REF] and of Theorem ?? (ex-Lehmer Conjecture) we deduce the following claims:

(1) Assertion 1: The stretch factors of the pseudo-Anosov elements of Mod(S g ) are Perron numbers which are not Mahler measures of algebraic numbers as soon as g is large enough.

(2) Assertion 2: The stretch factors of the pseudo-Anosov elements of Mod(S g,n ), where S g,n is an orientable surface with fixed genus g and n marked points, are Perron numbers which are not Mahler measures of algebraic numbers as soon as n is large enough.

Let S be a connected finite type oriented surface. Leininger [START_REF] Leininger | On Groups Generated by Two Positive Multi-Twists: Teichmüller Curves and Lehmer's Number[END_REF] considers subgroups of Mod(S) generated by two positive multi-twists; a multi-twist is a product of positive Dehn twists about disjoint essential simple closed curves. Given A and B two isotopy classes of essential 1-manifolds on S, we denote by T A , resp. T B , the positive multi-twist which is the product of positive Dehn twists about the components of A, resp. of B. Theorem 6.10 (Leininger). Any pseudo-Anosov element f ∈ T A , T B has a stretch factor which satisfies:

λ(f ) ≥ λ L (Lehmer's number).
(6.17)

The realization occurs when S has genus 5 (with at most one marked point), A = A Lehmer , B = B Lehmer given by Figure 1 in [START_REF] Leininger | On Groups Generated by Two Positive Multi-Twists: Teichmüller Curves and Lehmer's Number[END_REF] up to homeomorphism, and f conjugate to (T A T B ) ±1 . Leininger's Theorem 6.10 is strikingly reminiscent of McMullen's Theorem 6.8. The following questions are formulated in §9.1 in [339]:

• Q1: Which Salem numbers occur as dilatation factors of pseudo-Anosov automorphisms?

• Q2: Is there some topological condition on a pseudo-Anosov which guarantees that its dilatation factor is a Salem number? and denote λ (k,l) > 1, resp. λ (x,y,z) > 1, its dominant root. Related to the minimization problem is the one for orientable pseudo-Anosovs. The minimal dilatation factor for orientable pseudo-Anosov elements of Mod(S g ) is denoted by δ + g . The minimal dilatation problem for δ + g is open in general. For g = 2, Zhirov [START_REF] Zhirov | On the Minimum Dilatation of Pseudo-Anosov Diffeomorphisms of a Double Torus[END_REF] obtains δ + 2 = δ 2 . For g = 1, δ + 1 = δ 1 holds. From [START_REF] Hironaka | Small Dilatation Mapping Classes Coming From the Simplest Hyperbolic Braid[END_REF], [START_REF] Lanneau | On the Minimum Dilatation of Pseudo-Anosov Homeomorphisms on Surfaces of Small Genus[END_REF], δ g < δ + g for g = 4, 6, 8. Hironaka [START_REF] Hironaka | Small Dilatation Mapping Classes Coming From the Simplest Hyperbolic Braid[END_REF] obtains: (i) δ g ≤ λ (g+1,3) if g ≡ 0, 1, 3, 4 (mod 6) and g ≥ 3, (ii) δ g ≤ λ (g+1,1) if g ≡ 2, 5 (mod 6) and g ≥ 5, (iii) lim sup g→∞ gLog δ g ≤ Log 3+ √ 5 2

. Kin and Takasawa [START_REF] Kin | Pseudo-Anosovs on Closed Surfaces Having Small Entropy and the Whitehead Sister Link Exterior[END_REF] complement and improve these inequalities. They show:

(i) δ g ≤ λ (g+2,1) if g ≡ 0, 1, 5, 6 (mod 10) and g ≥ 5, (ii) δ g ≤ λ (g+2,2) if g ≡ 7, 9 (mod 10) and g ≥ 7;

for g ≡ 2, 4 (mod 10), under the assumption g + 2 ≡ 0 (mod 4641), then they prove:

(i) δ g ≤ λ (g+2,3) if gcd(g + 2, 3) = 1, (ii) δ g ≤ λ (g+2,7) if 3|(g + 2)
and gcd(g + 2, 7) = 1, (iii) δ g ≤ λ (g+2,13) if 21|(g + 2) and gcd(g + 2, 13) = 1, (iv) δ g ≤ λ (g+2,17) if 273|(g + 2) and gcd(g + 2, 17) = 1.

For g = 8 and 13 they obtain the sharper upper bounds: δ 8 ≤ λ (18,17,7) < λ (9,1) and δ 13 ≤ λ [START_REF] Amoroso | Small Points on Rational Subvarieties of Tori, Commentarii Math[END_REF][START_REF] Amoroso | On Fields with the Property (B)[END_REF][START_REF] Amoroso | Sur des Polynômes de Petites Mesures de Mahler[END_REF] < λ [START_REF] Amoroso | Le Théorème de Dobrowolski en Dimension Supérieure[END_REF][START_REF] Agoh | On the Relative Class Number of Special Cyclotomic Fields[END_REF] . For orientable pseudo-Anosovs, Lannneau and Thiffeault [START_REF] Lanneau | On the Minimum Dilatation of Pseudo-Anosov Homeomorphisms on Surfaces of Small Genus[END_REF] obtain δ + 3 = λ (3,1) , δ + 4 = λ (4,1) , and the following lower bounds for g = 6 to 8: (i) δ + 6 ≥ λ (6,1) , (ii) δ + 7 ≥ λ (9,2) and δ + 8 ≥ λ [START_REF] Amoroso | Sur des Polynômes de Petites Mesures de Mahler[END_REF][START_REF] Aaber | Closed Surface Bundles of Least Volume[END_REF] . Hironaka [START_REF] Hironaka | Small Dilatation Mapping Classes Coming From the Simplest Hyperbolic Braid[END_REF] gives the upper bounds: and, from [START_REF] Lanneau | On the Minimum Dilatation of Pseudo-Anosov Homeomorphisms on Surfaces of Small Genus[END_REF], the equality: δ + 8 = λ (8,1) . Kin and Takasawa [START_REF] Kin | Pseudo-Anosovs on Closed Surfaces Having Small Entropy and the Whitehead Sister Link Exterior[END_REF] give the better upper bounds:

(i) δ + g ≤ λ (g+1,3) if g ≡ 1,
(i) δ + g ≤ λ (g+2,2
) if g ≡ 7, 9 (mod 10) and g ≥ 7, (ii) δ + g ≤ λ (g+2,4) if g ≡ 1, 5 (mod 10) and g ≥ 5.

Moreover they prove: δ + 7 = λ (9,2) (Aaber and Dunfeld [START_REF] Aaber | Closed Surface Bundles of Least Volume[END_REF] obtain it independently) and δ 5 < δ + 5 . The realization of the minimal dilatations is a basic question, with the uniqueness problem, considered by many authors: associated with least volume [START_REF] Aaber | Closed Surface Bundles of Least Volume[END_REF], [START_REF] Kin | Entropy Versus Volume for Pseudo-Anosovs[END_REF], braids [START_REF] Cho | The Minimal Dilatation of a Genus-2 Surface[END_REF], [START_REF] Ham | The Minimum Dilatation of Pseudo-Anosov 5-Braids[END_REF], [START_REF] Hironaka | A Family of Pseudo-Anosov Braids With Small Dilatation[END_REF], [START_REF] Kin | An Asymptotic Behavior of the Dilatation for a Family of Pseudo-Anosov Braids[END_REF], [START_REF] Kin | Pseudo-Anosov Braids with Small Entropy and the Magic 3-Manifold[END_REF], [START_REF] Lanneau | On the Minimum Dilatation of Braids on Punctured Discs[END_REF], monodromies [START_REF] Farb | The Lower Central Series and Pseudo-Anosov Dilatations[END_REF], [START_REF] Kin | Entropy Versus Volume for Pseudo-Anosovs[END_REF], [START_REF] Kin | Pseudo-Anosovs on Closed Surfaces Having Small Entropy and the Whitehead Sister Link Exterior[END_REF], Coxeter graphs and Coxeter elements [START_REF] Leininger | On Groups Generated by Two Positive Multi-Twists: Teichmüller Curves and Lehmer's Number[END_REF], [START_REF] Hironaka | Small Dilatation Mapping Classes Coming From the Simplest Hyperbolic Braid[END_REF], [START_REF] Shin | Algebraic degrees of Stretch Factors in Mapping Class Groups[END_REF], quotient families of mapping classes [START_REF] Hironaka | Quotient Families of Mapping Classes[END_REF], self-intersecting curves [START_REF] Dowdall | Dilatation Versus Self-Intersection Number for Point-Pushing Pseudo-Anosov Homeomorphisms[END_REF], homology of mapping tori [START_REF] Agol | Pseudo-Anosov Stretch Factors and Homology of Mapping Tori[END_REF]. There exists several constructions of small dilatation families, e.g. by Hironaka [START_REF] Hironaka | Small Dilatation Pseudo-Anosov Mapping Classes[END_REF], [START_REF] Hironaka | Mapping Classes Associated to Mixed-Sign Coxeter Graphs[END_REF], McMullen [START_REF] Mcmullen | Polynomial Invariants for Fibered 3-Manifolds and Teichmuller Geodesics for Foliations[END_REF], Dehornoy [START_REF] Dehornoy | Small Dilatation Homeomorphisms as Monodromies of Lorenz Knots[END_REF], with Lorenz knots.

6.4. Knots, links, Alexander polynomials, homology growth, Jones polynomials, lenticularity of zeroes, lacunarity. Constructions of Alexander polynomials of knots and links are given in [START_REF] Kawauchi | A Survey on Knot Theory[END_REF], [START_REF] Murasugi | Classical Knot Invariants and Elementary Number Theory[END_REF], [START_REF] Rolfsen | Knots and Links[END_REF], [START_REF] Seifert | Über das Geschlecht von Knoten[END_REF], [START_REF] Turaev | Introduction to Combinatorial Torsions[END_REF]. Silver and Williams in [START_REF] Silver | Mahler Measure of Alexander Polynomials[END_REF] (reported in [START_REF] Smyth | Seventy Years of Salem Numbers[END_REF] § 4.2 for an overview) investigate the Mahler measures of various Alexander polynomials of oriented links with d components in a homology 3-sphere; they obtain theorems on limits of Mahler measures and Mahler measures of derivatives of d-variate Mahler measures by performing 1/q surgery (q ∈ N) on the dth component, allowing q → ∞. In particular they consider the topological realizability of the small Mahler measures and limit Mahler measures on various examples.

For Pretzel links Hironaka ([263], [START_REF] Hironaka | Lehmer's Problem, McKay's Correspondence, and 2, 3, 7, Topics in Algebraic and Noncommutative Geometry[END_REF], [START_REF] Hironaka | What is ... Lehmer's number?[END_REF], [START_REF] Ghate | The Arithmetic and Geometry of Salem numbers[END_REF] p. 308) solves the minimization problem for the subclass of Salem numbers defined in Theorem 6.1 by Theorem 6.11 (Hironaka [263]). Let p 1 , . . . , p d positive integers. Then the Alexander polynomial of the (p 1 , . . . , p d , -1, . . . , -1)-pretzel link (Coxeter link), where the number of -1's is d -2, with respect to a suitable orientation of its components, is ∆ p1,...,p d (-x). (

Lehmer's polynomial of the variable "-x" is the Alexander polynomial of the (-2, 3, 7)-pretzel knot and the (-2, 3, 7)-pretzel knot is equivalent to the (2, 3, 7, -1)-pretzel knot. Theorem 6.2 follows from Theorem 6.11. The Mahler measure of the (2, 3, 7, -1)-pretzel knot is the minimum of the set of Mahler measures of Alexander polynomials of (suitably oriented) (p 1 , . . . , p d , -1, . . . , -1)pretzel links, over all d in 2N + 1.

It is natural to find counterparts of Lehmer's problem in Topology where several polynomial invariants [START_REF] Franks | Braids and the Jones Polynomial[END_REF], [START_REF] Freyd | A New Polynomial Invariant of Knots and Links[END_REF], [START_REF] Jones | A Polynomial Invariant for Knots via von Neumann Algebras[END_REF], [START_REF] Smyth | The Mahler Measure of Algebraic Numbers: A Survey[END_REF] § 14.6, were associated to knots, links and braids, for which the notions of convergence and "limit" can be defined (as in [START_REF] Champanerkar | On the Mahler Measure of Jones Polynomials Under Twisting[END_REF], [START_REF] Dehornoy | On the Zeroes of the Alexander Polynomial of a Lorenz Knot[END_REF], [START_REF] Hironaka | Salem-Boyd sequences and Hopf plumbing[END_REF], [START_REF] Silver | Mahler Measure, Links and Homology Growth[END_REF]) in addition to Alexander polynomials. Indeed a Theorem of Seifert [START_REF] Seifert | Über das Geschlecht von Knoten[END_REF] asserts that (i) any monic reciprocal integer polynomial P (x), (ii) which satisfies |P (1)| = 1, is the Alexander polynomial of (at least) one knot, and conversely; Burde [START_REF] Burde | Alexanderpolynome Neuwirthscher Knoten[END_REF] extended it to fibered knots (cf Hironaka [START_REF] Hironaka | Salem-Boyd sequences and Hopf plumbing[END_REF]). A Theorem of Kanenobu [START_REF] Kanenobu | Module d'Alexander des Noeuds Fibrés et Polynôme de Hosokawa des Lacements Fibrés[END_REF] asserts that any reciprocal monic integer polynomial P (x) is, up to multiples of x -1, the Alexander polynomial of a fibered link. Let us recall that infinitely many knots may possess the same polynomial invariants (Morton [386], Kanenobu [START_REF] Kanenobu | Infinitely Many Knots With the Same Polynomial Invariant[END_REF]).

Periodic homology and exponential growth. The r-fold cyclic covering X r (K) of a knot K ⊂ S 3 admits topological invariants, i.e. homology groups H 1 (X r (K), Z), which are also invariants of the knot K. To K is associated a sequence of Alexander polynomials (∆ i ), i ≥ 1, in a single variable, such that ∆ i+1 |∆ i . Likewise, to an oriented link with d components is associated a sequence of Alexander polynomials in d variables. In both cases, the first Alexander polynomial of the sequence is usually called the Alexander polynomial of the knot K, resp. of the link. For a knot K Gordon [START_REF] Gordon | Knots Whose Branched Cyclic Coverings Have Periodic Homology[END_REF] proved that the first Alexander invariant λ 1 (t) = ∆ 1 (t)/∆ 2 (t) satisfies the following equivalence :

λ 1 (t)|(t n -1) ⇐⇒ H 1 (X r (K), Z) ∼ = H 1 (X r+n (K), Z) for all r. (6.22)
The equivalence (6.22) is an analogue of Kronecker's Theorem. Gordon used the Pierce numbers of the Alexander polynomial of K, for which a linear recurrence is expected as in [START_REF] Lehmer | Factorization of Certain Cyclotomic Functions[END_REF], [START_REF] Einsiedler | Primes in Sequences Associated to Polynomials (after Lehmer)[END_REF]. Gordon obtained periods which are not prime powers and how to find all of them for knots of a given genus. Theorem 6.12 (Gordon). There exists a knot K of genus g for which H 1 (X r (K), Z) has proper period n if and only if n = 1, or n = lcm{m i | i = 1, 2, . . . , r}, where the m i 's are all distinct, each has at least two distinct prime factors, and r i=1 Φ(m i ) ≤ 2g. Departing from "Kronecker's Theorem" Gordon conjectured that when some zero of ∆ 1 (t) is not a root of unity, then the order of H 1 (X r (K), Z) grows exponentially with r. This conjecture was proved by Riley [START_REF] Riley | Growth of Order of Homology of Cyclic Branched Covers of Knots[END_REF], with p-adic methods, and González-Acuña and Short [START_REF] González-Acuña | Cyclic Branched Coverings of Knot Complements and Homology Spheres[END_REF]. Both used the Gel'fond-Baker theory of linear forms in the logarithms of algebraic numbers. Silver and Williams [START_REF] Silver | Mahler Measure, Links and Homology Growth[END_REF] extended the conjecture of Gordon and its proof for knots, where the "finite order of H 1 (X r (K), Z)" is replaced by the "order of the torsion subgroup of H 1 (X r (K), Z)", and for links in S 3 . They identified the torsion subgroups with the connected components of periodic points in a dynamical system of algebraic origin [START_REF] Schmidt | Dynamical Systems of Algebraic Origin[END_REF], connected the limit with the logarithmic Mahler measure (for any finite-index subgroup Λ ⊂ Z d , the number of such connected components is denoted by b Λ and Λ := {|v| | v ∈ Λ \ {0}} is the norm of the smallest nonzero vector of Λ; cf [START_REF] Silver | Mahler Measure, Links and Homology Growth[END_REF] for the definitions): Theorem 6.13 (Silver-Williams [START_REF] Silver | Mahler Measure, Links and Homology Growth[END_REF]). Let l = l 1 ∪ . . . ∪ l d be an oriented link of d components having nonzero Alexander polynomial ∆, in d variables. Then

lim sup Λ →∞ 1 |Z d /Λ| Log b Λ = Log M(∆) (6.23)
where "lim sup" is replaced by "lim" if d = 1.

Let M be a finitely generated module over Z[Z n ] and M its (compact) Pontryagin dual. For any subgroup Λ ⊂ Z n of finite index, let b Λ be the number of connected components of the set of elements of M fixed by actions of the elements of Λ. Le ([334], Theorem 1) proved a conjecture of K. Schmidt [START_REF] Schmidt | On Periodic Expansions of Pisot Numbers and Salem Numbers[END_REF] on the growth of the number b Λ ; as a consequence Le generalized ([334], Theorem 2) Silver Williams's Theorem 6.13 on the growth of the homology torsion of finite abelian covering of link complements, with the logarithmic Mahler measure of the first nonzero Alexander polynomial of the link. In each case, since the growth is expressed by the logarithmic Mahler measure of the (first nonzero) Alexander polynomial, Lehmer's problem amounts to establishing a universal minorant > 0 of the exponential base. For non-split links in S 3 , that is in the nonabelian covering case, Le [START_REF] Le | Homology Torsion Growth and Mahler Measure[END_REF] generalized Theorem 6.13 using the L 2 -torsion, i.e. the hyperbolic volume in the rhs part of (6.23) instead of the logarithmic Mahler measure of the 0th Alexander polynomial; in such a case the minimality of Mahler measures would find its origin in the minimality of hyperbolic volumes [START_REF] Le | Hyperbolic Volume and Homology Torsion[END_REF].

The growth of the homology torsion depends upon the (nonzero) logarithmic Mahler measure of the Alexander polynomial(s) of a knot or a link. Hence the geometry of zeroes of Alexander polynomials is important for the minoration of the homology growth [START_REF] Ghate | The Arithmetic and Geometry of Salem numbers[END_REF]. At this step, let us briefly mention the importance of other studies on the roots of Alexander polynomials: (i) monodromies and dynamics of surface homeomorphisms [START_REF] Hironaka | On Coxeter Mapping Classes and Fibered Alternating Links[END_REF], [START_REF] Rolfsen | Knots and Links[END_REF], (ii) knot groups: factorization and divisibility [START_REF] Murasugi | Classical Knot Invariants and Elementary Number Theory[END_REF], (iii) knot groups: orderability (Perron Rolfsen), (iv) statistical models (Lin Wang).

Applying solenoidal dynamical systems theory to knot theory enabled Noguchi [START_REF] Noguchi | Zeros of the Alexander Polynomial of Knot[END_REF] to prove that the dominant coefficient a n of the Alexander polynomial ∆ K (t) = n i=0 a i t i , a 0 a n = 0, of a knot K, α i being the zeroes (counted with multiplicities) of ∆ K (t), satisfies (| • | p is the p-adic norm normalized by |p| p = 1/p on Q p ):

Log |a n | = p<∞ |αi|p>1 Log |α i | p (6.24)
He proved that the distribution of zeroes measures a "distance" of the Alexander module from being finitely generated as a Z-module, and that the growth of order of the first homology of the r-fold cyclic covering X r (K) branched over K is related to the zeroes by

lim r→∞,|H1(•)| =0 Log |H 1 (X r (K); Z)| r = p≤∞ |αi|p>1 Log |α i | p . (6.25)
Therefore the leading coefficient of ∆ K (t) is closely related to the homology growth and the p-adic norms of the zeroes α i . A link, or a knot, is said to be alternating if it admits a diagram where (along every component) the strands are passed under-over. In 2002 Hoste (Hirasawa and Murasugi [START_REF] Hirasawa | Various Stabilities of the Alexander Polynomials of Knots and Links[END_REF] ) stated the following conjecture: let K be an alternating knot and ∆ K (t) its Alexander polynomial. If α is a zero of ∆ K (t), then e(α) > -1.

Hoste's Conjecture is proved in some cases: cf [START_REF] Hironaka | On Coxeter Mapping Classes and Fibered Alternating Links[END_REF], [START_REF] Lyubich | On Zeros of the Alexander Polynomial of an Alternating Knot[END_REF], [START_REF] Stoimenow | Log-Concavity and Zeros of the Alexander Polynomial[END_REF], [START_REF] Stoimenow | Hoste's Conjecture and Roots of Link Polynomials[END_REF]. The problem of the geometry and the boundedness of zeroes of the (knot and link) Alexander polynomials is difficult and related to two other conjectures on the coefficients of these polynomials, namely the Fox's trapezoidal Conjecture and the Log-concavity Conjecture [START_REF] Koseleff | On Alexander-Conway Polynomials of Two-Bridges Links[END_REF], [START_REF] Stoimenow | Log-Concavity and Zeros of the Alexander Polynomial[END_REF], [START_REF] Stoimenow | Hoste's Conjecture and Roots of Link Polynomials[END_REF].

In his studies of Lorenz knots [START_REF] Dehornoy | Les Noeuds de Lorenz[END_REF], [START_REF] Dehornoy | On the Zeroes of the Alexander Polynomial of a Lorenz Knot[END_REF], [START_REF] Dehornoy | Small Dilatation Homeomorphisms as Monodromies of Lorenz Knots[END_REF], Dehornoy obtained the following much more precise statement on the geometry of the zero locus (g is the smallest genus of a surface spanning the knot; the braid index b is the smallest number of strands of a braid whose closure is the knot): Theorem 6.14 (Dehornoy [145]). Let K be a Lorenz knot. Let g denote its genus and b its braid index. Then the zeroes of the Alexander polynomial of K lie in the annulus {z ∈ C | (2g) -4/(b-1) ≤ |z| ≤ (2g) 4/(b-1) }. (6.26)

The Alexander polynomial of a Lorenz knot reflects an intermediate step between signatures and genus [START_REF] Dehornoy | On the Zeroes of the Alexander Polynomial of a Lorenz Knot[END_REF]. A certain proportion of zeroes lie on the unit circle and are controlled by the ω-signatures (Gambaudo and Ghys, cited in [START_REF] Dehornoy | On the Zeroes of the Alexander Polynomial of a Lorenz Knot[END_REF]). The other zeroes lie within a certain distance from the unit circle and are controlled by the house of the Alexander polynomial, which is the modulus of the largest zero. The problem of the minimality of the house of this Alexander polynomial is reminiscent of the Schinzel-Zassenhaus Conjecture if it were expressed as a function of its degree. For Lorenz knots this house is interpreted as follows: it is the growth rate of the associated homological monodromy (for details, cf [START_REF] Dehornoy | On the Zeroes of the Alexander Polynomial of a Lorenz Knot[END_REF] § 2). Figure 3.3 in [START_REF] Dehornoy | On the Zeroes of the Alexander Polynomial of a Lorenz Knot[END_REF] shows two examples of Lorenz knots, with respective braid index and genus (b, g) = (40, 100) and = (100, 625); interestingly, the distribution of zeroes within the annulus (6.26) appears angularly fairly regular (in the sense of Bilu's Theorem [START_REF] Bilu | Limit Distribution of Small Points on Algebraic Tori[END_REF]) but exhibit lenticuli of zeroes in the angular sector arg(z) ∈ [π -π/2, π + π/2]. Such lenticuli do exist for integer polynomials of small Mahler measure, of the variable "-x", and are shown to be at the origin of the minoration of the Mahler measure in the problem of Lehmer in the present note. Though Dehornoy did not publish (yet) further on the Mahler measures of the Alexander polynomials of Lorenz knots, in particular in the way (b, g) tends to infinity, it is very probable that such polynomials are good candidates for giving small Mahler measures together with a topological interpretation of the houses.

The above examples suggest that the Alexander polynomials of Lorenz knots are not Salem polynomials, though no proof seems to exist.

Before Le [START_REF] Le | Homology Torsion Growth and Mahler Measure[END_REF], [START_REF] Le | Hyperbolic Volume and Homology Torsion[END_REF], Boyd and Rodriguez-Villegas [START_REF] Boyd | Mahler's Measure and Invariants of Hyperbolic Manifolds, in Number Theory for the Millenium[END_REF] [90] [START_REF] Boyd | Mahler's Measure and the Dilogarithm (I)[END_REF] studied the connections between the Mahler measure of the A-polynomial of a knot and the hyperbolic volume of its complement. A-polynomials were introduced in hyperbolic geometry by Cooper et al [START_REF] Cooper | Plane Curves Associated to Character Varieties of 3-Manifolds[END_REF] (are not Alexander polynomials, though "A" is used in homage to Alexander). The irreducible factors of A-polynomials have (logarithmic) Mahler measures which are shown to be finite sums of Bloch-Wigner dilogarithms [START_REF] Gangl | Classical and Elliptic Polylogarithms and Special Values of L-series, The Arithmetic and Geometry of Algebraic Cycles[END_REF], [START_REF] Zagier | The Dilogarithm Function[END_REF], of algebraic numbers. The values of such dilogarithms are related to Chinburg's Conjecture. Several examples are taken by the authors to investigate Chinburg's Conjecture and its generalization refered to as Boyd's question (cf also Ray [START_REF] Ray | Relations between Mahler's Measure and Values of L-Series[END_REF]). Chinburg's Conjecture [START_REF] Boyd | Mahler's Measure, Hyperbolic Geometry and the Dilogarithm[END_REF] is stated as follows: for each negative discriminant -f there exists a polynomial P = P f ∈ Z[x, y] and a nonzero rational number r f such that Log M

(P ) = r f f √ f 4π L(2, χ f ). (6.27)
Boyd's question is stated as follows: for every number field F having a number of complex embeddings equal to 1 (i.e. r 2 = 1), does there exist a polynomial P = P F ∈ Z[x, y] and a rational number r F such that Log M(P ) = r F Z F ?, where ζ F is the Dedeking zeta function of F and

Z F = 3 |disc(F )| 3/2 ζ F (2) 2 2n-3 π 2n-1 ; (6.28)
the starting point being (Smyth [START_REF] Smyth | On Measures of Polynomials in Several Variables[END_REF]): for f = 3, Log M(1 + x + y) = 3 √ 3 4π L(2, χ 3 ). Jones polynomials of knots and links, lacunarity in coefficient vectors. Let L be a hyperbolic link and, for m ≥ 1, denote by L m the link obtained from L by adding m full twists on n strands [START_REF] Rolfsen | Knots and Links[END_REF], [START_REF] Champanerkar | On the Mahler Measure of Jones Polynomials Under Twisting[END_REF]. By Thurston's hyperbolic Dehn surgery, the volume Vol(S 3 \ L m ) converges to Vol(S 3 \ (L ∪ U )), as m tends to infinity, where U is an unknot encircling n strands of L such that L m is obtained from L by a -1/m surgery on U . More generally, let m = (1, m 1 , . . . , m s ), for s ≥ 1, and L m := L m1,...,ms the multi-twisted link obtained from a link diagram L by a -1/m i surgery on an unknot U i , for i = 1, . . . , s. In the following theorem convergence of Mahler measures has to be taken in the sense of the Boyd Lawton's Theorem 2.5. Theorem 6.15 (Champanerkar -Kofman). (i) The Mahler measure M(V Lm (t)) of the Jones polynomial of L m converges to the Mahler measure of a 2-variable polynomial, as m tends to infinity;

(ii) the Mahler measure M(V Lm (t)) of the Jones polynomial of L m converges to the Mahler measure of a (s + 1)-variable polynomial, as m tends to infinity.

In [START_REF] Champanerkar | On the Mahler Measure of Jones Polynomials Under Twisting[END_REF] Theorem 2.4, Champanerkar and Kofman [START_REF] Champanerkar | On the Mahler Measure of Jones Polynomials Under Twisting[END_REF] extended Theorem 6.15 to the convergence of the Mahler measures of colored Jones polynomials J N (L m , t) and J N (L m , t) for fixed N , as m, resp. m, tends to infinity; here coloring means by the N -dimensional irreducible representation of SL 2 (C) with the normalization of J 2 (L m , t) as J 2 (L m , t) = (t 1/2 + t -1/2 )V Lm (t), resp. for m. They proved that the limit lim m→∞ M(J N (L m , t)), resp. for m, is the Mahler measure of a multivariate polynomial. What smallness of limit Mahler measures can be reached by this construction, and what are the corresponding geometrical realizations?

In [START_REF] Champanerkar | On the Mahler Measure of Jones Polynomials Under Twisting[END_REF] (Theorem 2.5 and Corollary 3.2) Champanerkar and Kofman obtain the following theorem. Theorem 6.16 (Champanerkar -Kofman). Let N ≥ 1 be a fixed integer. With the above notations, (i) let {γ i,m } be the set of distinct roots of the Jones polynomial J N (L m , t). Then lim inf κ→∞ #{γ i,m | m ≤ κ} = ∞, and for any > 0, there exists an integer q such that the number of such roots satisfies

#{γ i,m | ||γ i,m | -1| ≥ } < q , (6.29)
(ii) for m sufficiently large, the coefficient vector of the Jones polynomial J N (L m , t) has nonzero fixed blocks of integer digits separated by gaps (blocks of zeroes) whose length increases as m tends to infinity.

In addition to the relative limitation of the multiplicities of the roots, Theorem 6.16 means that, in the annulus 1 -< |z| -1 < 1 + , the clustering of the roots occurs, up to q of them (densification), and is associated with a moderate lacunarity ("gappiness" in the sense of [START_REF] Verger-Gaugry | On Gaps in Rényi β-Expansions of Unity for β > 1 an Algebraic Number[END_REF]) of the Jones polynomials which increases with m. This Theorem has been extended to other Jones polynomials by these authors [START_REF] Champanerkar | On the Mahler Measure of Jones Polynomials Under Twisting[END_REF] and followed previous experimental observations. From Theorem 6.15 and Theorem 6.16 it is likely that such Jones polynomials lead to very small multivariate Mahler measures, at least are good candidates.

Other families of Jones polynomials, their zeroes and their limit distributions, were investigated, for which interesting limit Mahler measures may be expected: e.g. Chang and Shrock [START_REF] Chang | Zeros of Jones Polynomials for Families of Knots and Links[END_REF], Wu and Wang [START_REF] Wu | Zeroes of the Jones Polynomial[END_REF], Jin and Zhang [START_REF] Jin | Zeros of the Jones Polynomials for Families of Pretzel Links[END_REF], [START_REF] Jin | Jones Polynomials and Their Zeros for a Family of Links[END_REF], [START_REF] Jin | Zeros of the Jones Polynomial for Multiple Crossing-Twisted Links[END_REF], related to models in statistical physics. The moderate lacunarities occurring in the coefficient vectors of Jones polynomials were studied by Franks and Williams [START_REF] Franks | Braids and the Jones Polynomial[END_REF] in the context of polynomial invariants associated with braids, knot and links which generalize Alexander polynomials and Jones polynomials [START_REF] Franks | Braids and the Jones Polynomial[END_REF], [START_REF] Freyd | A New Polynomial Invariant of Knots and Links[END_REF], [START_REF] Jones | A Polynomial Invariant for Knots via von Neumann Algebras[END_REF], [START_REF] Murasugi | Classical Knot Invariants and Elementary Number Theory[END_REF].

6.5. Arithmetic Hyperbolic Geometry. Leininger's constructions in [START_REF] Leininger | On Groups Generated by Two Positive Multi-Twists: Teichmüller Curves and Lehmer's Number[END_REF] give the dilatation factors of pseudo-Anosovs as spectral radii of hyperbolic elements in some Fuchsian groups. The minimality of the Salem numbers as dilatation factors is defined in a more general context (Neuman and Reid [START_REF] Neumann | Arithmetic of Hyperbolic Manifolds[END_REF], Maclachlan and Reid [START_REF] Mclachlan | The Arithmetic of Hyperbolic 3-manifolds[END_REF], Ghate and Hironaka [START_REF] Ghate | The Arithmetic and Geometry of Salem numbers[END_REF] p. 303). Theorem 6.17 (Neuman -Reid). The Salem numbers are precisely the spectral radii of hyperbolic elements of arithmetic Fuchsian groups derived from quaternion algebras.

Arithmetic hyperbolic groups are arithmetic groups of isometries of hyperbolic n-space H n . Vinberg and Shvartsman [START_REF] Vinberg | Discrete Groups of Motions of Spaces of Constant Curvature[END_REF] p. 217 have defined the large subclass of the arithmetic hyperbolic groups of the simplest type, in terms of an admissible quadratic form over a totally real number field K. This subclass includes all arithmetic hyperbolic groups in even dimensions, infinitely many widecommensurability classes of hyperbolic groups in all dimensions [START_REF] Mclachlan | Commensurability Classes of Discrete Arithmetic Hyperbolic Groups[END_REF], and all noncocompact arithmetic hyperbolic groups in all dimensions. Isometries of H n are either elliptic, parabolic or hyperbolic. An isometry γ ∈ H n is hyperbolic if and only if there is a unique geodesic L in H n , called the axis of γ, along which γ acts as a translation by a positive distance l(γ) called the translation length of γ.

The following theorems generalize previous results of Neumann and Reid [START_REF] Neumann | Arithmetic of Hyperbolic Manifolds[END_REF] in dimension 2 and 3 and show the important role played by the smallest Salem numbers: Theorem 6.18 (Emery-Ratcliffe -Tschantz [START_REF] Emery | Salem Numbers and Arithmetic Hyperbolic Geometry[END_REF]). Let Γ be an arithmetic group of isometries of H n , n ≥ 2, of the simplest type defined over a totally real algebraic number K. Let Γ (2) be the subgroup of Γ of finite index generated by the squares of the elements of Γ. Let γ be a hyperbolic element of Γ, and let λ = e l(γ) . If n is even or γ ∈ Γ (2) , then λ is a Salem number such that K ⊂ Q(λ + λ -1 ) and deg

K (λ) ≤ n + 1. Conversely, if λ ∈ T , K is a subfield of Q(λ + λ -1
) and n such that deg K (λ) ≤ n + 1, then there exists an arithmetic group Γ of isometries of H n of the simplest type defined over K and a hyperbolic element γ ∈ Γ such that λ = e l(γ) . Theorem 6.19 (Emery -Ratcliffe -Tschantz [START_REF] Emery | Salem Numbers and Arithmetic Hyperbolic Geometry[END_REF]). Let Γ be an arithmetic group of isometries of H n , n ≥ 2 odd, of the simplest type defined over a totally real algebraic number K. Let Γ (2) be the subgroup of Γ of finite index generated by the squares of the elements of Γ. Let γ be a hyperbolic element of Γ, and let λ = e l(γ) . Then λ is a Salem number which is square-rootable over K.

Conversely, if λ ∈ T , K is a subfield of Q(λ + λ -1 ) and n an odd positive integer such that deg K (λ) ≤ n + 1, and λ is square-rootable over K, then there exists an arithmetic group Γ of isometries of H n of the simplest type defined over K and a hyperbolic element γ ∈ Γ such that √ λ = e l(γ) .

6.6. Salem numbers and Dynamics of Automorphisms of Complex Compact Surfaces. Let X be a compact Kähler variety and f an automorphism of X. The automorphism f induces an invertible linear map f * on H * (X, C), resp. H * (X, R), H * (X, Z), which preserves the Hodge decomposition, the intersection form, the Kähler cone. Iterating f provides a dynamical system to which real algebraic integers ≥ 1 are associated. The greatest eigenvalue of the action of f on H * (X, C) is usually called the maximal dynamical degree of f . This terminology is the same as the one used for the β-shift in the present note, but the notions are different. The maximal dynamical degree of f is denoted by λ(f ); it is related to the topological entropy h top (f ) of f by Log λ(f ) = h top (f ) by a Theorem of Gromov and Yomdin [247][534]. Saying that an automorphism is of positive entropy is equivalent to saying that its maximal dynamical degree is > 1. In particular if X is a surface the characteristic polynomial of f * on H 2 (X, Z) is a (not necessarily irreducible) Salem polynomial (McMullen [START_REF] Mcmullen | Dynamics on K3 Surfaces: Salem Numbers and Siegel Disks[END_REF]); the maximal dynamical degree λ(f ) of f is the spectral radius of f * on H 1,1 (X) and is a Salem number. Salem numbers are deeply linked to the geometry of the surface. Among all complex compact surfaces [START_REF] Barth | Compact Complex Surfaces[END_REF], Cantat [111] [112] showed that, if X is a complex compact surface for which there exists an automorphism of X having a positive entropy, then there exists a birational morphism from X to a torus, a K3 surface, a surface of Enriques, or the projective plane. Therefore it suffices to consider complex tori (Oguiso and Truong [405], Reschke [START_REF] Reschke | Salem Numbers and Automorphisms of Complex Surfaces[END_REF] 2017), Enriques surfaces (Oguiso [START_REF] Oguiso | Salem Polynomials and the Bimeromorphic Automorphism Group of a Hyper-Kähler Manifold, Selected Papers on Analysis and Differential Equations[END_REF], [START_REF] Oguiso | A Remark on Dynamical Degrees of Automorphisms of HyperKähler Manifolds[END_REF]), and K3 surfaces (Gross and McMullen [249], McMullen [START_REF] Mcmullen | Dynamics on K3 Surfaces: Salem Numbers and Siegel Disks[END_REF], Oguiso [START_REF] Oguiso | The Third Smallest Salem Number in Automorphisms of K3 Surfaces, Algebraic Geometry in East Asia-Seoul[END_REF], Shimada [START_REF] Shimada | Automorphisms of Supersingular K3 Surfaces and Salem Polynomials[END_REF]) if X is not rational. The restriction to compact Kähler surfaces is justified by the fact that the topological entropy of all automorphisms vanishes on compact complex surfaces which are not Kähler (Cantat [112]). The existence of an automorphism of positive entropy is a deep question [START_REF] Brandhorst | Dynamics of Supersingular K3 Surfaces and Automorphisms of Salem Degree 22[END_REF], [START_REF] Brandhorst | Automorphisms of Salem Degree 22 on Supersingular K3 Surfaces of Higher Artin Invariant[END_REF], [START_REF] Cantat | Automorphisms of Surfaces: Kummer Rigidity and Measure of Maximal Entropy[END_REF], [START_REF] Esnault | Automorphisms of Elliptic K3 Surfaces and Salem Numbers of Maximal Degree[END_REF], [START_REF] Mcmullen | Dynamics on K3 Surfaces: Salem Numbers and Siegel Disks[END_REF], [START_REF] Oguiso | Pisot Units, Salem Numbers and Higher Dimensional Projective Manifolds with Primitive Automorphisms of Positive Entropy[END_REF], [START_REF] Oguiso | Explicit Examples of Rational and Calabi-Yau Threefolds with Primitive Automorphisms of Positive Entropy[END_REF].

On each type of surface, what are the Salem numbers which appear? In this context the problem of Lehmer can be formulated by asking what are the minimal Salem numbers which occur, per type of surface, and the corresponding geometrical realizations.

In [START_REF] Mcmullen | From Dynamics on Surfaces to Rational Points on Curves[END_REF] McMullen gives a general construction of K3 surface automorphisms f from unramified Salem numbers, such that, for every such automorphism f , the topological entropy Log λ(f ) is positive, together with a criterion for the resulting automorphism to have a Siegel disk (domains on which f acts by an irrational rotation). The Salem polynomials involved, of the respective dynamical degrees λ(f ), have degree 22, trace -1 and are associated to an even unimodular lattice of signature [START_REF] Agoh | On the Relative Class Number of Special Cyclotomic Fields[END_REF][START_REF] Amoroso | Distribution des Points de Petite Hauteur Normalisée dans les Groupes Multiplicatifs[END_REF] on which f acts as an isometry, by the Theorem of Torelli. The surface is non-projective to carry a Siegel disk.

McMullen [START_REF] Mcmullen | Dynamics on Blowups of the Projective Plane[END_REF] (Theorem A.1) proved that Lehmer's number (denoted by λ 10 ) is the smallest Salem number that can appear as dynamical degree of an automorphism of a complex compact surface: h(f ) ≥ Log λ 10 = 0.162357 . . . . (6.30) He gave a geometrical realization of Lehmer's number in [START_REF] Mcmullen | Dynamics on Blowups of the Projective Plane[END_REF] on a rational surface (cf also Bedford and Kim [START_REF] Bedford | Periodicities in Linear Fractional Recurrences: Degree Growth of Birational Surface Maps[END_REF]), in [START_REF] Mcmullen | K3 Surfaces, Entropy and Glue[END_REF] on a nonprojective K3 surface, in [START_REF] Mcmullen | Automorphisms of Projective K3 Surfaces with Minimum Entropy[END_REF] on a projective K3 surface. On the contrary Oguiso [START_REF] Oguiso | The Third Smallest Salem Number in Automorphisms of K3 Surfaces, Algebraic Geometry in East Asia-Seoul[END_REF] proved that Lehmer's number cannot be realized on an Enriques surface. In [START_REF] Mcmullen | Automorphisms of Projective K3 Surfaces with Minimum Entropy[END_REF] McMullen proved that the value Log λ d arises as the entropy of an automorphism of a complex projective K3 surface if Brandhorst and González-Alonso [START_REF] Brandhorst | Automorphisms of Minimal Entropy on Supersingular K3 Surfaces[END_REF] completed the above "realizability" list with the value d = 12 (Theorem 1.2 in [START_REF] Mcmullen | Automorphisms of Projective K3 Surfaces with Minimum Entropy[END_REF]). For projective surfaces, the degree of the Salem number is bounded by the rank of the Néron-Severi group; for K3 surfaces in characteristic zero it is at most 20, due to Hodge theory. In positive characteristic the rank 22 is possible (case of supersingular K3 surfaces) [START_REF] Brandhorst | Automorphisms of Salem Degree 22 on Supersingular K3 Surfaces of Higher Artin Invariant[END_REF] [START_REF] Yu | Elliptic Fibrations on K3 Surfaces and Salem Numbers of Maximal Degree[END_REF]. Therefore all such Salem numbers, when less than 1.3, are listed in Mossinghoff's list in [START_REF] Mossinghoff | Known Polynomials with Small Mahler Measure Through Degree 180[END_REF], the list being complete up to degree 44.

d = 2,
Reschke [START_REF] Reschke | Salem Numbers and Automorphisms of Complex Surfaces[END_REF], [START_REF] Reschke | Salem Numbers and Automorphisms of Abelian Surfaces[END_REF], gave a necessary and sufficient condition for a Salem number to be realized as dynamical degree of an automorphism of a complex torus, with degrees 2, 4 or 6; moreover he investigated the relations between the values of the Salem numbers and the corresponding geometry and projectiveness of the tori. Zhao [START_REF] Zhao | Nombres de Salem et Automorphismes à Entropie Positive de Surfaces Abéliennes et de Surfaces K[END_REF] extended the method of Reschke for tori endowed with real structures, showing that it suffices to consider real abelian surfaces. Zhao classified such real abelian surfaces into 8 types according to the number of connected components and the simplicity of the underlying complex abelian surface. For each type the set of Salem numbers which can be realized by real automorphisms is determined. Zhao [START_REF] Zhao | Nombres de Salem et Automorphismes à Entropie Positive de Surfaces Abéliennes et de Surfaces K[END_REF] proved that Lehmer's number cannot be realized by a real K3 surface.

Dolgachev [START_REF] Dolgachev | Salem Numbers and Enriques Surfaces[END_REF] investigated automorphisms on Enriques surfaces of dynamical degrees > 1 which are small Salem numbers, of small degree 2 to 10 (Salem numbers of degree 2 are quadratic Pisot numbers). The method does not allow to conclude on the minimality of the Salem numbers. The author uses the lower semi-continuity properties of the dynamical degree of an automorphism g of an algebraic surface S when (S, g) varies in an algebraic family.

In positive characteristic Brandhorst and González-Alonso [START_REF] Brandhorst | Automorphisms of Minimal Entropy on Supersingular K3 Surfaces[END_REF] proved that the values Log λ d arise as the entropy of an automorphism of a supersingular K3 surface over a field of characteristic p = 5 if and only if d ≤ 22 is even and d = 18, giving in their Appendix B the list of Salem numbers λ d of degree d and respective minimal polynomials. They develop a strategy to characterize the minimal Salem polynomials, in particular their cyclotomic factors, for various realizations in supersingular K3 surfaces having Artin invariants σ ranging from 1 to 7, in characteristic 5. Yu [START_REF] Yu | Elliptic Fibrations on K3 Surfaces and Salem Numbers of Maximal Degree[END_REF] studied the maximal degrees of the Salem numbers arising from automorphisms of K3 surfaces, defined over an algebraically closed field of characteristic p, in terms of the elliptic fibrations having infinite automorphism groups, and Artin invariants.

Oguiso and Truong [START_REF] Oguiso | Salem Numbers in Dynamics on Kähler Threefolds and Complex Tori[END_REF] Dinh, Nguyen and Truong [START_REF] Dinh | Comparion of Dynamical Degrees for Semi-Conjugate Meromorphic Maps[END_REF], [START_REF] Dinh | On the Dynamical Degrees of Meromorphic Maps Preserving a Fibration[END_REF], investigated the structure of compact Kähler manifolds, in dimension ≥ 3, from the point of view of establishing relations between non-trivial invariant meromorphic fibrations, pseudo-automorphisms f and the dynamical degrees λ k (f ). Lehmer's problem can be formulated by asking when the first dynamical degree λ 1 (f ) is a Salem number, what minimal value for λ 1 (f ) can be reached and what are the possible geometrical realizations for the minimal ones.

Appendix -Standard notations

The notations used by several authors in the above sections can be found directly in the corresponding articles and books which are quoted in the text. We just report the standard notations in the following. obtained Let P (X) ∈ Z[X], m = deg(P ) ≥ 1. The reciprocal polynomial of P (X) is P * (X) = X m P ( 1 X ). The polynomial P is reciprocal if P * (X) = P (X). When it is monic, the polynomial P is said unramified if |P (1)| = |P (-1)| = 1. If P (X) = a 0 m j=1 (X -α j ) = a 0 X m + a 1 X m-1 + . . . + a m , with a i ∈ C, a 0 a m = 0, and roots α j , the Mahler measure of P is The absolute Mahler measure of P is M(P ) 1/ deg(P ) , denoted by M(P ). The Mahler measure of an algebraic number α is the Mahler of its minimal polynomial P α : M(α) := M(P α ). For any algebraic number α the house α of α is the maximum modulus of its conjugates, including α itself; by Jensen's formula the Weil height h(α) of α is Log M(α)/ deg(α). By its very definition, M(P Q) = M(P )M(Q) (multiplicativity). A Perron number is either 1 or a real algebraic integer θ > 1 such that the Galois conjugates θ (i) , i = 0, of θ (0) := θ satisfy: |θ (i) | < θ. Denote by P the set of Perron numbers. A Pisot number is a Perron number > 1 for which |θ (i) | < 1 for all i = 0. The smallest Pisot number is denoted by Θ = 1.3247 . . . , dominant root of X 3 -X -1.

(7.2)

A Salem number is an algebraic integer β > 1 such that its Galois conjugates β (i) satisfy: |β (i) | ≤ 1 for all i = 1, 2, . . . , m -1, with m = deg(β) ≥ 1, β (0) = β and at least one conjugate β (i) , i = 0, on the unit circle. All the Galois conjugates of a Salem number β lie on the unit circle, by pairs of complex conjugates, except 1/β which lies in the open interval (0, 1). Salem numbers are of even degree m ≥ 4.

The set of Pisot numbers, resp. Salem numbers, is denoted by S, resp. by T. If τ ∈ S or T, then M(τ ) = τ . A j-Salem number [START_REF] Kerada | Une Caractérisation de Certaines Classes d'Entiers Algébriques Généralisant les Nombres de Salem[END_REF] [454], j ≥ 1, is an algebraic integer α such that |α| > 1 and α has j -1 conjugate roots α (q) different from α, satisfying |α (q) | > 1, while the other conjugate roots ω satisfy |ω| ≤ 1 and at least one of them is on the unit circle. We call the minimal polynomial of a j-Salem number a j-Salem polynomial. Salem numbers are 1-Salem numbers. A Salem number is said unramified if its minimal polynomial is unramified. We say that two Salem numbers λ and µ are commensurable if there exists positive integers k and l such that λ k = µ l . Commensurability is an equivalence relation on T . Let λ ∈ T , K a subfield of Q(λ + λ -1 ), and P λ,K the minimal polynomial of λ over K; we say that λ is square-rootable over K if there exists a totally positive element α ∈ K and a monic reciprocal polynomial q(x), whose even degree coefficients are in K and odd degree coefficients are in √ αK such that q(x)q(-x) = P λ,K (x 2 ). A Garsia number is an algebraic integer of norm ±2 such that all of the roots of its minimal polynomial are strictly greater than 1 in absolute value [START_REF] Hare | Some Comments on Garsia Numbers[END_REF].

The set of algebraic numbers, resp. algebraic integers, in C, is denoted by Q, resp. O Q . The nth cyclotomic polynomial is denoted by Φ n (z). For any postive integer n, let [n] := 1 + x + x 2 + . . . + x n-1 . The (naïve) height of a polynomial P is the maximum of the absolute value of the coefficients of P . Let A be a countable subset of the line; the first derived set A (1) of A is the set of the limit points of nonstationary infinite sequences of elements of A; the k-th derived set A (k) of A is the first derived set of A (k-1) , k ≥ 2.

For x > 0, x , {x} and x denotes respectively the integer part, resp. the fractional part, resp. the smallest integer greater than or equal to x. For β > 1 any real number, the map T β : [0, 1] → [0, 1], x → {βx} denotes the β-transformation. With T 0 β := T β , its iterates are denoted by T n k , |z| ≤ 1, is the kth-polylogarithm function [START_REF] Gangl | Classical and Elliptic Polylogarithms and Special Values of L-series, The Arithmetic and Geometry of Algebraic Cycles[END_REF], [START_REF] Lewin | Polylogarithms and Associated Functions[END_REF], [START_REF] Zagier | The Dilogarithm Function[END_REF]. For x > 0, Log + x denotes max{0, Log x}. Let F be an infinite subset of the set of nonzero algebraic numbers which are not a root of unity; we say that the Conjecture of Lehmer is true for F if there exists a constant c F > 0 such that M(α) ≥ 1 + c F for all α ∈ F.

  is a real number for which the Rényi β-expansion of unity starts by d β (1) = 0.10 m 1 . . ., then the dynamical degree of β is by definition dyg(β) := m + 2. The respective dynamical degrees dyg(β) of the last two Salem numbers β are 7 and 12, with Parry polynomials of respective degrees 20 and 75.

Conjecture 4 . 4 .

 44 (Multiplicative Lehmer problem) For any integer n ≥ 1, there exists a real number c

(4. 23 )

 23 where D = [K ab (P ) : K ab ].

1 0

 1 Log |F (e 2πit1 , . . . , e 2πitn )|dt 1 . . . dt n(4.32)

  ) r = 0 for any α ∈ Q × and any root ζ of unity. With the usual absolute value | • | on Q, h(α r/s ) = | r s |h(α), and h is a norm on the Q-vector space G. Let Y denote the totally disconnected, locally compact, Hausdorff space of all places y of Q. Let B be the Borel σ-algebra of Y . For any number field k ⊂ Q such that k/Q is Galois and any place v of k, denote Y (k, v) := {y ∈ Y | y|v} so that Y = allplaces v of k Y (k, v) (disjoint union).(5.8)

. 13 )For p = 1

 131 Conjecture 5.2 is exactly the classical Lehmer Conjecture. Moreover, Fili and Miner ([214], Proposition 4.1) proved that, for p = ∞, Conjecture 5.2 is exactly the classical Conjecture of Schinzel-Zassenhaus.

1

 1 and a ij = 0 otherwise. Assume that Γ is a tree. Denote by W Γ the Weyl group of Γ, generated by the reflections σ 1 , σ 2 , . . . , σ n and ΦΓ := σ 1 • σ 2 • . . . • σ n ∈ W Γ the Coxeter transformation of Γ. The Coxeter polynomial of Γ is the characteristic polynomial of the Coxeter transformation Φ Γ : R n → R n : cox Γ (x) := det(x • Id n -Φ Γ ) ∈ Z[x].(6.1)Coxeter(1934) showed remarkable properties of the roots of the Coxeter polynomials. Coxeter polynomials were extensively studied for Γ any simply laced Dynkin diagram A n , D n and E n . For Γ = E n , Gross, Hironaka and McMullen[START_REF] Gross | Cyclotomic Factors of Coxeter Polynomials[END_REF] have obtained the factorization of Coxeter polynomials cox Γ (x) as products of cyclotomic polynomials and irreducible Salem polynomials. In particular, cox E10 (x) := x 10 + x 9 -x 7 -x 6 -x 5 -x 4 -x 3 + x + 1 (6.2)

Theorem 6 . 7 (

 67 McMullen). For any Coxeter system (W, S), we have inf Sn λ(s σ(1) s σ(2) . . . s σ(n) ) ≥ β(W, S). (6.10) Theorem 6.8 (McMullen). There are 38 minimal hyperbolic Coxeter systems (W, S), and among these the infimum inf β(W, S) is Lehmer's number.

  = Log (λ(f )) be the entropy of f , as logarithm of the stretch factor λ(f ) > 1 (or dilatation; the dilatation measures the dynamical complexity), (iii) L(k, g) := min{h(f ) | f : S g → S g and κ(f ) ≥ k}.(6.11)

  M(P ) := |a 0 | m j=1 max{1, |α j |}. (7.1)

  β (T j-1 β ) for j ≥ 1. A real number β > 1 is a Parry number if the sequence (T (j) β (1)) j≥1 is eventually periodic; a Parry number is called simple if in particular T

  = 0 for some integer q ≥ 1. The set of Parry numbers is denoted by P P . The terminology chosen by Parry in[START_REF] Parry | On the β-expansions of Real Numbers[END_REF] has changed: β-numbers are now called Parry numbers, in honor of W. Parry.The Mahler measure of a nonzero polynomialP (x 1 , . . . , x n ) ∈ C[x 1 , . . . , x n Log |P (x 1 , . . . , x n )| dx 1 x 1 . . . dx n x n ,whereT n = {(z 1 , . . . , z n ) ∈ C n | |z 1 | = . . . = |z n | = 1} is the unit torus in dimension n. If n = 1, by Jensen's formula, it is given by (7.1). A function f : R → R is said quasiperiodic if it is the sum of finitely many periodic continuous functions. The function, defined for k ≥ 2, Li k (z) = ∞ n=1z n

  .34) Dobrowolski, then McKee and Smyth [354] obtained minorants of M(P ) for the reciprocal polynomials P (z) = z n D A (z + 1/z) where D A is the characteristic polynomial of an integer symmetric n × n matrix A; McKee and Smyth obtained M(P ) = 1 or M(P ) ≥ 1.176280 . . . (Lehmer's number) solving the problem of Lehmer for the family of such polynomials.Dobrowolski (2008) proved that many totally real integer polynomials P cannot be represented by integer symmetric ma-

trices A, disproving a conjecture of Estes and Guralnick. Dubickas and Konyagin

[START_REF] Dubickas | On the Number of Reducible Polynomials of Bounded Naïve Height[END_REF]

[184], then Dubickas and Sha

[START_REF] Dubickas | Counting and Testing Dominant Polynomials[END_REF]

  10 5 10 5 10 5 10 5 10 7 ) ω 10 5 10 9 10 5 10 17 10 7 10 6 10 6 10 7 10 12 ) ω 6 10 6 10 10 10 16 10 12 10 7 10 12 0 16 10 10 10 6 10 8 ) ω

	dyg deg		β	P β,P		d β (1)
	5	3	θ -1 5	= 1.324717	5		0.10
	6	18		1.29567	22		0.1(0 4 10 6 ) ω
	6	10	1.293485	12		0.1(0 4 10 6 ) ω
	6	24	1.291741	24 irr.		0.1(0 4 10 11 10 6 ) ω
	6	26	1.286730	30		0.1(0 4 10 17 10 6 ) ω
	6	34	1.285409	38		0.1(0 4 10 25 10 6 ) ω
	6	30	1.285235	45		0.1(0 4 10 32 10 6 ) ω
	6	44	1.285199	66		0.1(0 4 10 54 10 6 ) ω
	6	6	θ -1 6	= 1.285199	6 irr.		0.10
	7 0.1(0 5 7 26 1.285196 44 26 1.281691 .. 0.1(0 5 7 8 1.280638 20	0.1(0 5 10 7 ) ω
	7	10	1.261230	14		0.1(0 5 10 7 ) ω
	7	24	1.260103	28		0.1(0 5 10 13 10 7 ) ω
	7	18	1.256221	36		0.1(0 5 10 21 10 7 ) ω
	7	7	θ -1 7	= 1.255422	7 irr.		0.10
	8 0.1(0 8 18 1.252775 120 12 1.240726 48	0.1(0 6 10 11 10 11 10 8 ) ω
	8	20	1.232613	41		0.1(0 6 10 24 10 8 ) ω
	8	8	θ -1 8	= 1.232054	8 irr.		0.10
	9	10	1.216391	18		0.1(0 7 10 9 ) ω
	9	9	θ -1 9	= 1.213149	9 irr.		0.10
	10	14	1.200026	20		0.1(0 8 ) ω
	10	10	θ -1 10 = 1.197491 10 irr.		0.10
	11	9	θ -1 11 = 1.184276	11		0.10
	12	10	1.176280	75	Lehmer's

number : 0.1(0 10 18 10 12 10 18 10 12 ) ω 12 12 θ -1 12 = 1.172950 12 irr. 0.10 10 1 Table 1. Smallest Salem numbers β < 1.3 of degree ≤ 44, which are Parry numbers, computed from the "list of Mossinghoff" [389]. In Column 1 is reported the dynamical degree of β. Column 4 gives the degree dP of the Parry polynomial P β,P of β; P β,P is reducible except if "irr." is mentioned.

  Log |P s (x 1 , . . . , x n )|}

	dx 1 x 1	. . .	dx n x n	,
			(5.19)
	the multiple Mahler measure by m(P 1 , . . . , P s

  .3) Cooley, McKee and Smyth[START_REF] Cooley | Non-Bipartite Graphs of Small Mahler Measure[END_REF] [START_REF] Mckee | Salem Numbers, Pisot Numbers, Mahler Measures and Graphs[END_REF] [START_REF] Mckee | Integer Symmetric Matrices of Small Spectral Radius and Small Mahler Measure[END_REF] [355] studied Lehmer's problem from various constructions of finite graphs. They prove ([START_REF] Cooley | Non-Bipartite Graphs of Small Mahler Measure[END_REF] Theorem 1 and Figures1 to 3) that every connected non-bipartite graph that has Mahler measure smaller than the golden mean 1.618 . . . is one of the following type: (i) an odd cycle, (ii) a kite graph, (iii) a balloon graph, or (iv) one of the eight sporadic examples Sp a , . . . , Sp h . 6.2. Growth series of groups, Coxeter groups, Coxeter systems. Let G be an infinite group.

•

  Q3 (Lehmer's problem for dilatation factors): Is there an > 1 such that if f is a pseudo-Anosov automorphism in a finite co-area Teichmüller disk stabilizer, then λ(f ) ≥ ? Since dilatations factors of pseudo-Anosovs are Perron numbers and not necessarily Mahler measures of algebraic numbers (cf Assertions 1 and 2 above), Leininger's Theorem 6.10 and McMullen's Theorem 6.8 are addressed to the set of Salem numbers and suggest that Lehmer's number is actually the smallest Salem number in this set; meaning first that Lehmer's Conjecture is true for Salem numbers.Letf k,l (t) := t 2k -t k+l -t k -t k-l + 1, (6.18) resp. f x,y,z (t) := t x+y-z -t x -t y -t x-z -t y-z + 1,(6.19)

  4, 6, 8, 10 or 18, but not if d = 14, 16 or d ≥ 20. (6.31)