the final twist in endocytic membrane scission Many endocytic uptake events depend on the 'pinchase' activity of dynamin. By measuring the orientation of single gold nanorods, a new study reveals that invaginated clathrin-coated endocytic pits undergo a strong rotational twist prior to or concomitant with their detachment.

Ludger Johannes 1 and cesar augusto Valades-cruz 1,2 E ndocytosis is a cellular process by which extracellular materials such as nutrients and signalling molecules are internalized via membrane-bound carriers. A large variety of mechanisms involving 1 or independent of2 the conventional clathrin coat contribute to the construction of endocytic pits from which cargo-loaded endocytic carries detach by membrane scission. For the latter, the large GTPase dynamin has attracted much attention as a 'pinchase' even when other fission mechanisms coexist 3 . Dynamin structure and function have been studied from model membrane systems and cells in culture to whole organisms, and it has been concluded that dynamin favours the detachment of endocytic carriers by forming a helical assembly around the neck of an invaginated endocytic pit and squeezing it (Fig. 1) 4 . Several models of how this might be achieved have been proposed, of which the two-stage model 5 and the ratchet/constriction model 6,7 are the most documented. According to the two-stage model, coordinated GTP hydrolysis across the helix favours a strongly constricted state that then leads to the rupture of the neck and the release of the endocytic carrier. According to the ratchet/constriction model, dynamin functions more like a molecular motor to make adjacent dynamin helices progressively slide over each other, eventually also leading to a sufficiently constricted state to favour fission.

New tools are needed to study predictions from both models in the cellular context at the site and time of scission, which has been achieved in the Article by Cheng et al. in this issue of Nature Cell Biology 8 . To dissect the final stages of clathrin-mediated endocytosis in cells in culture, the authors used a newly developed single-particle orientation and rotational tracking technique termed 3D-SPORT (Fig. 1a).

Information on the orientation of anisotropic imaging probes (e.g., fluorescent dipolar emitters or gold nanorods) can be retrieved by different techniques, including polarized fluorescence microscopy or defocused dark-field imaging as used in the current study 8 . Polarization is a property of light that can be exploited with the purpose of increasing the full scope of information that can be obtained by photonic imaging. Some polarization microscopy developments have focused on encoding information on orientation and wobbling into the shape of a single molecule's point spread function (PSF) 9,10 . Other techniques use ratiometric intensity measurements between different polarization projections 11,12 . Taking advantage of the plasmon-resonanceassociated birefringence of gold nanorods ✉ e-mail: ludger.johannes@curie.fr (i.e., the scattering of light that is polarized along their main axis) and a Nomarski-type of differential interference contrast microscope for defocused dark-field imaging, it is possible to retrieve the orientation of these gold nanorods 8 .

The authors of the current study have coupled gold nanorods to a well-established endocytic cargo protein, the iron transporter transferrin 8 . Using 3D-SPORT in living cells, it was possible to track the gold nanorods in 3D with a time resolution of about 20 ms and a spatial precision of around 10 nm and to determine at the same time their orientation changes with a precision of less than 2 degrees (Fig. 1). Furthermore, the authors' 3D-SPORT setup also allows two additional color channels to be collected simultaneously, here for the imaging of dynamin and clathrin in genome-edited SK-MEL-2 melanoma cells. Very clearly, this approach represents a technical tour de force.

In these experiments, the detachment point of the endocytic carrier was deduced from three main criteria: a peak of dynamin accumulation, a sudden movement of the carrier into the cells' interior space, and the gain of active rotational motion resulting from the loss of a rigid link to the plasma membrane. Clathrin stayed for a few seconds on the inward moving carriers before being shed, as expected for canonical clathrin-dependent endocytosis events 1 .

The groundbreaking contribution of the current study comes from analysis of the rotational behaviour of the transferrin-coupled gold nanorod in relation to accumulation of clathrin and dynamin and the detachment point. The authors described a succession of stages at which variable degrees of rotation were observed that likely represent the molecular events that occur when the clathrin-coated pit is built (i.e., initial membrane curvature and clathrin recruitment, cargo immobilization, progressive invagination of the nascent pit, and dynamin recruitment leading to the squeezing of the neck region). The most dramatic event was observed around the detachment point: a sharp right-handed super twist movement of the cargo-loaded gold nanorods that preceded membrane fission and endocytic vesicle release (Fig. 1b,c). The average twisting angle (130°) was large, and the duration of the twisting movement was very short (0.28 s). This will be key for the discussion on the implications of the super twist discovery for the mechanism of dynamin function.

The authors next considered an important control experiment. The K44A mutant of dynamin has reduced affinity for GTP, partially inhibiting its GTPase and fission activities 13 . In cells that expressed this mutant, not only was endocytosis blocked as expected, but also strikingly the super twist was absent. It could therefore be concluded that GTP hydrolysis by dynamin is required for the molecular rearrangements that lead to the sudden rotational movement of the super twist in the context of membrane fission and endocytosis.

The most important contribution of the current study is the demonstration that dynamin also produces torque in the cellular context. This had been discovered in vitro 6 , but whether it was also true at sites of endocytosis had remained unexplored. Is the dynamin riddle thereby solved? Not quite yet, as a number of outstanding questions remain. One concerns the model-two-stage or ratchet/constrictionthat describes best the molecular mechanism of function of dynamin. Notably, the abruptness of the super twist movement tends to argue in favour of the two-stage model. However, as observed by the authors themselves, possible sub-steps as predicted by the ratchet model might be lost if these were even faster than the time and angular resolution of the current 3D-SPORT setup. Molecular simulations may help to close this gap in future studies. Another important question concerns the structure of the dynamin helices before and after the super twist. Different helical states of dynamin (i.e., non-constricted, constricted, and super-constricted) have been described 14 , and the knowledge of those that feed into super twisting and that result from it would likely provide critical clues to the underlying molecular events. Finally, the authors describe the existence of a static phase that precedes the super twist 8 , which points to a specific regulation or external triggering whose molecular nature remains to be unraveled.

Beyond its exciting contribution to the understanding of the molecular mechanism of dynamin function in the cellular context, the current study more generally highlights the power of orientation-sensitive microscopy techniques and the larger contribution that these techniques might make in the field of membrane biology. Other forms of orientation-sensitive microscopy include polarization-based fluorescence imaging techniques, using polarization splitting or PSF engineering. In principle, these allow enrichment of the information from light microscopy imaging by monitoring how the environment of anisotropic probe-tagged biological macromolecules, notably proteins and lipids, affects their rotational and/or orientation freedom (e.g., when they get trapped into endocytic pits). An implementation of orientation-sensitive microscopy with multi-angle total internal reflection fluorescence microscopy can be designed to increase the axial resolution on fixed samples, or when coupled with single-particle tracking photoactivated localization microscopy to achieve super-resolution orientation imaging on live cells. Orientation studies as presented in the current study could even be extended to tissue imaging using polarized nonlinear microscopy, such as second-harmonic generation, of metal nanostructures 15 .

Fig. 1 |

 1 Fig. 1 | 3D-SPORT on dynamin's function in endocytosis. a, Microscopy on a patch of plasma membrane for concomitant analysis of gold nanorod particle orientation and fluorescent light from dynamin and clathrin. b, Dynamin-driven super twist in the sequence of events leading to membrane scission. Scission occurs when GTP is hydrolyzed to GDP + Pi. The arrow indicates time (t). c, From live cell experiments, the orientation of single gold nanorod particles is measured in function of time, here for the stages (i) and (ii) as indicated in b before and after the super twist, respectively. The 3D orientation of the gold nanorod particle is depicted as the polar angle θ and the azimuth angle ϕ.

Cellular and Chemical Biology unit, Institut Curie, U1143 INSERM, UMR3666 CNRS, PSL Research University, Paris, France.

Serpico -STED Team, Inria Centre Rennes-Bretagne Atlantique, Institut Curie, UMR144 CNRS, PSL Research University, Rennes, France.

competing interests

The authors declare no competing interests.


Denys Dutykh
email: denys.dutykh@univ-smb.fr

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

ON A CLASS OF LACUNARY ALMOST NEWMAN POLYNOMIALS MODULO p AND DENSITY THEOREMS

Keywords: Lacunary integer polynomial, zeroes, factorization, Lehmer's problem, Chebotarev density theorem, Frobenius density theorem, number of zeroes modulo p. 2020 Mathematics Subject Classification: 11C08, 11R09, 11R45, 12E05, 13P05

The reduction modulo p of a family of lacunary integer polynomials, associated with the dynamical zeta function ζ β (z) of the β -shift, for β > 1 close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in F p and their factorizations, via Kronecker's Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density theorems (Frobenius, Chebotarev, Serre, Rosen). These questions arise from the search for the existence of integer polynomials of Mahler measure > 1 less than the smallest Salem number 1.176280. Explicit connection with modular forms (or modular representations) of the numbers of zeroes of these polynomials in F p is obtained in a few cases. In general it is expected since it must exist according to the Langlands program.

1. Introduction 2. Dynamical zeta function of the β -shift and Lehmer's problem 3. An analog of Uniform Distribution Theorems -Proof of Theorem 1.1 4. Factorization, large gaps and Conjecture "B" 5. On the lower bound in Serre's density theorem 6. Counting solutions mod p and letting p tend to infinity 6.1. Trinomials -1 + z + z n mod p and Newforms 6.2. Densities and lacunarity 7. Appendix: The expression of the dynamical zeta function ζ β (z) of the β -shift when β > 1 is close to 1

INTRODUCTION

In this paper we will address a number of arithmetic questions on a class of lacunary integer polynomials which have all their coefficients in {0, 1} except the constant term equal to -1. We have called such polynomials almost Newman polynomials in [START_REF] Dutykh | On the Reducibility and the Lenticular Sets of Zeroes of Almost Newman Lacunary Polynomials[END_REF], by comparison with Newman polynomials which all have their coefficients in {0, 1}. Recall that, with probability one, Newman polynomials are irreducible [BV].

This class is the following. For n ≥ 2, we denote by B the class of lacunary polynomials

f (x) := -1 + x + x n + x m 1 + x m 2 + . . . + x m s
where s ≥ 0, m 1n ≥ n -1, m q+1m q ≥ n -1 for 1 ≤ q < s, and by B n those whose third monomial is exactly x n , so that

B = ∪ n≥2 B n .
The case "s = 0" corresponds to the trinomials G n (z) := -1 + z + z n . This class admits a very special type of lacunarity. This class appears naturally in the study of the existence of reciprocal integer polynomials having small Mahler measure [Sm]: in the Appendix we recall how this class of polynomials is arising from the dynamical zeta function ζ β (z) of the β -shift (equivalently from the Parry Upper function f β (z)), when β > 1 is close to one. Recall that the Mahler measure of a nonzero algebraic number β , of minimal polynomial (i) |} =: M(P β ). The search for non-trivial Mahler measures of reciprocal integer polynomials smaller than the smallest Salem number known 1.176280 . . . (Lehmer's number) is of particular interest in the Problem of Lehmer [Sm] [START_REF] Verger-Gaugry | A Panorama on the Minoration of the Mahler Measure: from the Problem of Lehmer to its Reformulations in Topology and Geometry[END_REF]. When β > 1 is close to one, it is shown in [START_REF] Dutykh | Alphabets, rewriting trails and periodic representations in algebraic bases[END_REF] [VG21] that a subcollection of zeroes (the lenticular zeroes, cf Section 2) of f β (z) is at the origin of a nontrivial (universal) minorant of M(β ). By Hurwitz Theorem, the zeroes of f β (z) in the open unit disk of C are limit points of zeroes of polynomials of the class B, which are its polynomial sections. It is the reason why this class of lacunary almost Newman polynomials is important for the Problem of Lehmer.

P β (X ) = a 0 X m + a 1 X m-1 + . . . + a m = a 0 ∏ i (X -α (i) ) ∈ Z[X ], is M(β ) = |a 0 | ∏ i max{1, |α
By analogy, there is interest in studying the zeroes of f (z) modulo p, for any f ∈ B and any prime number p. In the present note, we start the investigation of (i) the number of zeroes N p ( f ) of the polynomials f ∈ B in F p , i.e. modulo a prime number p, as a function of p, n, and the type of lacunarity of f characterized by the sequence (m j ) j=1,...,s , (ii) the subcollection of prime numbers p for which N p ( f ) is equal to zero or is maximal, (iii) the asymptotics of the averages of N p ( f ) when p tends to infinity.

For a nonzero integer polynomial f (X ) ∈ Z[X ] we denote by c( f ) the greatest common divisor of its coefficients. Let Z

[X ] c := { f ∈ Z[X ] \ {0} : c( f ) = 1}. Let N be the map Z[X ] c → (N) P , f → (N p ( f )) p∈P . Obviously, for f 1 , f 2 ∈ Z[X ] c , N ( f 1 f 2 ) = N ( f 1 ) + N ( f 2 )
since, for any p ∈ P, N p ( f 1 f 2 ) = N p ( f 1 ) + N p ( f 2 ). Now, integrating N over the set of prime numbers P and taking the limit average with respect to π(x), which is as usual the number of primes p ≤ x, gives the following result (proof in Section 3).

Theorem 1.1. Let f ∈ Z[X ] c . If f = ∏ i∈J f ν i i is the decomposition of f into irreducible factors, with all ν i ≥ 1, then (1.1) lim x→∞ 1 π(x) ∑ p≤x N p ( f ) = ∑ i∈J ν i
which is the number of irreducible factors in the decomposition of f .

By the Langlands program closed formulas are expected between the values N p ( f ) and the coefficients a(n) of Newforms ∑ a(n)q n . Key properties of Newforms, geometrical objects attached to the coefficients, can be found e.g. in [CS] [O].

Let us observe that (1.1) is an analog of classical Theorems of Uniform Distribution [KN] [Str] (cf Section 3).

Therefore the present study on the class B starts a concrete investigation not only of the quantities N p ( f ), f ∈ B, and the associated Newforms, but also of their limit averages by (1.1), which corresponds to the factorization of f . Remark 1.2. In the literature, following Lehmer's work, there are two "Conjectures of Lehmer", which a priori are completely independent. The first one is evoked in [Sm] [VG21P], and is concerned by a universal minorant of the Mahler measure of reciprocal nonzero algebraic integers which are not roots of unity. It is of concern in this study. The second one is related to the τ-function of Ramanujan. The present work, though trying to correlate the Problem of Lehmer modulo p to modular forms (Newforms) has no objectives to try to link the two Conjectures. The authors have no idea if the two Conjectures are linked.

The factorization of f ∈ B as a function of its reductions modulo p is a deep question [LS] [SL]. It is well-known that a monic integer polynomial is irreducible over the rationals if it is irreducible modulo some prime. The converse is not true in general [Bl]. Even if f ∈ B is irreducible a first question is about the density of the set of prime numbers p such that N p ( f ) = 0. Indeed Guralnick, Schacher and Sonn [GSS], then Gupta [Ga], have shown the existence of irreducible integer polynomials which are reducible modulo all primes. Whether the factorization of f modulo p contains linear factors is a basic question. If it is so, it is natural to say that the number N p ( f ) can take a priori any value between 1 and the maximal value deg( f ).

By Conjecture ARC (Conjecture (2.2) below) only 75% of the polynomials f in B are irreducible. General irreducibility criteria for the polynomials f in B are missing. Theorem 1.1 (ii) (Kronecker's Average Value Theorem), applied to each irreducible f ∈ B, says

lim p→∞ 1 π(x) ∑ p≤x N p ( f ) = 1, preventing the values N p ( f ) = deg( f ) to occur often, since the sum ∑ p≤x N p ( f ) behaves like x/ log(x) at infinity. If f ∈ B is irreducible and deg( f ) is a prime number, then N ( f ) = (N p ( f )
) p∈P is such that infinitely many N p ( f ) are equal to 0 [SL]. Examples of the distribution of values N p (S j ) of the (non-reciprocal parts of the) polynomial sections S j (x) of the Parry Upper function f τ (x) = -1/ζ τ (x), where τ is Lehmer's number (cf Section 2 for its definition), are given in Section 5.

The density Theorems of Frobenius and Chebotarev [Cev] play a role in the frequencies of values taken by N p ( f ) as p varies in general. For the class B we summarize this in Theorem 1.3 below. The question of the non-existence of factors of degree 1 (the "N p ( f ) = 0" case) in the factorization of f modulo p, is partially answered by a general theorem of Serre (Serre's Density Theorem [Se]), which gives a positive lower bound on the density of such primes (reported in Theorem 1.3 (i)).

The density is taken in the following sense: a subset S of the set of primes P has density

c if lim x→∞ number of p ∈ S with p ≤ x π(x) = c.
The limit need not exist. When it exists the natural density of S is said to be defined; then it is denoted by δ (S).

Theorem 1.3. Let n ≥ 3. Let f (x) := -1 + x + x n + x m 1 + x m 2 + . . . + x m s ∈ B where s ≥ 0, m 1 -n ≥ n -1, m q+1 -m q ≥ n -1 for 1 ≤ q < s.
The polynomial f is assumed irreducible. Denote by G the Galois group of f (x) and g := #G. Then (i) the set P 0 := {p ∈ P | N p ( f ) = 0} is infinite, has a density and its density satisfies

δ (P 0 ) ≥ 1 m s ,
with strict inequality if m s is not a power of a prime, (ii) the set

P max := {p ∈ P | N p ( f ) = deg( f )} is infinite and has density δ (P max ) = 1 g .
Theorem 1.3 (i) is due to Serre; the proof of (i) is given in [Se]. The statement of (ii) is Corollary 2 in Rosen [Ro]. This Corollary 2 is a consequence of a Theorem of Frobenius (Theorem 2 in [Ro]).

In Section 4 we continue the direct study of the factorization of the polynomials f in the class B, initiated in [START_REF] Dutykh | On the Reducibility and the Lenticular Sets of Zeroes of Almost Newman Lacunary Polynomials[END_REF]. Indeed this study has left open the problem of the existence of reciprocal non-cyclotomic components in the factoring of any such f . The main theorem on the factorization of all f ∈ B is Theorem 2.1 [START_REF] Dutykh | On the Reducibility and the Lenticular Sets of Zeroes of Almost Newman Lacunary Polynomials[END_REF]. We recall in Section 2 the Asymptotic Reducibility Conjecture ("ARC") which states that the probability of finding a polynomial f in B which is irreducible is 3/4, and the Conjecture "B", which states the non-existence of a reciprocal non-cyclotomic component in the factorization of a polynomial f in B. We revisit Conjecture B using a Theorem of Schinzel and a recent new lower bound by Sawin, Shusterman and Stoll [SSS] for large gaps. We prove that Conjecture B is valid on some infinite subclasses of B. For doing this, we consider the new bound for large gaps given in [SSS] as a new critical value above which Conjecture B is always true. Then, at intermediate lacunarity, for moderate gaps below this critical value, we show numerically that Conjecture B is also true. The subclasses considered are families of pentanomials in B, chosen from the quadrinomials studied by Finch and Jones [FJ].

The Problem of Lehmer modulo p, adressed to the class B, with its asymptotics when p tends to infinity, also calls for understanding the interplay between the asymptotics of N p ( f ) and the peculiar lacunarity of f , for any f ∈ B. The importance of the problem of the factorization of lacunary polynomials was outlined in a series of papers by Schinzel [Sch69] [Sch76] [Sch78] [Sch83] [Sch00]. The link between lacunarity, say the geometry of the gappiness, and irreducibility of any f ∈ B, is emphasized in Corollary 1.4 in [SSS], as follows.

Theorem 1.4. [SSS] Let f ∈ B, and write it as a polynomial f (x) = g N (x) = d(x) + x N c(x -1 ) as in (4.1) and (4.3). Under the assumptions of Schinzel's Theorem 4.2, the set of N > deg c + deg d such that f is irreducible is the complement of the union of a finite set with a finite union of arithmetic progressions.

In Section 6 the quantities N p ( f ), p tending to infinity, are studied for trinomials, following [Se], as functions of the coefficients of q-expansions and correlated to Newforms and modular forms. Some basic questions about the densities of ps such that N p ( f ) is congruent to a fixed integer modulo some integers are asked in Serre's general context [Se1] [Se2].

To outline the novel strategy of the paper, this note initiates the study of the quantities N p ( f ), p any prime number, and also p tending to infinity, for any f in the class B, and concomittantly the factorization of such f s. This difficult problem is tackled by the simplest cases of f s in B, which are trinomials, as in Section 6; and by showing on some subfamilies of B, as in Section 4, that factorization occurs with the absence of reciprocal non-cyclotomic components, which is conjectured to be the general rule.

DYNAMICAL ZETA FUNCTION OF THE β -SHIFT AND LEHMER'S PROBLEM

Let us recall standard definitions. A complex number α is an algebraic integer if there exists a monic polynomial R(X

) ∈ Z[X ] such that R(α) = 0. If R is the minimal polynomial of α and is reciprocal, i.e. satisfies X deg R R(1/X ) = R(X ), then α is called reciprocal. If α is reciprocal, α and 1/α are conjugated. If the minimal polynomial of α is not reciprocal, α is called non-reciprocal. If α = 1, or if α > 1 and the conjugates α (i) = α of α satisfy: |α (i) | < α, then α is said to be a Perron number.
Let n ≥ 3 be a fixed integer. Selmer [Sr] has shown that the trinomials -1

+ x + x n ∈ Z[x]
are irreducible if n ≡ 5 (mod 6), and, for n ≡ 5 (mod 6), are reducible as product of two irreducible factors whose one is the cyclotomic factor x 2x + 1, the other factor (-1 + x + x n )/(x 2x + 1) being nonreciprocal of degree n -2. We denote by θ n the unique zero in (0, 1) of the trinomial -1 + x + x n . The inverses θ -1 n > 1 are non-reciprocal algebraic integers which are Perron numbers, and constitute a decreasing sequence (θ -1 n ) n≥3 tending to 1 + . The smallest Mahler measure known is Lehmer's number τ, root > 1 of

x 10 + x 9 - x 7 -x 6 -x 5 -x 4 -x 3 + x + 1, such that: θ -1 12 < τ = 1.176280 . . . < θ -1 11 .
Here n is equal to 12. The search for reciprocal algebraic integers β s, of Mahler measure M(β ) ≤ 1.176280, in the intervals (θ -1 n , θ -1 n-1 ), n ≥ 13, having a minimal polynomial for which there is no

Z-minimal integer polynomial P β (X ) such that (2.1) P β (X ) = P β (X r )
for some integer r ≥ 2 is of importance in the problem of the non-trivial minoration of the Mahler measure (Section 2 in [START_REF] Verger-Gaugry | A Panorama on the Minoration of the Mahler Measure: from the Problem of Lehmer to its Reformulations in Topology and Geometry[END_REF]). The existence of very small Mahler measures is still a mystery.

Let us assume the existence of such a reciprocal algebraic integer β > 1. It is canonically associated with, and characterized by, two analytic functions:

(i) its minimal polynomial function, say z → P β (z), which is monic and reciprocal; denote d := deg P β , H := the (naïve) height of P β , (ii) the Parry Upper function f β (x) at β -1 , which is the generalized Fredholm deter- minant of the β -transformation T β (Section 3 in [START_REF] Verger-Gaugry | A proof of the Conjecture of Lehmer[END_REF]). It is a (infinite) power series with coefficients in the alphabet {0, 1} except the constant term equal to -1, with distanciation between the exponents of the monomials:

f β (x) := -1 + x + x n + x m 1 + x m 2 + . . . + x m s + . . . where m 1 -n ≥ n -1, m q+1 -m q ≥ n -1 for q ≥ 1. β -1 is the unique zero of f β (x) in the unit interval (0, 1). The analytic function f β (z) is related to the dynamical zeta function ζ β (z) of the β -shift (Section 3 in [VG21]
; [FLP]) by:

f β (z) = -1/ζ β (z).
Since β is reciprocal, with the two real roots β and 1/β , the series f β (x) is never a polynomial, by Descartes's rule on sign changes on the coefficient vector. The algebraic integer β is associated with the infinite sequence of exponents (m j ). Let us observe that all the polynomial sections

S s (x) := -1 + x + x n + x m 1 + x m 2 + . . . + x m s , s ≥ 1, of f β (x) are polynomials of the class B n .
The principal motivation to study the class B for itself in the present note comes from the peculiar form of these polynomial sections.

The polynomials f of the class B are often irreducible by the following conjecture, formulated in [START_REF] Dutykh | On the Reducibility and the Lenticular Sets of Zeroes of Almost Newman Lacunary Polynomials[END_REF]. ) which are irreducible is given by the limit, assumed to exist,

Asymptotic Reducibility Conjecture (ARC) Let n ≥ 2 and N ≥ n. Let B (N) n denote the set of the polynomials f ∈ B n such that deg( f ) ≤ N. Let B (N) := 2≤n≤N B (N) n . The proportion of polynomials in B = ∪ N≥2 B (N
(2.2) lim N→∞ #{ f ∈ B (N) | f irreducible} #{ f ∈ B (N) }
and its value is expected to be 3 4 .

Let us recall the generic factorization of the polynomials f ∈ B [DVG18] which generalizes that of the trinomials -1 + x + x n by Selmer [Sr].

Theorem 2.1 (Dutykh Verger-Gaugry). For any f ∈ B n , n ≥ 3, denote by f (x) = A(x)B(x)C(x) = -1 + x + x n + x m 1 + x m 2 + . . . + x m s , where s ≥ 1, m 1 -n ≥ n -1, m j+1 -m j ≥ n -1 for 1 ≤ j < s,

the factorization of f

where A is the cyclotomic part, B the reciprocal noncyclotomic part, C the nonreciprocal part. Then (i) the nonreciprocal part C is nontrivial, irreducible, and never vanishes on the unit circle, (ii) if β > 1 denotes the real algebraic integer uniquely determined by the sequence (n, m 1 , m 2 , . . . , m s ) such that 1/β is the unique real root of f in (θ n-1 , θ n ), the nonreciprocal polynomial -C * (X ) of C(X ) is the minimal polynomial of β , and β is a nonreciprocal algebraic integer.

From numerous experiments on the class B by Monte-Carlo in [START_REF] Dutykh | On the Reducibility and the Lenticular Sets of Zeroes of Almost Newman Lacunary Polynomials[END_REF], the components "B" were never observed. The following conjecture is reasonable to formulate. It will be partially proved in Section 4.

Conjecture B The reciprocal non-cyclotomic part B of any

f ∈ B n , n ≥ 3, is always trivial.
Let us mention now the link with Lehmer's problem. Lehmer's number 1.176280 is the smallest Mahler measure known. By a theorem of Smyth [Sm] the Mahler measure of a nonzero algebraic integer which is not reciprocal, not a root of unity, is ≥ 1.3247..., dominant root of X 3 -X -1 and smallest Pisot number. Then the Mahler measures of algebraic integers which are in the range (1, 1.3247) arise from reciprocal algebraic integers which are not roots of unity. This is the main reason why Conjecture B is important to investigate, about the possible existence of reciprocal parts in the factors of the polynomials of the class B.

Our attention is focused on the search for hypothetical reciprocal algebraic integers β > 1 for which M(β ) ∈ (1, 1.176280) and (2.1) is satisfied, that is when n is large in Conjecture B. Using intermediate alphabets and periodic representations of Q(β ) in the algebraic basis β , it was shown in [START_REF] Dutykh | Alphabets, rewriting trails and periodic representations in algebraic bases[END_REF] that the relation between f β and P β is a relation of identification on the subcollection of lenticular zeroes of f β . The definition of a lenticular zero is given in [START_REF] Dutykh | On the Reducibility and the Lenticular Sets of Zeroes of Almost Newman Lacunary Polynomials[END_REF], where many examples are proposed.

The lenticular zeroes of f β are peculiar zeroes, off the unit circle. Let us briefly recall what is a lenticular zero of f β . Many examples of lenticular zeroes are given in [START_REF] Dutykh | On the Reducibility and the Lenticular Sets of Zeroes of Almost Newman Lacunary Polynomials[END_REF]. The following theorem is Theorem 4 in [START_REF] Dutykh | On the Reducibility and the Lenticular Sets of Zeroes of Almost Newman Lacunary Polynomials[END_REF].

Theorem 2.2. Assume n ≥ 260. There exist two positive constants c n and c

A,n , c A,n < c n , such that the roots of any f ∈ B n , f (x) = -1 + x + x n + x m 1 + x m 2 + . . . + x m s , where s ≥ 1, m 1 -n ≥ n -1, m j+1 -m j ≥ n -1 for 1 ≤ j < s, lying in -π/18 < arg z < +π/18 either belong to {z ∈ C : ||z| -1| < c A,n n }, or to {z ∈ C : ||z| -1| ≥ c n n }.
The lenticulus of zeroes ω of f is then defined as

L β := {ω ∈ C : f (ω) = 0, |ω| < 1, - π 18 < arg ω < + π 18 , ||ω| -1| ≥ c n n }
where 1/β ∈ L β is the positive real zero of f . If a zero of f belongs to L β we say that it is a lenticular zero of f .

More precisely if Ω is a lenticular zero, then

f β (Ω) = 0 =⇒ P β (Ω) = 0.
3. AN ANALOG OF UNIFORM DISTRIBUTION THEOREMS -PROOF OF THEOREM 1.1

Recall [KN] [Str] that we say a sequence

(x n ) n≥1 ⊆ [0, 1) s is uniformly distributed on [0, 1) s if for each box B ⊆ [0, 1) s which is a cartesian product of intervals contained in [0, 1), of volume |B | we have lim N→∞ 1 N #{1 ≤ n ≤ N : x n ∈ B} = |B | .
Equivalently, for continuous f : [0, 1) s → C we have

(3.1) lim N→∞ 1 N N ∑ n=1 f (x n ) = [0,1) s f (t)dt.
In the present context, the left-hand side of (1.1) is the analog of the left-hand side of (3.1). The right-hand side of (1.1) is deduced from Kronecker's Average Value Theorem. This Theorem was announced by Kronecker at the Academy of Sciences, Berlin, in 1880, without proof. It has been given a proof by Rosen in [Ro].

FACTORIZATION, LARGE GAPS AND CONJECTURE "B"

The factorization of integer lacunary polynomials has received a lot of attention, eg Prasolov's book [Pr], Schinzel [START_REF] Schinzel | Reducibility of polynomials and covering systems of congruences[END_REF] Proving Conjecture B on all f ∈ B amounts to checking Conjecture B when the gaps in f do not obey the conditions of Theorem 4.2 and Corollary 4.4, that is at small gappiness. Examples of pentanomials with moderate gappiness are given in Table 1.

Let us now state Theorem 4.2 and Corollary 4.4, and make precise the critical bounds

N 1 , N 2 , N 3 , N 4 .
For a polynomial u = ∑ r i=0 a i x i ∈ Z[x], we define u as the squared Euclidean length of its coefficient vector:

u := r ∑ i=0 |a i | 2 ,
and u * denotes its reciprocal polynomial, as

u * (x) = x deg u u(1/x) = r ∑ i=0 a r-i x i .
We say that u is reciprocal when u * = u. It is said to be non-reciprocal if u * = u.

From (4.1) the two integers N 1 and N 2 can be defined: let T := max{deg c, deg d}, and τ be the smallest Salem number known, 1.176280, unique real positive root > 1 of

x 10 + x 9 -x 7 -x 6 -x 5 -x 4 -x 3 + x + 1 (Lehmer's polynomial). Denote (4.2) N 1 := deg c + deg d + 2T log τ log( c + d ) if T ≤ 27, T (log(6T )) 3 log( c + d ) otherwise; N 2 := deg c + exp 5 16 • 2 ( c + d ) 2 2 + max{2, (degc) 2 , (deg d) 2 } c + d . Definition 4.1. A pair (c, d) of polynomials c, d ∈ Z[x] with c(0), d(0) = 0 is Capellian when -d(x)/c(x -1
) is a pth power in Q(x) for some prime p or d(x)/c(x -1 ) is 4 times a fourth power in Q(x).

The following Theorem 4.2 is Theorem 1.2 in [SSS], a revisited formulation of a theorem of Schinzel (Theorem 74 in [START_REF] Schinzel | Polynomials with special regard to reducibility[END_REF]); the upper bound N 2 in item (ii) can be found in [START_REF] Schinzel | Reducibility of lacunary polynomials. I[END_REF]. Capellian. Then (i) there is a bound N 0 depending only on c and d such that for N > N 0 , the nonreciprocal part of g N is irreducible, (ii) the bound N 0 satisfies N 0 ≤ N 2 .

Theorem 4.2 (Schinzel). Let c, d ∈ Z[x] with c(0), d(0) = 0. Assume that c = ±d, that gcd Z[x] (c * , d) = 1 and (c, d) is not
Remark 4.3. Theorem 4.2 is probably the best theorem on the subject in general. However, in the case of the class B, a direct application of Ljunggren's tricks provide a better solution, which is Theorem 2.1, obtained recently by the authors in [START_REF] Dutykh | On the Reducibility and the Lenticular Sets of Zeroes of Almost Newman Lacunary Polynomials[END_REF].

Filaseta, Ford and Konyagin [FFK] have shown that the upper bound N 2 in (ii) can be replaced by the better upper bound

N 3 := deg c + 2 max{5 4( c + d +t)-15 , T (5 2( c + d +t)-8 + 1/4)}
where T = max{deg c, deg d} and t is the number of terms in c plus the number of terms in d.

Under some assumptions, Sawin, Shusterman and Stoll [SSS] show that the upper bound N 2 in (ii) can still be improved by replacing N 2 by

N 4 := (1 + deg c + deg d)2 c + d .
The bound N 4 is considerably smaller than N 3 , and the bound N 2 given by Schinzel is extremely large. For instance, for the pentanomial f (x) = -1 + x + x 5 + x 14 + x 100 , the bounds are: N 1 = 292 (cf Corollary 4.4 for its use), N 2 = 1.54 • 10 4553919 , N 3 = 6 • 10 17 whereas N 4 = 480. The authors in [SSS] also suggest an algorithm to improve further the value N 4 , by replacing the exponential 2 c + d by a polynomial function of c + d . This algorithm is useful in some cases.

The integer N 1 defined in (4.2) is introduced in [SSS] and both N 1 and N 2 are used when applying numerically the following statement (which can be found in [SSS]).

Corollary 4.4. Under the assumptions of Theorem 4.2, if N > max{N 0 , N 1 }, then the noncyclotomic part of g N is irreducible.

Now let f (x) = -1 + x + x n + x m 1 + . . . + x m j-1 + x m j + . . . + x m s ∈ B,
with n ≥ 3, s ≥ 1. It is easy to write it under the general form g N as above.

Define m 0 = n for coherency. With the distanciation rules we have missing monomials, those between x n = x m 0 and x m 1 , between x m 1 and x m 2 , ..., and between x m s-1 and x m s . Let us fix an integer j ∈ {1, 2, . . ., s} and write f as

(4.3) f (x) = d(x) + x m j x m s -m j c(x -1 ) with d(x) = -1 + x + x n + x m 1 + . . . + x m j-1 and c(x) = x m s -m j + x m s -m j+1 + x m s -m j+2 + . . . + x m s -m s-1 + 1.
The two polynomials c and d are fixed. Let us make the link with (4.1). We consider the family of polynomials deduced from f by taking arbitrarily sizes of the "hole" left between the monomials x m j-1 and x m j , as follows. We consider the subcollection

f ( j) = { f N (x) := d(x) + x N c(x -1 ) | N ≥ m s } ⊂ B n ,
and are interested in the irreducibility of f N for large N. Let us note that f m s = f . The superscript ( j) means that this family is "associated" with f and its jth "hole". Let us observe that the integer c + d is the number of monomials of any f N in f ( j) . It is an invariant of the family: for any f N in f ( j) , the integer N 3 is the same, and is equal to

N 3 := deg c + 2 max{5 8( c + d )-15 , T (5 4( c + d )-8 + 1/4)}.
Example.-We consider infinite families of pentanomials which present a variable gappiness at the last monomial. The polynomial c is taken equal to 1. When c = 1, then assumptions of Theorem 4.2 are satisfied. The pentanomials are defined below. When applying Corollary 4.4 to the polynomials f N ∈ f ( j) for j = m 2 and N > max{N 4 , N 1 }, then Conjecture B is valid for all the f N ∈ f ( j) . In Table 1 we check the validity of Conjecture B for the intermediate values of N in the range m 1 + (n -1) ≤ N ≤ N 4 .

Theorem 4.5 (Finch -Jones).

Let d ∈ B, d(x) = -1 + x + x n + x m 1 .
Let e 1 = gcd(m 1 , n -1), e 2 = gcd(n, m 1 -1). The quadrinomial d(x) is irreducible over Q if and only if

m 1 ≡ 0 (mod 2 e 1 ), n ≡ 0 (mod 2 e 2 ).
In Table 1 the Conjecture B is tested on all the pentanomials f (

x) = d(x) + x N c(x -1 ) with d(x) = -1 + x + x n + x m 1 , c(x -1 ) = 1,
and for N in the range {m 1 + n -1, m 1 + n, . . ., N 4 }. The quadrinomials d(x) are chosen to be either irreducible (in which case they are labelled "+") or reducible (in which case they are labelled "-") after Finch-Jones's Theorem 4.5. The pentanomials f (x) = 1+x+x n +x m 1 +x N obtained are either irreducible, or have cyclotomic factors Φ k (x) with k = 3, 6, 9, 10, 12, 18, 24 or 30. No reciprocal noncyclotomic factor appears in the factorizations. Remark 4.8. The other infinite families of pentanomials whose first quadrinomial is given in the list of Table 1 present different values of N 1 and N 4 , and may be studied in the same way, with respect to Conjecture B In the continuation of search for the conditions of existence of very small Mahler measures M(β ) > 1 of reciprocal algebraic integers β > 1, close to 1, it should be noticed that the gaps of the Parry Upper functions

f β (x) := -1 + x + x n + x m 1 + x m 2 + . . . + x m s + . . .
where m 1n ≥ n -1, m q+1m q ≥ n -1 for q ≥ 1, are never large by the following asymptotic upper bound [START_REF] Verger-Gaugry | On Gaps in Rényi β -expansions of Unity for β > 1 an Algebraic Number[END_REF]:

lim sup j→∞ m j+1 m j ≤ log M(β ) log β .
Consequently the polynomial sections of f β (z) have asymptotically a moderate gappiness, which is the intermediate domain of study for the non-existence of reciprocal noncyclotomic factors. If the domain of very large gaps is covered by Corollary 4.4, the existence of non-zero reciprocal algebraic integers β > 1 would lead to the difficult domain of intermediate gappinesses.

ON THE LOWER BOUND IN SERRE'S DENSITY THEOREM

In this paragraph we show on examples that the lower bound given by Serre in Theorem 1.3 (i) of the density of the set P 0 is far from being sharp for the polynomials of the class B.

We consider the set of the polynomial sections arising from ζ τ (z) with τ = 1.176280 Lehmer's number.

Let

f τ (x) = -1 + x + x 12 + x 31 + x 44 + x 63 + x 86 + x 105 + x 118 + . . . = -1/ζ τ (x)
S 0 = -1 + x + x 12 , irreducible, S 1 = -1 + x + x 12 + x 31 , reducible, = (x 2 + 1)(x 4 -x 2 + 1)C 1 (x), S 2 = -1 + x + x 12 + x 31 + x 44 , irreducible, S 3 = -1 + x + x 12 + x 31 + x 44 + x 63 , irreducible, S 4 = -1 + x + x 12 + x 31 + x 44 + x 63 + x 86 , irreducible, S 5 = -1 + x + x 12 + x 31 + x 44 + x 63 + x 86 + x 105 , reducible, = Φ 3 (x)Φ 4 (x)Φ 12 (x)C 5 (x), S 6 = -1 + x + x 12 + x 31 + x 44 + x 63 + x 86 + x 105 + x 118 , irreducible.
The numbers of solutions of the non-reciprocal parts of S j , C j (x) ≡ 0 mod p with p ≤ 43, are given in Table 2, for 0 ≤ j ≤ 6. In this table, on each line, the frequencies of zeroes are substantially higher than the ones deduced from the general bounds 1/ deg(C j ) of Theorem 1.3 (i). TABLE 2. Values of the quantities N p (S j ) for all primes p in the range {2, 3, . . . , 43}, where S j is the jth polynomial section of the Parry Upper function f τ (x) = -1/ζ τ (x), and τ Lehmer's number, if S j is irreducible. When S j is not irreducible the quantity N p (S j ) represented is replaced by N p (C j ) where C j is the non-reciprocal part of S j . The contributions of the cyclotomic parts modulo p, removed from the lines j = 1 and j = 5 are indicated underneath (calculated with PARI/GP). j 2 3 5 7 11 13 17 19 23 27 31 37 41 43 0 0 0 0 0 2 0 2 1

1 1 0 1 0 1 1 0 0 0 0 0 0 1 1 1 1 0 1 2 1 2 0 1 0 1 0 0 1 1 0 0 2 2 2 0 3 1 0 0 1 0 0 1 1 1 1 1 0 1 1 4 0 0 1 0 0 1 2 2 0 1 0 3 0 1 5 0 0 0 0 0 1 1 0 0 2 1 0 0 0 6 0 0 0 1 0 1 1 1 1 0 1 0 3 1 N p (X 2 + X + 1) 0 1 0 2 0 2 0 2 0 0 2 2 0 2 N p (X 2 + 1)
1 0 2 0 0 2 2 0 0 2 0 2 2 0 N p (X 4 -X 2 + 1) 0 0 0 0 0 4 0 0 0 0 0 4 0 0

COUNTING SOLUTIONS MOD p AND LETTING p TEND TO INFINITY

For understanding the quantities N p ( f ), f ∈ B, and possibly relate them to the coefficients of some power series, by the global correspondence Langlands program, we follow the general strategy of Serre in [Se] [Se1] [Se2]. The starting point is the study of the roots of the trinomials -1 + x + x n mod p, then of the quadrinomials of B mod p, whose first three terms are -1 + x + x n , etc, then ideally all the elements of the class B.

On one side the numbers N p ( f ), f ∈ B, are correlated to the factorization of the polynomials f via Kronecker's Average Value Theorem 1.1. On the other side the numbers N p ( f ) are related to questions of modularity and geometry. 6.1. Trinomials -1 + z + z n mod p and Newforms. The case n = 2:

f (x) = -1 + x + x 2 .
The discriminant of f is 5. The polynomial f has a double root mod 5; then N 5 ( f ) = 1. If p = 2, 5, the roots of f in F p are (1 ± √ 5)/2. If 5 is a square mod p then there are two roots,

N p ( f ) = 2. If not N p ( f ) = 0.
For p and q two distinct odd prime numbers, define the Legendre symbol as q p = +1 if r 2 ≡ q mod p for some integer r, -1 otherwise.

The law of quadratic reciprocity says

q p p q = (-1) p-1 2 q-1 2 .
Then 5 is a square mod p if and only if p ≡ ±1 mod 5. Therefore

N p (-1 + x + x 2 ) = 0 if p ≡ ±2 mod 5, 2 if p ≡ ±1 mod 5.
We deduce the distribution of values of N p (-1 + x + x 2 ) in the first column of Table 3. This distribution seems fairly regular. The probability limit for each value 0 and 2 is 1/2, but there is a Chebyshev bias [RS], mentioned in Serre [Se], which slightly shifts the probability distribution to 0 preferentially. It is observed in Table 4. This bias occurs for polynomials f ∈ B and will be studied elsewhere.

Using the change of variable x to -x, the results of [Se] section 5.2, can be directly applied to the trinomials -1 + x + x 2 , as follows.

Let us consider the q-series F = ∑ ∞ m=0 a m q m , defined by F = qq 2q 3 + q 4 1q 5 = qq 2q 3 + q 4 + q 6q 7q 8 + q 9 + . . .

Then

N p (-1 + x + x 2 ) = a p + 1 for all prime numbers p.

Note that the coefficients (a m ) of F have the property to be strongly multiplicative, in the sense:

a rq = a r a q for all integers r, q ≥ 1.

The corresponding Dirichlet series [LMFDB] is the L-series

∞ ∑ m=1 a m m s = ∏ p 1 - p 5 p -s -1 .
TABLE 3. Values of the numbers N p (-1 + x + x n ) for all primes p in the range {2, 3, . . . , 101} and n in the range {2, 3, . . . , 15} (calculated with PARI/GP).

p n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 1 0 1 0 1 0 1 0 1 0 1 0 1 5 1 0 0 1 1 0 0 1 1 0 0 1 1 0 7 0 0 1 2 0 1 0 0 1 2 0 1 0 0 11 2 1 1 1 0 1 1 0 0 1 2 1 1 1 13 0 1 1 2 1 0 0 0 0 2 0 1 0 1 17 0 1 2 1 1 0 0 1 0 1 2 1 1 0 19 2 0 0 3 1 0 1 1 0 3 1 0 1 1 23 0 1 1 0 0 2 2 2 0 0 1 0 1 1 29 2 1 1 1 2 0 0 2 0 1 1 0 1 0 31 2 2 0 0 1 1 1 1 1 3 0 1 1 0 37 0 1 2 0 1 0 0 0 0 2 1 0 1 1 41 2 0 1 1 0 0 0 0 1 0 0 0 0 1 43 0 1 0 1 1 2 0 1 0 2 1 0 0 1 47 0 3 0 0 2 2 1 0 2 2 0 2 0 2 53 0 1 2 0 1 0 0 1 0 2 0 0 0 0 59 2 0 1 1 1 2 1 0 2 1 0 0 2 1 61 2 1 1 1 1 1 1 1 0 2 0 1 0 1 67 0 3 2 0 1 0 0 0 0 3 0 1 1 1 71 2 0 0 0 0 0 1 1 1 1 0 1 2 1 73 0 1 0 1 0 0 0 0 2 3 0 0 1 0 79 2 1 2 0 2 1 1 0 2 2 1 1 0 0 83 0 1 4 0 3 3 2 1 0 1 0 0 2 1 89 2 1 1 1 0 0 0 1 0 1 1 1 2 0 97 0 0 1 0 1 1 2 0 0 3 1 1 1 0 101 2 0 0 1 0 0 0 0 0 1 1 1 0 0 The case n = 3: the discriminant of f (x) = -1+x+x 3 is -31. Modulo 31, the polynomial f has a double root and a simple root. Hence N 31 ( f ) = 2. For p = 31, one has (6.1)

N p ( f ) = 0 or 3 if p 31 = +1, 1 if p 31 = -1.
This explains the values reported in the corresponding column in Table 3.

The case n = 4: using the change of variable x to -x, the case of the trinomial -1+x+x 4 may be deduced from the case of -1xx 4 . In Section 5.4 of [Se] Serre gives the values N p (-1x + x 4 ) from coefficients of Newforms. Let us summarize the expressions he obtains, for the trinomials -1 if p can be written as 7x 2 + 5xy + 11y 2 , 0 or 2 if p 283 = -1. A complete determination of N p ( f ) can be obtained via a Newform F = ∑ ∞ m=0 a m q m of weight 1 and level 283 [LMFDB] whose first hundred terms are given in Crespo [Cr]:

+ x + x 4 . The discriminant of f (x) = -1 + x + x 4 is -283. Modulo
F = q + i √ 2q 2 -i √ 2q 3 -q 4 -i √ 2q 5 + 2q 6 -q 7 -q 9 + 2q 10 + q 11 i √ 2q 12 + q 13 -i √ 2q 14 -2q 15 -q 16 -i √ 2q 18 + i √ 2q 19 + i √ 2q 20 + i √ 2q 21 + i √ 2q 22 -q 23 -q 25 + i √ 2q 26 + q 28 -q 29 -2i √ 2q 30 + i √ 2q 31 -i √ 2q 32 -i √ 2q 33 + i √ 2q 35
+q 36 -2q 38i √ 2q 39 + q 41 -2q 42 + . . .

Then one has:

N p ( f ) = 1 + (a p ) 2 - p 283
for all primes p = 283.

This explains the values reported in the corresponding column in Table 3.

The case n = 7: the Galois group G of the trinomial -1 + x + x 7 is S 7 [CMS]. Applying Theorem 1.3 (ii) gives 1.984 • 10 -4 = (#G) -1 for the density of primes p such that N p (-1 + x + x 7 ) = 7. Indeed, the only two prime numbers ≤ 10 5 realizing the maximality identity N p (-1 + x + x 7 ) = 7 are: p = 41143 and p = 82883. By Theorem 1.3 (i) the density of prime numbers p such that N p (1 + x + x 7 ) = 0 exists and is above 1/7. This is compatible with the corresponding column in Table 3.

The general case n: following [Se] there should exist formulas for the numbers N p ( f ) coming from the coefficients of q-series, newforms, etc. 6.2. Densities and lacunarity. Let n ≥ 3 and f

(x) = (-1 + x + x n ) + x m 1 + x m 2 + . . . + x m s ∈ B.
To align the statements on the presentation of Serre [Se1] [Se2], let us adopt the geometric language of schemes. The system of algebraic equations defining the variety is reduced to one equation. Denote by A = Z[x]/( f ) the finitely generated ring over Z. Let X = Spec(A). The solutions mod p of f (x) ≡ 0 correspond to the elements x ∈ X (F p ) in the fiber. We denote N p ( f ) by N X (p). The polynomial f is fixed and p varies. Theorem 6.1. For any integer q ≥ 1, any γ ∈ Z/qZ, the set P γ := {p | N X (p) ≡ γ mod q} has a density. This density is a rational number.

Proof. [Se1] [Se2].

For q = 2 the question of the density of P γ amounts to understand when N X (p) is even, and when N X (p) is odd. In general the set P γ is empty, is a finite set or has a density which is > 0. A case when the density is > 0 is of topological origin and comes from the topology of the complex space X (C), resp. of the real space X (R). Denote by χ(X (C)), resp χ c (X (R)), in Z, the Euler characteristic of X (C), resp. the Euler characteristic with compact support of X (R). Theorem 6.2. For any integer q ≥ 1,

(i) δ P γ > 0 for γ = χ(X (C)), (ii) δ P γ > 0 for γ = χ c (X (R)).
Proof. [Se1] [Se2].

The decomposition of f as the sum of the trinomial part

-1 + x + x n
and the perturbation term

x m 1 + x m 2 + . . . + x m s for m 1 -n ≥ n-1, m j+1 -m j ≥ n-1 for 1 ≤ j ≤ s-1 suggests to define Y = Spec(Z[X ]/(-1+ x + x n ))
and to study N Y (p) for p tending to infinity. A first question is about the comparison between N X (p) and N Y (p), and when p tends to infinity.

Question 1: For any integer q ≥ 1, any γ ∈ Z/qZ has The importance of the class of polynomials B comes from the formulation of the dynamical zeta function ζ β (z) of the β -shift when β > 1 is close to 1. Let us recall the main steps, leaving the details for the reader.

The notion of dynamical zeta function was introduced by M. Artin and B. Mazur [START_REF] Artin | On Periodic Points[END_REF] in 1965. Let h : V → V be a diffeomorphism of a compact manifold V , such that its iterates h k all have isolated fixed points. Then they defined (7. [START_REF] Artin | On Periodic Points[END_REF] ζ β (z) := exp

∞ ∑ n=1 #{x ∈ V | h n (x) = x} n z n ,
and showed that for a dense set of diffeomorphisms h of V the power series of such an expression converged in a neighbourhood of z = 0. The dynamical zeta function of a dynamical system, when defined, is an analytic function which concentrates a lot of information on the dynamical system, and therefore is a powerful tool (Pollicott [Po]).

From Theorem 2 in [START_REF] Baladi | Zeta Functions and Transfer Operators for Piecewise Monotone Transformations[END_REF], concerning the Rényi-Parry dynamical numeration system (V = [0, 1], h = T β ), where β > 1 and T β : x → β x mod 1 is the β -transformation [ITi] [Ls], we deduce Theorem 7.1. Let β ∈ (1, θ -1

2 ). Then, the Artin-Mazur dynamical zeta function [Ti] and Ito Takahashi [ITi] obtained the reformulation of (7.2) as follows.

Theorem 7.2. Let β > 1 be a real number. Then

(7.3) ζ β (z) = 1 -z N (1 -β z) ∑ ∞ n=0 T n β (1) z n
where N, which depends upon β , is the minimal positive integer such that T N β (1) = 0; in the case where T j β (1) = 0 for all j ≥ 1, "z N " has to be replaced by "0". Up to the sign, the expansion of the power series of the denominator (7. The total ordering < on (1, +∞) is uniquely in correspondence with the lexicographical ordering < lex on Rényi expansions of 1 by the following Proposition, which is Lemma 3 in Parry [Py]

  [Sch69] [Sch78] [Sch00], or Filaseta and his collaborators[DFV] [F] [FFN] [FM]. Finding irreducibilty criteria is an important topic, and many problems remain open.Consider polynomials of the form (4.1)g N (x) = x N c(x -1 ) + d(x),where c and d are fixed polynomials in Z[x] with c(0), d(0) = 0. We are interested in the irreductibility of g N for large N. Such polynomials have already appeared in different contexts, eg[DFV] [HVW] [FM][FFK] [Sch67]. From Schinzel[START_REF] Schinzel | Polynomials with special regard to reducibility[END_REF] [Sch69] and following Sawin, Shusterman and Stoll[SSS] we first recall the general statements of Theorem 4.2 and Corollary 4.4, concerned by the factorization of g N for N large. The step after, which is important to understand the factorization of any f ∈ B containing large gaps, is to recognize such f as a g N , and apply these general theorems to f . Of course only the f ∈ B having large gaps are concerned by these statements. It is expected that the polynomials f ∈ B which possess a small gappiness, though not concerned by these statements, have the same factorization properties.What is the main conclusion at large gaps? Corollary 4.4 implies that any f ∈ B would have an irreducible non-cyclotomic component. But, in view of Theorem 2.1, since the non-cyclotomic part of f (x) is B(x)C(x) and that C(x) always exist and is irreducible, it means that the reciprocal non-cyclotomic component B(x) is trivial. So to say, Conjecture B is true at large gaps on B.

  Proposition 4.7. The family of pentanomials f (x) = -1 + x + x 5 + x 14 + x N , N ≥ 18, of B 5 , admits the bounds N 1 = 292, N 4 = 480. All the polynomials of this family satisfy Conjecture B.

  be the Parry Upper function at Lehmer's number τ = 1.176280 . . .. All the polynomial sections belong to the subclass B 12 . Denote

  283, f has one double root and two simple roots. Then N 283 ( f ) = 3. If p = 283, one has N p ( f ) =    0 or 4 if p can be written as x 2 + xy + 71y 2 , 1

P 2 :

 2 γ := {p | N X (p) -N Y (p) ≡ γ mod q} a density? Question What is the role of n in the sets {p | N X (p) -N Y (p) ≡ γ mod q}, in particular when n becomes very large? 7. APPENDIX: THE EXPRESSION OF THE DYNAMICAL ZETA FUNCTION ζ β (z) OF THE β -SHIFT WHEN β > 1 IS CLOSE TO 1

  of periodic points of period dividing n, is nonzero and meromorphic in {z ∈ C : |z| < 1}, and such that 1/ζ β (z) is holomorphic in {z ∈ C : |z| < 1}, Let β > 1 be a real number. Denote A β := {0, 1, 2, . . ., ⌈β -1⌉}, where ⌈β -1⌉ denotes the upper integer part of β -1. If β is not an integer, then ⌈β -1⌉ = ⌊β ⌋ which is the usual integer part of β . Using ergodic theory, Takahashi

  3) is the Parry Upper function f β (z) at β . It satisfies(7.4) (i) f β (z) = -1z N ζ β (z)in the first case,(7.5) (ii) f β (z) = -1 ζ β (z)in the second case, and, denoting by t 1 ,t 2 , . . . ∈ A β the coefficients in(7.6) -1 + t 1 z + t 2 z 2 + t 3 z 3 + . . . = f β (z) = -(1 -β z) z n , f β (z) is such that 0.t 1 t 2 t 3 . . .is the Rényi β -expansion of unity d β (1). The Parry Upper function f β (z) has no zero in {z ∈ C : |z| ≤ 1/β } except z = 1/β which is a simple zero.

TABLE 1 .

 1 Numerical verification of the Conjecture B for all low degree pentanomials f (x) = -1 + x + x n + x m 1 + x N for which m 1 + (n -1) ≤ N ≤ N 4 . For each experiment number, the polynomials f (x) are either irreducible or have cyclotomic factors in {Φ k } (e.g. for Exp. num. equal to 1, only Φ k=3 and Φ k=6 are encountered). Newman polynomial since all the coefficients are in {0, 1}. By the Odlyzko-Poonen Conjecture[BV] it is irreducible with probability 1. But it has no zero in the interval [0, 1], which is not the case of the non-reciprocal part of d(x) ∈ B by Theorem 2.1. Therefore, with probability one, c * cannot be the non-reciprocal part of d. Therefore, with probability one, it is an irreducible cyclotomic polynomial, which is a cyclotomic factor of Applying Corollary 4.4, the numerical investigation reported in Table2allows to complement completely the study of Conjecture B at large gaps by the one at intermediate lacunarity, and gives a proof to the following result.

	Exp. num. n m 1 Quad. irred. N 4 {Φ k ( x )} B-conj. 1 3 5 + 192 {3, 6} 2 3 6 + 224 {3, 6} 3 3 7 + 256 {10} 4 3 8 -288 {3} 5 3 9 + 320 {3, 10} 6 3 10 + 352 {18} 7 3 11 + 384 {3, 6} 8 3 12 -416 {3, 6} 9 3 13 + 448 / 0
	10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27	3 17 4 7 4 8 4 9 4 10 4 11 4 12 4 13 4 17 5 9 5 10 5 11 5 12 5 13 5 14 5 15 5 16 5 17	+ --+ ---+ + + + + + + + + -+	576 {3, 6, 10} 256 {6, 9} 288 / 0 320 {9} 352 / 0 384 {6} 416 / 0 448 {6, 24} 576 {6} 320 {3, 6, 12} 352 {6} 384 {6} 416 {3, 6, 12} 448 {6} 480 {6} 512 {3, 6} 544 {6, 30} 576 {6}
	Remark 4.6. The assumptions of Theorem 4.2 are not strong and are compatible with Con-
	jecture B. Indeed, if f (x) = d(x) + x N c

* (x) belongs to B, then c = ±d always, the couple (c, d) is never Capellian. What about the assumption gcd Z[x] (c * , d) = 1? The polynomial c * (x) is a d. At worst, gcd Z[x] (c * , d) would be a cyclotomic factor of f with probability one.

TABLE 4 .

 4 Chebyshev bias for -1 + x + x 2 mod p.

	x = 101 1001 10001

.

  Proposition 7.3. Let α > 1 and β > 1. If the Rényi α-expansion of 1 is d α (1) = 0.t ′ + . . . and the Rényi β -expansion of 1 is d β (1) = 0.t 1 t 2 t 3 . . . ,

	1 t ′ 2 t ′ 3 . . . ,	i.e. 1 = α 3 i.e. 1 = t ′ 1 α + t ′ 2 t ′ 3 α 2 + t 1 β + t 2 t 3 β 2 + β

3 + . . . ,

Acknowledgements References

ACKNOWLEDGEMENTS

We would like to thank Florent Jouve for a very helpful discussion and the referees for their useful comments.

(1) = 0.10 n-1 1 and d θ -1 n (1) = 0.10 n-2 1, Proposition 7.3 implies that the condition is sufficient. It is also necessary: d β (1) begins as 0.10 n-1 1 for all β such that θ -1 n+1 ≤ β < θ -1 n . For such β s we write d β (1) = 0.10 n-1 1u with digits in the alphabet A β = {0, 1} common to all β s, that is

, which characterizes uniquely the base of numeration β , readily implies h 0 = 0 and h k = 1 and n k ≥ n -1 for all k ≥ 1.

The polynomials of the class B are all the polynomial sections of the power series f β (z) for β in the interval (1, (1 + √ 5)/2). Indeed, from (7.7) and (7.6), the power series in (7.6) takes the form:

-1 + x + x n + x m 1 + x m 2 + . . . + x m s + . . .

with the distanciation conditions: m 1n ≥ n -1, m q+1m q ≥ n -1 for 1 ≤ q.


L Couëdel

V Nosenko

M Rubin-Zuzic

S Zhdanov

Y Elskens

T Hall

A V Ivlev

Full melting of a two-dimensional complex plasma crystal triggered by localized pulsed laser heating

Keywords: numbers: 52, 27, Lw

The full melting of a two-dimensional plasma crystal was induced in a principally stable monolayer by localized laser stimulation. Two distinct behaviors of the crystal after laser stimulation were observed depending on the amount of injected energy: (i) below a well-defined threshold, the laser melted area recrystallized; (ii) above the threshold, it expanded outwards in a similar fashion to mode-coupling instability induced melting, rapidly destroying the crystalline order of the whole complex plasma monolayer. The reported experimental observations are due to the fluid modecoupling instability which can pump energy into the particle monolayer at a rate surpassing the heat transport and damping rates in the energetic localized melted spot, resulting in its further growth. This behavior exhibits remarkable similarities with impulsive spot heating in ordinary reactive matter.

I. INTRODUCTION

Complex (or dusty) plasmas are weakly ionized gases containing solid micro-or nanoparticles (often referred to as dust particles). Due to interactions with electrons and ions in the ambient plasma, dust accumulates a net electric charge [START_REF] Bouchoule | Dusty Plasmas: Physics, Chemistry and Technological impacts in Plasma Processing[END_REF][START_REF] Shukla | Introduction to dusty plasma[END_REF][START_REF] Morfill | [END_REF]. In laboratory low-pressure gas discharges, this charge is negative due to the high mobility of the electrons. By injecting monosized spherical microparticles in the sheath of a capacitively coupled radiofrequency (cc-rf) discharge, it is possible for particles to form a monolayer. This is a result of all the microparticles levitating at the same height in the sheath region above the lower electrode where the electric field is strong enough to balance gravity and ensure stiff confinement [4,5]. Under specific conditions, they form strongly coupled ordered structures known as two-dimensional (2D) complex plasma crystals [6][7][8][9][10][11].

2D complex plasmas are often used as model systems for studies of generic phenomena occurring in liquids and crystals at the kinetic (particle) level [START_REF] Morfill | [END_REF]. In such systems, it is indeed possible to obtain complete information about the state of the whole system of particles in the kinetic (x, v)-space. This offers an important advantage for the investigation of collective processes occurring in strongly coupled media including melting and recrystallization [12][13][14][15], mass and heat diffusion [16], solitons and shocks [START_REF] Samsonov | Physical Review E[END_REF].

In 2D complex plasmas, as in any strongly coupled 2D system, two in-plane wave modes can be sustained. In crystals, one of these modes is longitudinal, while the * lenaic.couedel@usask.ca other is transverse, with both having an acoustic dispersion [18]. Since the strength of the vertical confinement due to the sheath electric field is finite, there is a third fundamental wave mode with negative optical dispersion associated with the out-of-plane oscillations [19,20]. The propagation of dust-lattice (DL) waves in these 2D lattices is often used as a diagnostic tool to determine parameters of the plasma crystal since the wave dispersion depends directly on them [8,[21][22][23].

In the sheath of a cc-rf discharge, the ion stream coming from the bulk plasma and directed toward the electrode is focused downstream of each negatively charged particle of the monolayer and creates a perturbed region called the "ion (or plasma) wake". Ion wakes exert an attractive force on the neighboring particles. They can thus be considered as an (external) "third body" in the interparticle interaction resulting in non-reciprocal particle pair interactions [24][25][26][27]. Under certain conditions, this can lead to the formation of an unstable hybrid mode and the melting of the crystalline monolayer when the out-of-plane mode crosses the in-plane compressional mode: the mode coupling instability (MCI) [28][29][30][31][32]. The hybrid mode has clear fingerprints: critical angular dependence, a mixed polarization, distinct thresholds [30], and synchronization of the particle motion [33]. Wake-induced mode coupling is also possible in liquid complex plasma monolayers [31]. Remarkably, in such cases, the confinement and dust particle density thresholds, which are important features of the MCI in 2D complex plasma crystals, disappear and the instability has a higher growth rate. Consequently, conditions exist for which both the crystalline and the fluid states are viable, meaning no crossing of the modes in the crystal state and MCI growth rate high enough in the fluid state to prevent crystallization. This explains why, in many experiments, the crystallization of a monolayer is achieved only by increasing the gas pressure and/or the rf power (confinement strength is an increasing function of the discharge power [34]). In addition, it is in principle possible to trigger sporadic melting of a stable crystal which is not too far from the crystalline MCI threshold by applying a sufficiently strong mechanical perturbation.

In this article, we demonstrate experimentally that wake-mediated resonant mode coupling can be induced in a principally stable 2D plasma crystal (no crossing of the modes) levitating in the sheath of a cc-rf discharge through an external excitation mechanism. Localized laser stimulation of the monolayer can trigger MCI-induced melting of the stable crystal if the injected energy is sufficient to create a melted spot in which the excess energy due to the fluid MCI is high enough to overcome neutral damping and heat conduction in the rest of the crystal. This causes a further increase in temperature leading to the extension of the melted area over the whole monolayer in a kind of uncontrolled positive feedback. This uncontrollable melting exhibits similarities with impulsive spot heating in ordinary matter. The experiments were carried out in a cc-rf glow dis-charge at 13.56 MHz (modified GEC chamber). A sketch of the setup is presented in Fig. 1. The argon pressure p Ar was between 0.6 Pa and 1 Pa, and the forward rf power P RF was between 10 W and 20 W. In the bulk discharge, the electron temperature was T e = 2.5 eV and the electron density was n e = 2 × 10 9 cm -3 at p Ar =0.66 Pa and P RF =20 W [35]. Melamine-formaldehyde spherical particles with a diameter of 9.19 ± 0.09 µm were levitated in the plasma sheath above the lower rf electrode and formed a horizontal monolayer, up to 60 mm in diameter. The dust particle suspension was illuminated by a horizontal laser sheet. The particles were imaged through a window at the top of the chamber by a 4 Megapixel Photron FASTCAM Mini WX100 camera at a speed of 250 frames per second. The particle horizontal coordinates, x and y, and velocities, v x and v y , were then extracted with sub-pixel resolution in each frame. An additional side-view camera (Basler Ace ACA640-100GM) was used to confirm that we were indeed working with a single layer of particles. More details can be found in previous publications [30,32,36].

A SpectraPhysics Millennia PRO 15sJ solid-state 532nm laser with a continuous-wave (cw) output power of 0.30-2 W was focused for a short period of time on the crystal. Particles in the laser spot were accelerated and the crystal was melted locally. At the crystal surface, the laser spot had an elliptical shape with its major axis in the direction of propagation and the illuminated area was S L ≃ 0.32 mm 2 . The duration of the laser pulse was varied from 0.05 s to 0.65 s.

Longitudinal

Transverse

III. RESULTS

A. Characterization of the stable crystal

Prior to laser excitation, a stable 2D crystal with respect to the mode coupling instability (no crossing of the eigenmodes [30]) was levitated above the rf electrode. In the set of experiments presented in this article, the following discharge conditions were chosen: p Ar = 1.04 Pa and P RF = 12 W. For each run, the same microparticles were used and, in case of melting induced by the laser excitation, the monolayer was recrystallized by increasing argon pressure and rf power. The discharge conditions were then restored to the chosen values and the crystalline monolayer was allowed to relax a few minutes before the next laser pulse.

From the particle positions, the mean interparticle distance ∆ = 415 ± 10 µm (number density n 2D = 675 ± 20 cm -2 ) was obtained by calculating the pair correlation function and measuring the position of the first peak. Using particle velocities, the in-plane compressional and shear particle current fluctuation spectra of the crystalline monolayer were computed. In Fig. 2, the spectra are shown for a propagation angle θ = 0 • (see inset of Fig. 1). No fingerprints of the unstable hybrid mode could be detected and only the eigenmodes were present. By fitting simultaneously the in-plane compressional and shear current fluctuation spectra by the theoretical curves of a 2D Yukawa crystal [29], the microparticle charge Q = -13800e ± 500e, where e is the elementary charge, the screening length λ D = 408 ± 47 µm and the screening parameter κ = 1.0 ± 0.15 were obtained. The vertical confinement frequency was ∼ 25 Hz which is in perfect agreement with a crystal far from the MCI threshold.

B. Propagation of the melting front: a threshold behavior

The laser beam was focused on the crystal so that only a few particles were accelerated during the pulse (N ≤ 50). It created, at a few ∆ in front of the laser spot position in the direction of laser light propagation, a small melted spot (1 -4 mm in diameter) containing particles having high velocities. Depending on the pulse duration and the laser power, the melted area recrystallized or expanded over the whole crystalline monolayer. By tracking the particles during and after the laser pulse, the evolution of the particle kinetic energies was measured, and the energy injected by the laser into the crystal was estimated. At the end of the laser pulse, the total kinetic energy of the particles in the camera field of view (FoV) was at its maximum. However, the particles in the melted spot had trajectories directed along the direction of the laser beam. By looking at the first and second moments of the velocity distribution function in x and y, thermalization due to collisions with other particles was TABLE I. Laser power P l , pulse duration ton, the sum Kt of the kinetic energy of the particles in the camera FoV, and, depending on whether the crystal recrystallized at the end of the laser pulse or melted, the transition time ttr before recrystallization or the inverse of energy growth rate γ -1 g , respectively. The boldface entries are the experiments for which a full melting of the monolayer was induced by the laser pulse.

Exp.

P l ton Kt(ton) Kt(ton completed after t th = 0.5 s (similar Maxwellian velocity distribution centered on 0 in x and y [37]).

+ t th ) ttr, γ -1 g # (W) (s) (keV) (keV) (
In Table I, the sum of the kinetic energies of the particles K t = i K i (t) (total energy) in the camera FoV at the end of the laser pulse (K t (t on )) and after thermalization (K t (t on +t th )) are listed for the different runs. As expected, at a given laser power, the total energy increased with the pulse duration. The energy after thermalization was slightly smaller due to damping by the neutral background and heat transport in the whole crystalline suspension (which was not entirely comprised in the FoV). Before the laser pulse, the average kinetic temperature of the particles was estimated at T 0 = 0.13 ± 0.05 eV, giving a total energy in the field of view K t < 800 eV. Thus the measured total energy just after the laser pulse corresponds roughly to the energy injected by the pulse into the crystal since it was always much higher than a few keV. In Fig. 3, the state of the monolayer at the end of the experiment (recrystallized or fully melted) is plotted as a function of the total energy after thermalization. As can be seen, when K t (t on + t th ) reached = 40 ± 1 keV (corresponding to t on ≥ 500 ms at P l = 0.85 W), the crystalline monolayer fully melted. The observed step function is a clear sign of a threshold phenomenon.

K (t +t ) (keV) on th t

In Figs. 4 and5, we illustrate the difference between two experiments showing either recrystallization or melting (Exps. 19 and 20, respectively; see Table I). In Fig. 4, the monolayer Voronoi map (colored according to |Ψ 6 |) and the kinetic energy map are presented for two experiments using the same laser pulse setting. Here, Ψ 6 (j) = (1/N ) N l=1 e iθ jl , where N is the number of closest neighbors and θ jl is the angle of vector linking the jth to the lth particle with respect to the x-axis. This order parameter |Ψ 6 | represents the deformation of a hexagonal cell from a perfect hexagon (|Ψ 6 | = 1). As can be seen, before the laser pulse (t = 0 s), the crystalline monolayer was cold in both cases with T 0 ≃ 0.13 eV and the crystals were almost perfect with only a few dislocations and a few grain boundaries. At the end of the laser pulse (t = 0.55 s), a melted spot (high kinetic energy and low values of |Ψ 6 |) could be seen in the center of the frame. In both cases, compressional waves traveled preferentially in the direction of the laser beam (positive y direction) due to the non-thermalized motion of the excited particles. At t = 2 s > t on + t th , the particles were thermalized and compressional waves emerged from the hot spot isotropically. It was observed that, in both cases, the melted spot became slightly larger than at t = 0.55 s due to heat transport in the crystalline monolayer. However, the spot in experiment 20 showed a slightly larger I).

Each map is a square of 24.6×24.6 mm 2 .

melted spot and the particle kinetic energies were slightly higher. This observation is confirmed in Fig. 5, in which the time evolution (after the laser pulse) of the radius of the melted zone (MZ) and the evolution of the total kinetic energy in the MZ and the camera FoV are plotted. In this article, the MZ is defined as the area in which the mean kinetic energy is greater than 30 eV [38]. As can be seen in both experiments, most of the kinetic energy in the FoV is carried by the particles in the MZ, though in experiment 19, the injected energy and the radius of the MZ were slightly smaller. In both cases the kinetic energy and MZ radius initially increased, but in experiment 19 the total energy inside the MZ became stable, meaning that the kinetic energy began to decrease. On the contrary, K MZ continued to grow in the case of experiment 20. After ∼ 2.25 s, the MZ started to shrink in experiment 19 and the total energy started to decrease while both parameters kept growing in the case of experiment 20 (the energy growth seemed exponential). Finally, in the case of experiment 19, after ∼ 3.1 s, the total energy exhibited an exponential decay while the MZ was rapidly shrinking, denoting recrystallization. In experiment 20, the energy kept growing and the MZ eventually extended to the whole monolayer. After 3.5 s, the radius of the MZ grew quasi linearly with a velocity v MZ ∼ 2 mm/s. A snapshot of the monolayer during the final stage of both experiments can be seen in Fig. 4 I). (b) Evolution of the radius of the melted zone as a function of time.

transition period before the recrystallization sets in for injected energy below the melting threshold. For the purpose of analysis, the transition time t tr is defined as the time before K t exhibits a clear exponential decay after the end of the laser pulse [39]. In experiment 19, the transition time was 3.13 s (see Fig. 5a). As can be seen in the inset of Fig. 3, the transition time gets longer as the input energy gets closer to the induced-melting energy threshold (experimental values are listed in Table I). For the experiments with injected energy above the threshold the energy growth looked nearly exponential. In Table I, the measured growth times γ -1 g are listed [40]. The average measured value was γ -1 g = 1.1 ± 0.2 s.

IV. DISCUSSION

As reported in the previous section, the laser-induced melting of the crystal exhibited a clear energy threshold: above a well-defined amount of energy injected by the laser, the full melting of the crystalline monolayer was triggered. The mechanism by which the monolayer melted can be understood when taking into account the existence of the MCI in fluid monolayers. It was indeed shown by Ivlev et al. [31] that the wake-induced coupling of wave modes always occurs in 2D fluid complex plasmas and, contrary to MCI onset in 2D complex plasma crystals, there is no confinement-density threshold. This means that for sufficiently low damping rates the energy input from the fluid MCI (i.e. energy transferred by the ion flow to the crystal) can prevent crystallisation of the monolayer. Moreover, if a large enough melted spot is induced in the crystalline monolayer, the energy input from the localized fluid MCI will not be dissipated rapidly enough (through neutral damping and heat conduction in the crystal), causing a further increase in temperature triggering the expansion of the melted area in an uncontrolled positive feedback. This mechanism will lead to the total destruction of the crystalline monolayer as observed for regular MCI-induced melting of 2D complex plasma crystals [30-32, 41, 42]. Such an "explosive melting" of a 2D complex plasma crystal has been recently studied in detail by Yurchenko et al. [42] and was found to be similar to impulsive spot heating and thermal explosion in ordinary matter [43].

In the presented experiments, the laser pulse created a localized melted spot in the crystal where the fluid MCI exists. The spatial temperature distribution T (r, t) in a continuous reactive medium, including damping, can be described by the following heat equation [42,43]

∂T ∂t = Q(T ) Cn 2D - 2γ d C (T -T 0 ) + χ r ∂ ∂r r ∂T ∂r (1)
with the initial condition

T (r, 0) = T 0 + E l 2πw 2 MZ n 2D C exp - r 2 2w 2 MZ (2)
where E l is the energy injected by the laser pulse, w MZ is the width of the melted zone, C ≃ 2 -3 is the heat capacity per particle, γ d is the Epstein damping rate (for our experimental conditions, γ d ≃ 1.21 ± 0.13 s -1 [44]), and χ is the thermal diffusivity.

In Eq. ( 1), the first term in the r.h.s. is the heat input due to the fluid MCI. Since fluid MCI essentially develops when the correlations between the particles are largely destroyed (so that a horizontal layer, uniform and of infinite extent, can be considered [31]), it supposes a significant overheating compared to the melting temperature T m of a 2D complex plasma crystal (T m ∼ 10-20 eV in our condition [45,46]). Consequently, the MCI heat source term requires the introduction of an activation temperature T a well above T m . It was indeed demonstrated that pair correlations are destroyed when the monolayer temperature is well above the melting temperature [47,48]. Accordingly, the fluid MCI heat source can be roughly modeled as a reaction rate governed by the Arrhenius law

Q(T ) Cn 2D = γ mci T ∞ C e -Ta/T , (3)
which can be approximated by [42] Q(T )

Cn 2D = 0, T < T a , γmciT∞ C , T ≥ T a , (4)
where γ mci T ∞ is the saturated fluid MCI heat source with the fluid MCI growth rate γ mci and the saturation temperature T ∞ ∼ 1 -3 keV. In a recent study on the melting front propagation in MCI-induced 2D crystal melting, the activation temperature was found to be T a ∼ 150 eV [42]. The saturation temperature T ∞ arises from the thermal spreading of the monolayer which inhibits the mode crossing and consequently the fluid MCI growth rate [31]. Since T 0 ≪ T a , we can neglect T 0 . Eq. ( 1) can be then written in a dimensionless manner

∂Θ ∂τ = λe -1/Θ -ΓΘ + 1 r ∂ ∂r r ∂Θ ∂r , (5)
where Θ = T /T a , r = r/r * with r

* 2 = E l /n 2D CT a , τ = t/t * with t * = r * 2 /χ. The initial condition becomes Θ(r, 0) = 1 2π w2 MZ exp - r2 2 w2 MZ , (6)
with wMZ = w MZ /r * . The normalization 2π 0 ∞ 0 Θ(r, 0)rdrdϕ = 1 determines the characteristic length scale. Thus, in its dimensionless form, the problem is characterized by two numbers

λ = γ mci T ∞ E l C 2 χn 2D T 2 a (7)
and

Γ = 2γ d E l C 2 χn 2D T a = 2γ d γ MCI T a T ∞ λ . (8)
Eq. ( 5) is similar to the one describing impulsive spot heating and thermal explosion in ordinary matter with the addition of a dimensionless damping coefficient Γ [42,43]. When there is no damping and the heating of a reactive medium occurs within a sufficiently localized spot (so it can be accurately described with a Dirac distribution), the subsequent thermal evolution is characterized only by λ which identifies a bifurcation between two distinct regimes. When exceeding a critical threshold, the heat equation exhibits rapid temperature growth. For a 2D system, the critical value is λ cr (Γ = 0) = 9.94 [43]. The damping term increases the critical value: numerical estimates are, for example, λ cr (Γ = 1.5) ∼ 17 and λ cr (Γ = 2.5) ∼ 21. In our experimental conditions, we assume that the thermal diffusivity of the crystal was χ ∼ 10 mm 2 /s [16] and γ mci ∼ γ g ∼ 0.9 s -1 . By fitting the temperature distribution after laser excitation with a 2D Gaussian for experiments near threshold, the width of the melted zone was w MZ ∼ 1 -2 mm [START_REF]area defined by RMZ includes ∼ 90% of the energy, giving RMZ ∼ 2wMZ which is in agreement with the measured values[END_REF], which is noticeably smaller than the characteristic length r * ∼ 4.5 mm. These experimental parameters yield the value of Γ ∼ 2 -4 and, at threshold energy, λ ∼ 10 -30, in remarkable agreement with the theoretical estimate. Note that the width of the MZ has a significant effect on the energy threshold since at a given injected energy a wider spot results in a significant portion of the energy being distributed over particles with a kinetic temperature below the activation temperature. Moreover, for large melted spots (w MZ ≥ r * ), the problem is no longer characterized by a single dimensionless number and the threshold depends on the temperature of the melted spot and its size [43,[START_REF] Merzhanov | [END_REF].

In Fig. 6, the evolution of the temperature profile is presented for experiments 19 and 20 and the numerical solution of Eqs. ( 1) and ( 4). The input parameters have been chosen to reproduce qualitatively the experimental observations. As can be seen in all cases, the energy is transported from the MZ to the rest of the crystalline suspension. The instability provided energy to the microparticle suspension inside the fluid region which is partially transported in the crystal through heat conduction and partially dissipated through neutral damping. For these reasons, the size of the melted area can slightly increase even if the total input energy is below the required melting threshold. As reported in the previous section, when increasing the amount of injected energy while staying below threshold (for example, increasing the duration of the laser pulse), the crystallization is delayed. This behavior was also observed in the numerical solutions and can be explained by the amount of energy pumped by the fluid MCI into the melted spot which depends directly on its size. Indeed, crystallization can occur only after enough energy has been removed from the MZ. Consequently, this suppresses the fluid MCI heat flux (Fig. 6a andc). This transition period is longer for larger spots since a significantly larger amount of energy has to be removed due to the fluid MCI energy input. When the injected energy is slightly above the threshold, fluid MCI delivers more energy to the monolayer than what can be extracted away from the MZ which grows as a result (Fig. 6b andd). The melting pattern is then similar to the one observed in the regular (crystal) MCI induced melting of a crystal [29][30][31][32]42].

In all experiments in which the melting was successfully induced, the MZ radius grew at a constant velocity. For r large enough in Eq. ( 5), the last term of the r.h.s. tends asymptotically towards ∂ 2 Θ/∂r 2 , reducing the problem to one dimension. The reaction model [START_REF] Morfill | [END_REF] or (4) in the evolution equation ( 1) defines a bistable system, with two uniform stationary solutions (in the reduced version with T 0 = 0 and the rate equation (4), these states are 0 and Θ max = λ/Γ). With inhomogeneous boundary conditions, the front profile is a heteroclinic connection between these two states, propagating at constant velocity v MZ . As the MZ radius grows linearly, a self-similar solution Θ(r, τ ) = Θ(X), with X = r -ṽMZ τ , where ṽMZ is the dimensionless velocity of the melting front, can be obtained analytically using the approximation (4) [42] Θ

(X) = exp(-σ 1 X), X > 0, Θ max + (1 -Θ max ) • exp(-σ 2 X), X ≤ 0, (9)
with σ 1 = (ṽ MZ /2) + 4Γ + ṽ2 MZ /2 > 0 and σ 2 = (ṽ MZ /2) -4Γ + ṽ2 MZ /2 < 0. The boundary conditions are Θ(+∞) = 0 and Θ(-∞) = Θ max . The position of the melting front is given by Θ(0) = 1 and its velocity is ṽMZ = -∂ τ Θ/∂ X Θ. It exists only if the two partial derivatives are continuous, giving the condition

(1 -Θ max )σ 2 = σ 1 . (10)
The velocity of the melting front is then straightforwardly

obtained ṽMZ = λ -2Γ √ λ -Γ . ( 11
)
For λ > 2Γ (viz. Θ max > 2), the front velocity is positive, which corresponds to the melting of the monolayer, due to the fact that the final temperature is further away from the threshold T a than the cold crystal is (T ∞ -T a > T a -T 0 = T a ). For Γ < λ < 2Γ (viz. 1 < Θ max < 2), the front velocity is negative, corresponding to recrystallization due to the fact that the melted state equilibrium temperature is closer to the activation threshold than the cold crystal is. The case λ = 2Γ is the bifurcation case, for which the front would be stationary. It would be interesting to further investigate this range, recalling that model ( 4) is very crude and does not yield accurate values for the actual front velocity, though being qualitatively relevant. Under our experimental conditions, ṽMZ ∼ 0.5 -5 or v MZ ∼ 1 -10 mm/s, which is in agreement with our measured value. This velocity yields the spatial decay exponents for the front,

σ 1 = (λ -Γ)/ √ λ -Γ and σ 2 = ṽMZ -σ 1 = -Γ/ √ λ -Γ.
The slope of the front and the temperature difference Θ max yield an estimate Θ max /σ 1 = λ/(Γ √ λ -Γ) for its width. Given the experimental ranges of λ and Γ, the front width is Θ max /σ 1 ∼ 1 -3 or in physical units 4.5 -13.5 mm. This value is close to the one observed in Figs. 4 and6.

Finally, in the frame of our model, the evolution of the total kinetic energy in the monolayer K tot can be obtained

K tot (t) = 2πn 2D ∞ 0 T (r, t)rdr . (12)
Multiplying Eq. ( 1) by n 2D and integrating over space, one obtains

Ktot + 2γ d C K tot = S (13)
where the heat source S due to the fluid mode-coupling instability is (using the reaction model ( 4))

S = 2π ∞ 0 Q(T ) C rdr ≃ 2πn 2D γ mci T ∞ C ra(t) 0 rdr ≃ πn 2D γ mci T ∞ C r 2 a (t) = πn 2D γ mci T ∞ C v 2 MZ t 2 , ( 14
)
where r a (t) is the front radius, such that T (r a , t) = T a . The solution of Eq. ( 13) is trivial and one immediately sees that, in the limit t → ∞, K tot ∝ t 2 . This result cannot be confirmed at the present time by experimental observations since the video recordings of the laserinduced melting were not long enough. In addition, using the reaction model ( 4), the stage of exponential growth reported in the experiments is lost. It has indeed been reported that, during MCI-induced melting in crystals, the microparticles kinetic temperature in the MZ grows exponentially until it saturates [32]. Moreover, in the present study, the total kinetic energy growth in the FoV seems to be nearly exponential (even though the videos were not long enough to recover the kinetic energy evolution over a significant energy interval to draw a firm conclusion). However, using the reaction model (3), the stage of exponential growth can be recovered. Indeed, the "source term" in ( 5) is Σ(Θ) = λ exp(-1/Θ) -ΓΘ .

For λ > Γ exp(1) (which is always the case for λ > λ cr ), the source term Σ has two nonzero real roots: the unstable one Θ 1 = -1/W -1 (-Γ/λ) < 1 and the stable one Θ 2 = -1/W 0 (-Γ/λ) > 1 where W k is the k th branch of the Lambert W function. The maximum of Σ occurs at Θ gm = -1/(2W 0 (-0.5 Γ/λ)) such that Θ 1 < Θ gm < Θ 2 . Since Σ ′ (Θ 1 ) > 0, the stage of exponential growth can be explained by the lower, unstable root. In the early stage of the instability, the dimensionless heat equation can be approximated by

∂Θ ∂τ - 1 r ∂ ∂r r ∂Θ ∂r ≃ Σ ′ (Θ 1 ) • (Θ -Θ 1 ) . ( 16
)
Given the experimental ranges of λ and Γ, the unstable root is 0.28 < Θ 1 < 0.34 and the instability growth rate is 5 < Σ ′ (Θ 1 ) < 8. Using our experimental parameters, the time scale is t * = 2 -3 s giving a growth rate 1.7 s -1 < γ g < 4 s -1 . This range slightly overestimates the reported experimental value but, given the simplicity of our reaction rate model, it is in remarkable agreement. It nevertheless calls for more investigations and better modeling of the fluid MCI heat source in order to obtain the full physical picture of the energy input due to the fluid MCI.

V. CONCLUSION

To conclude, we have experimentally demonstrated that wake-mediated resonant mode coupling can be in-duced in a 2D plasma crystal through a localized pulsed laser heating. Due to the co-existence of a stable crystalline state and a melted state experiencing fluid MCI, this heating can trigger a rapid full melting of the crystalline monolayer. In a stable crystal (with respect to crystalline MCI), an energy threshold was observed. Below the threshold, recrystallization was always observed with a transition period which grew longer when approaching the threshold. The localized fluid MCI could not pump enough energy to the microparticle suspension to maintain the fluid state and propagate through the crystal. Above the threshold, the full melting of the crystal was observed due to the activated fluid MCI (i.e. the crystalline order of the whole complex plasma monolayer was rapidly destroyed). Remarkable similarities with impulsive spot heating in ordinary reactive matter were reported.

In future investigations, detailed studies of the threshold for various crystal parameters and laser spot sizes will be performed. The influence of the temperature of the fluid state on the MCI growth rate will be of particular interest.

FIG. 1 .

 1 FIG. 1. (Color online) Sketch of the experimental setup. The left inset shows an elementary cell of the hexagonal lattice and the frame of reference chosen in this article. The orientation of the wave vector k is measured in respect to the x-axis. The right inset is a snapshot of the crystal after 0.25 s of laser heating. The red ellipse shows where the laser light beam interacts with the monolayer.

FIG. 2 .

 2 FIG. 2. (Color online) Longitudinal and transverse current fluctuation spectra at an angle θ = 0 • . The dotted black curves represent the fitted dispersion relations.

FIG. 3 .

 3 FIG. 3. (Color online) State of the monolayer (red, melted or black, unmelted) at the end of the experiment as a function of Kt(ton + t th ). Each experiment of Table I is indicated by an arrow of the appropriate color. A threshold at ∼40 keV is clearly identified. The inset shows the transition time before recrystalisation sets in for injected energies below the melting threshold. The blue line is a guide for the eye.

FIG. 4 .

 4 FIG. 4. (Color online) |Ψ6| maps (a,c) and energy maps (b,d) of the crystal during Exps. 19 (a,b) and 20 (c,d) (see TableI). Each map is a square of 24.6×24.6 mm 2 .

FIG. 5 .

 5 FIG. 5. (Colour online) (a) Evolution of the sum the kinetic energies of the particles in the field of view and in the melted zone as a function of time for Exps. 19 and 20 (see TableI). (b) Evolution of the radius of the melted zone as a function of time.

FIG. 6 .

 6 FIG. 6. (Color online) Evolution of the temperature profile. (a) Experiment 19: recrystallization, (b) Experiment 20: melting. Solution of Eqs. (1) using the reaction model (4): (c) slightly below threshold, (d) slightly above threshold. For the numerical profiles, the following parameters were used: χ = 9 mm 2 /s, γ d =1.2 s -1 , γMCI = 0.9 s -1 , T∞=1800 eV, Ta=105 eV, n2D=6.75 mm -2 , C = 2, wMZ=1.15 mm and (c) E l =40103 eV (λ = 24.2497, Γ = 3.7722) and (d) E l =40103.7 eV (λ = 24.2506, Γ = 3.7722).

ACKNOWLEDGMENTS

This work was partially supported by the French-German PHC PROCOPE Program (Project No. 35325NA/57211784). The authors would like to thank Prof. E. Thomas Jr and Dr I. Laut for fruitful discussions and suggestions. We would also like to thank the anonymous referees who have helped us to improve the article.