Evelia Rosa Garc

Í A Barroso

Pedro Daniel

Gonz Ález
email: pgonzalez@mat.ucm.es

P Érez

Patrick Popescu-Pampu

VARIATIONS ON INVERSION THEOREMS FOR NEWTON-PUISEUX SERIES

Keywords: Mathematics Subject Classification. 14B05 (primary), 14H20, 32S25 Branch, Characteristic exponents, plane curve singularities, hypersurface singularities, quasiordinary series, Lagrange inversion, Newton-Puiseux series

Let f px, yq be an irreducible formal power series without constant term, over an algebraically closed field of characteristic zero. One may solve the equation f px, yq " 0 by choosing either x or y as independent variable, getting two finite sets of Newton-Puiseux series. In 1967 and 1968 respectively, Abhyankar and Zariski published proofs of an inversion theorem, expressing the characteristic exponents of one set of series in terms of those of the other set. In fact, a more general theorem, stated by Halphen in 1876 and proved by Stolz in 1879, relates also the coefficients of the characteristic terms of both sets of series. This theorem seems to have been completely forgotten. We give two new proofs of it and we generalize it to a theorem concerning irreducible series with an arbitrary number of variables.

Let f px, yq be a polynomial with complex coefficients and without constant term. In his Treatise of fluxions and of infinite series [29, Sect. XXIX-XXXIII], Newton described an iterative method to compute a formal power series η P Crrtss, such that f px, ηpx 1{n qq " 0 for certain positive integer n. In his 1850 paper [START_REF] Puiseux | Recherches sur les fonctions algébriques[END_REF], Puiseux proved that the series produced by Newton's algorithm were convergent whenever one starts from a convergent series f px, yq P Crrx, yss. Since then, the series whose exponents are positive rational numbers with bounded denominators, be they convergent or not, are called either Newton-Puiseux series or Puiseux series. In the sequel we will use the first denomination.

In the years 1870, Smith [START_REF] Smith | On the higher singularities of plane curves[END_REF] and Halphen [START_REF] Halphen | Sur une série de courbes analogues aux développées[END_REF] realized that for several questions about the singularities of plane algebraic curves, a finite number of the exponents of a Newton-Puiseux series were more important than the others. Halphen called those special exponents characteristic. Their modern definition is the following one: if one looks at the sequence of exponents taken in increasing size, then the characteristic ones are exactly those at which jumps the lowest common denominator which may be used for the exponents up to that point. It is a basic fact that if f px, yq P Crrx, yss is irreducible, then all the associated Newton-Puiseux series have the same sequence of characteristic exponents. Let us call it the characteristic sequence of f px, yq relative to x.

Nowadays, one describes usually the importance of this notion as follows. Consider a branch C (that is, an irreducible germ of curve) on a germ of smooth complex analytic surface S. If px, yq are local coordinates on S and f px, yq P Crrx, yss is a defining function of C relative to those coordinates, then one may consider its characteristic sequence relative to x. This sequence is independent of px, yq once the y-axis is transversal to the branch C, that is, once its tangent does not coincide with the reduced tangent cone of C, which is a line. One speaks then of the generic characteristic sequence of C. Its main property is that it is a complete invariant of the embedded topological type of the branch in the ambient germ of surface (see for instance [START_REF] Wall | Singular points of plane curves[END_REF]Theorem 5.5.8]). In fact, most computations of other topological invariants of the pair pS, Cq are done in terms of its generic characteristic sequence.

It is nevertheless important to work also with Newton-Puiseux series computed relative to non-generic coordinate systems. For instance, another usual way to study the branch C is to perform its process of embedded resolution by blow-ups of points (see for instance [START_REF] Wall | Singular points of plane curves[END_REF]Chapter 3]). A basic problem is then to express the generic characteristic sequence of the strict transform of C obtained after one blow up in terms of that of C. If one starts from a generic Newton-Puiseux series ηpx 1{n q of C, then x ´1 ¨ηpx 1{n q is a Newton-Puiseux series of the strict transform of C. This series is generic for the strict transform if and only if the x-order of ηpx 1{n q is at least 2. In this case, it is immediate to get from it the generic characteristic sequence of the strict transform. But how to proceed when this is not the case?

One gets the following problem, in whose formulation we replaced for simplicity the strict transform by the initial branch: to compute the generic characteristic exponents of C in terms of those of a Newton-Puiseux series ηpx 1{n q of a defining function f px, yq, when C is tangent to the y-axis. But in this case C is necessarily transversal to the x-axis. Therefore, if ξpy 1{m q is a Newton-Puiseux series of f px, yq relative to y, that is, if f pξpy 1{m q, yq " 0, then its characteristic sequence is exactly the generic characteristic sequence of C. Consequently, it is enough to express the characteristic sequence of ξpy 1{m q in terms of that of ηpx 1{n q. Such an inversion theorem (called in this way because one inverts the roles of x and y in passing from ηpx 1{n q to ξpy 1{m q) is well-known and it is often attributed to Abhyankar's paper [START_REF] Abhyankar | Inversion and invariance of characteristic pairs[END_REF] of 1967 or to Zariski's paper [START_REF] Zariski | Studies in equisingularity III. Saturation of local rings and equisingularity[END_REF] of 1968. Proofs of this inversion theorem can be also found in [START_REF] Casas-Alvero | Singularities of plane curves[END_REF]Section 5.6], [START_REF] De Jong | Local analytic geometry[END_REF]Theorem 5.2.21], [START_REF] Popescu-Pampu | Approximate roots[END_REF]Proposition 4.3] and [START_REF] Abhyankar | Inversion and invariance of characteristic terms: Part I[END_REF]Page 111].

We were very surprised to discover that in his 1876 paper [20, page 91], Halphen had already formulated a stronger result than the previous inversion theorem. He did not provide a proof of it. As far as we know, the first proof was given by Stolz [START_REF] Stolz | Die Multiplicität der Schnittpunkte zweier algebraischer Curven[END_REF]Sect. 3] in 1879. For this reason, we will speak in the sequel about the Halphen-Stolz inversion theorem. It is stronger than the inversion theorem of Abhyankar-Zariski because it does not only provide formulae for the characteristic exponents of ξpy 1{m q in terms of those of ηpx 1{n q, but also for the corresponding coefficients. The previous papers of Halphen and Stolz seem to be forgotten, even though they were mentioned in Halphen's appendix [START_REF] Halphen | Étude sur les points singuliers des courbes algébriques planes. Appendix to G. Salmon's book Traité de géométrie analytique (courbes planes)[END_REF] to Salmon's treatise on plane algebraic curves, which is cited sometimes nowadays.

The aim of our paper is to extend the Halphen-Stolz inversion theorem to an arbitrary number of variables. We achieve this aim in Corollary 5.21 of our Inversion Theorem 5.20.

In order to arrive at those results, we give first two new proofs of the classical Halphen-Stolz inversion theorem (stated by us as Corollary 4.5 of Theorem 4.4). The first one is based on the relations between the coefficients associated to the irreducible exponents of an invertible power series (see Definition 3.2), those of its powers and those of its dual (see Proposition 3.16). A flow-chart representing our line of reasoning for this first proof is drawn in diagram (4.17). Our second proof uses a formula expressing all the coefficients of the Newton-Puiseux series ξpy 1{m q in terms of those of ηpx 1{n q (see Proposition 4.10). This formula, based on the Lagrange inversion theorem (see Theorem 4.9), generalizes the Halphen-Stolz inversion formula for the coefficients. It is the first proof which we extend into a proof of the several-variables case. We could have given directly the most general statements and proofs. We preferred to start explaining in a detailed way the classical case, because it served us as a model for building the general proof, and because we feel that in this way the paper is easier to read. Its title is inspired by the title of Griffiths' paper [START_REF] Griffiths | Variations on a theorem of Abel[END_REF].

As the generic characteristic sequence is crucial for understanding the embedded topology of complex plane branches, we expect that the associated coefficients could play an important role in problems related not only to their topology, but also to their analytical type. It is the main reason why we considered that it is important to bring to the attention of researchers the forgotten inversion theorem of Halphen-Stolz.

Zariski's proof of the inversion theorem in [START_REF] Zariski | Studies in equisingularity III. Saturation of local rings and equisingularity[END_REF] was obtained as an application of the theory of saturation of local rings, in connexion with the study of topological equisingularity. There exist other notions of saturation, for instance, the Lipschitz-saturation (see [START_REF] Pham | Saturation Lipschitzienne d'une algèbre analytique complexe et saturation de Zariski[END_REF][START_REF] Lipman | Relative Lipschitz-saturation[END_REF]) and also the presaturation of Campillo, which is better adapted to positive characteristic (see [START_REF] Campillo | On saturations of curve singularities (any characteristic)[END_REF][START_REF] Campillo | Arithmetical aspects of saturation of singularities[END_REF]). In the case of irreducible germs of quasiordinary hypersurface singularities, Zariski's results on saturation and Lipman's inversion theorem appear also in the combinatorial characterization of the embedded topological type of this class of singularities (see [START_REF] Gau | Embedded topological classification of quasi-ordinary singularities[END_REF][START_REF] Lipman | Topological invariants of quasi-ordinary singularities[END_REF]). We expect that these lines of research combined with our generalized inversion theorem will lead to a better understanding of the invariants of singularities of other classes of hypersurface germs.

The article is structured as follows. In Section 2 we recall basic facts about Newton-Puiseux series associated to plane branches and their characteristic exponents. In Section 3 we introduce the notions of irreducible and essential exponents of a series and we give some results relating the coefficients of certain pairs of invertible series. In Section 4 we explain our proofs of the Halphen-Stolz inversion theorem and of its generalization into an inversion formula for all coefficients, based on a Lagrange inversion formula. Finally, in Section 5 we prove our generalization concerning an arbitrary number of variables and we explain in which way it extends the inversion theorem of Lipman concerning the characteristic exponents of quasi-ordinary series. Note that we work always over a fixed algebraically closed field of characteristic zero.

Acknowledgements. This research was partially supported by the French grants ANR-12-JS01-0002-01 SUSI, Labex CEMPI ANR-11-LABX-0007-01 and the Spanish grants MTM2012-36917-C03-0, MTM2013-45710-C2-2-P, MTM2016-76868-C2-1-P, MTM2016-80659-P. We are grateful to Herwig Hauser and Hana Kováčová for their translation of parts of Stolz' paper. The third-named author is grateful to Mickaël Matusinski for the opportunity to explain our first proof of the Halphen-Stolz theorem at the Geometry seminar of the University of Bordeaux. We thank him and the anonymous referee for their remarks on a previous version of this article, which allowed us to improve our presentation. We also thank Antonio Campillo for the information he sent us about the uses of Abhyankar-Zariski inversion.

Newton-Puiseux series and their characteristic exponents

In this section we introduce the notations and vocabulary about power series with integral or rational exponents which will be used throughout the text. Among the series with rational exponents, we will be interested only in those with bounded denominators, called Newton-Puiseux series. We conclude the section introducing the well-known notion of characteristic exponents of a Newton-Puiseux series.

Throughout the paper N denotes the set of non-negative integers, N ˚denotes Nzt0u and K denotes a fixed algebraically closed field of characteristic zero.

The following definition explains the basic vocabulary and notations about power series with integer exponents which will be used in the sequel: Definition 2.1. The ring Krrtss of entire series consists of the formal power series in the variable t, with coefficients in K and exponents in N. We say that the elements of its field of fractions Kpptqq are meromorphic series. They are exactly the series with coefficients in K, exponents in Z and a finite number of terms with negative exponents. If η P Kpptqq and m P Z, we denote by rηs m P K the coefficient of the monomial t m in ηptq and by Spηq Ă Z its support, consisting of the exponents m with non-zero coefficients rηs m . Therefore, a meromorphic series η P Kpptqq may be written as:

ηptq " ÿ mPSpηq rηs m t m .
We will also use series with rational exponents, but such that their support has bounded denominators: Definition 2.2. A Newton-Puiseux series ψ in the variable x is a power series of the form ηpx 1{n q, where ηptq P Krrtss and n P N ˚. For a fixed n P N ˚, they form the ring Krrx 1{n ss. Its field of fractions is denoted Kppx 1{n qq.

One extends immediately to Newton-Puiseux series ψ the notion of support (which is a set with bounded denominators in the sense of Definition 3.1) and the notation rψs m for their coefficients (where now m P Q `).

Denote by:

Krrx 1{N ss :" ď nPN ˚Krrx 1{n ss
the local K-algebra of Newton-Puiseux series in the variable x.

The algebra Krrx 1{N ss is endowed with the natural order valuation:

ord x : Krrx 1{N ss ÝÑ Q `Y t8u
which associates to each series ψ " ηpx 1{n q P Krrx 1{n ss the minimum of its support. The dominating coefficient of a Newton-Puiseux series ψ is the coefficient of its term of exponent ord x pψq.

The field of fractions of the ring Krrx 1{N ss of Newton-Puiseux series is:

Kppx 1{N qq :" ď nPN ˚Kppx 1{n qq.
One has the following fundamental theorem (see for instance, [10, Chapter 5.1], [START_REF] Fischer | Plane algebraic curves[END_REF]Chapter 7], [9, Chapter 2.1] or [38, Chapter 2] for a proof), which explains the reason why we need to work with Newton-Puiseux series even if we are interested primarily in series with integral exponents: Theorem 2.3 (The Newton-Puiseux theorem). Any monic reduced polynomial f P Krrxssrys of degree n P N ˚has n roots in Krrx 1{N ss. If f is moreover irreducible, then those roots are precisely the series of the form:

(2.1) ψ ρ :" ηpρ ¨x1{n q, where ψ " ηpx 1{n q P Krrx 1{n ss is any one of them and ρ P K ˚varies among the multiplicative subgroup G n of pK ˚, ¨q of n-th roots of 1 P K ˚.

Remark 2.4. If f P Krrxssrys is a monic irreducible polynomial of degree n P N ˚and if ψ P Kppx 1{n qq is a root of f , then the field extension:

Kppxqq Ă Kppxqqrys{pf q » Kppxqqrψs " Kppx 1{n qq
is Galois. Its Galois group is isomorphic to the group G n , acting on Kppx 1{n qq by:

(2.2) pρ, x 1{n q Ñ ρ ¨x1{n , for all ρ P G n .

The series ψ ρ in Theorem 2.3 are precisely the conjugates of ψ under this action.

Given a Newton-Puiseux series ψ, there exists an infinite number of choices of n P N ˚such that ψ P Krrx 1{n ss. This is simply due to the fact that Krrx 1{n ss Ă Krrx 1{m ss whenever n divides m. One may get nevertheless a canonical choice of n P N ˚by asking it to be minimal : Definition 2.5. If ψ P Kppx 1{N qq, a representation ψ " ηpx 1{n q with ηptq P Krrtss and n P N ˚is called primitive if n is the lowest common denominator of the exponents of ψ.

Example 2.6. Assume that ψ " x 5{2 `x8{3 . Then ψ " ηpx 1{6 q, with ηptq " t 15 `t16 . This defines a primitive representation of ψ. Writing now ψ " η 1 px 1{12 q with η 1 ptq " t 30 `t32 , one gets a non-primitive representation. Let us consider a 6-th root of unity ρ P K ˚. Then:

ψ ρ " ηpρ x 1{6 q " ρ 15 x 5{2 `ρ16 x 8{3 " ρ 3 x 5{2 `ρ4 x 8{3 .
Among the exponents of a Newton-Puiseux series, several are distinguished by looking at the way they may be written as quotients of coprime integers: Definition 2.7. Let ψ P Krrx 1{N ss be a nonzero Newton-Puiseux series with zero constant term. The set Epψq of characteristic exponents of ψ consists of the elements of the support of ψ which, when written as quotients of integers, need a denominator strictly bigger than the lowest common denominator of the previous exponents. That is:

Epψq :" tl P Spψq | N l ¨l R Zu , where N l :" min

" N P N ˚| pSpψq X r0, lqq Ă 1 N Z * .
The sequence of characteristic exponents of ψ P Krrx 1{N ss is defined by writing the elements of Epψq in increasing order.

Example 2.8. Both Newton-Puiseux series x 5{2 `x8{3 and 2x ´x5{2 `x8{3 ´3x 7{2 `x23{6 have the same set t5{2, 8{3u of characteristic exponents.

Remark 2.9. According to Enriques and Chisini [11, page 375], it was Smith [START_REF] Smith | On the higher singularities of plane curves[END_REF] and Halphen [START_REF] Halphen | Sur une série de courbes analogues aux développées[END_REF] who discovered in the years 1870 that special exponents of a Newton-Puiseux series are particularly important if one wants to compute the intersection number of two plane branches starting from their Newton-Puiseux series. This information was repeated by Zariski [39, Ch. 1], but without citing anymore their precise papers. Those special exponents were called characteristic by Halphen in [20, Sect. 1.1], which is also the paper in which he stated his inversion theorem for both exponents and coefficients. This denomination remained, but with slightly variable meanings (see also Remarks 2.12 and 4.8). Let us mention that Smith did not name those special exponents (which he defined in [START_REF] Smith | On the higher singularities of plane curves[END_REF]Sect. 8]).

The set Epψq of characteristic exponents of ψ is necessarily finite, even if the series has infinite support. More precisely, if ψ P Krrx 1{n ss, then Epψq has at most as many elements as the number of factors of the prime factorisation of n. The set Epψq may also be characterized using the Galois action, as the set of orders ord x pψ ρ ´ψq, when ρ varies in G n zt1u (see for instance [START_REF] Wall | Singular points of plane curves[END_REF]Prop. 4.13]). Note that, because all the conjugates ψ ρ have the same support (by the explicit description of the Galois action recalled in Remark 2.4), they also have the same set of characteristic exponents, a fact implicitly used in Definition 2.11 below.

One has the following particular case of the Weierstrass preparation theorem (see for instance [START_REF] De Jong | Local analytic geometry[END_REF]Chap. 3.2] or [START_REF] Fischer | Plane algebraic curves[END_REF]Chap. 6]): Theorem 2.10. Let f P Krrx, yss be a series such that ord y pf p0, yqq " n P N ˚. Then, there exist a unique monic polynomial F P Krrxssrys of degree n and a unique unit ǫ P Krrx, yss such that:

(2.3) f " ǫ ¨F.
In addition, f is irreducible in Krrx, yss if and only if the polynomial F is irreducible in Krrxssrys.

This theorem allows us to introduce the following vocabulary:

Definition 2.11. Let f P Krrx, yss be an irreducible series such that ord y f p0, yq " n P N ˚. The polynomial F P Krrxssrys provided by Theorem 2.10 is called the Weierstrass polynomial associated to the series f P Krrx, yss relative to x. Then, the Newton-Puiseux series of f relative to x are the roots of the associated Weierstrass polynomial F P Krrxssrys, in the ring Krrx 1{N ss. If f is irreducible, then its characteristic exponents relative to x are the characteristic exponents of any one of those roots.

Remark 2.12. Let us explain how the previous algebraic notions apply in the geometrical setting of a branch, an irreducible germ of complex analytic plane curve C on a germ S of smooth complex analytic surface. Choose a local system of coordinates px, yq on S. Then, the branch C is defined by an irreducible series f P Crrx, yss. If the y-axis is transversal to the tangent line of the branch C, then one may show that the characteristic exponents of f relative to x are always the same (see for instance [START_REF] Wall | Singular points of plane curves[END_REF]Thm. 3.5 Zariski uses also a characteristic sequence pβ 0 , β 1 , ..., β g q of natural numbers instead of the sequence of generic characteristic exponents, which may then be reconstructed as ´β1 β0 , ..., βg β0 ¯(here β 0 is the multiplicity of the branch C, that is, the minimal degree of the monomials of f px, yq). We do not use the previous notations in this paper for two reasons: on one side we never need a genericity hypothesis on the coordinate system relative to C and on the other side we find the related notion of essential exponent relative to 1 (see Definition 3.6) better suited to a simple formulation of the Halphen-Stolz inversion theorem (see Remark 4.7).

A calculus for the irreducible terms of invertible entire series

In this section we introduce several notions allowing to study the supports of Newton-Puiseux series and the semigroups generated by them: their irreducible elements (see Definition 3.2) and their essential elements (see Definition 3.6) relative to an arbitrary natural number. We concentrate then on the entire series with non-zero constant terms. If φ is such a series, we introduce its dual q φ and we show that the coefficients of the monomials with irreducible exponents in the positive integral powers of φ and in the dual q φ may be deduced from those of φ by simple formulae (see Proposition 3.16). In Section 4 we will apply those formulae to the essential exponents relative to well-chosen natural numbers, in order to prove the Halphen-Stolz inversion theorem.

Next definition introduces vocabulary about the sets of rational numbers which may appear as supports of Newton-Puiseux series: Definition 3.1. A set with bounded denominators is a non-empty (possibly infinite) set E Ă Q such that there exists n P N ˚with E Ă 1 n N. We denote by N ˚E Ă Q `the semigroup generated by E, that is, the set of non-empty finite sums of elements of E. Analogously, we denote by Z E Ă Q the group generated by E.

Note that the semigroup N ˚E contains 0 (that is, it is a monoid for addition) if and only if E does. Given a set with bounded denominators, we will be interested in its irreducible elements: Definition 3.2. Assume that E Ă Q `is a set with bounded denominators. We denote by IrrpEq the set of irreducible elements of E, that is, the subset of E formed by those elements which cannot be written as sums of at least two elements of Ezt0u. The elements of E which are not irreducible are called reducible. If E is the support Spψq of a Newton-Puiseux series ψ, then we write also Irrpψq :" IrrpSpψqq, and we call it the set of irreducible exponents of ψ.

Remark 3.3. Let E Ă Q `be a set with bounded denominators. Notice that if E contains 0, then 0 is by definition an irreducible element of E. More generally, the minimum of E is always irreducible in E.

Example 3.4. Assume that E " t6, 15, 16, 21, 23u. Then IrrpEq " t6, 15, 16, 23u. Note that 21 is reducible in E, because it is equal to the sum 6 `15 and 6, 15 P E.

The sets of irreducible elements of E and of the semigroup it generates coincide: Lemma 3.5. Assume that E Ă Q `is a set with bounded denominators. Then:

(1) IrrpEq " IrrpN ˚Eq and this set is the minimal generating set of the semigroup N ˚E, relative to the inclusion partial order on the set of its generating sets. (2) The set IrrpEq is finite.

Proof. Multiplying E by a convenient rational number, we may restrict to the sets E Ă N whose elements are globally coprime.

(1) Both inclusions between IrrpEq and IrrpN ˚Eq are immediate to check, therefore we will assume from now on that the two sets are equal.

Let us prove the minimality property of IrrpN ˚Eq. Consider another generating set A of N ˚E and a P IrrpN ˚Eq. As A is generating, a may be written as a sum of elements of A. If this sum were non-trivial, then a would not be irreducible in N ˚E. Therefore a P A, which shows the desired inclusion IrrpN ˚Eq Ă A.

(2) In order to show that IrrpEq is finite, it is enough to show that IrrpN ˚Eq is finite. The semigroup N ˚E being generated by globally coprime elements, it has finite conductor, that is, there exists c P N such that all natural numbers greater than or equal to c belong to N ˚E (see [38, page 82]; in this case, the smallest such c is called the conductor of the numerical semigroup N ˚E).

But this implies that IrrpN ˚Eq Ă t0, 1, ..., 2c ´1u. Indeed, any element l ě 2c of the semigroup may be written in the form c `d, with d ě c, that is, as a non-trivial sum of elements of the semigroup.

We will be especially interested in particular sequences of irreducible elements of a given set E with bounded denominators: Definition 3.6. Let us consider a set E Ă Q `with bounded denominators and an integer p P N ˚. Then the sequence esspE, pq :" pesspE, pq l q l of essential elements of E relative to p is defined inductively by: ' esspE, pq 0 :" min E. ' If l ě 1, then the term esspE, pq l is defined if and only if E is not included in the group Ztp, esspE, pq 0 , ..., esspE, pq l´1 u. In this case: esspE, pq l :" min pEzZtp, esspE, pq 0 , . . . , esspE, pq l´1 uq .

The following basic property of this notion is a direct consequence of the definition:

Lemma 3.7. Assume that E Ă Q `has bounded denominators and take p, q P N ˚. Then:

q esspE, pq " ess pqE, qpq .

One has also:

Lemma 3.8. Assume that E Ă Q `has bounded denominators and that p P N ˚. Then the sequence of essential exponents esspE, pq is finite.

Proof. Lemma 3.7 implies that it is enough to consider the case where E Ă Z. Then one has by definition the strict inclusions:

Ztp, esspE, pq 0 , . . . , esspE, pq l´1 u Ĺ Ztp, esspE, pq 0 , . . . , esspE, pq l u, for all l ě 1 (for which the term esspE, pq l is defined). Any ascending chain of subgroups of Z being stationary, we deduce that the sequence of essential exponents is finite. ( is precisely the support of the second series considered in Example 2.8, whose sequence of characteristic exponents is `5 2 , 8 3 ˘. Note that its sequence of essential exponents relative to 1 may be obtained from the characteristic sequence by adjoining to it as initial term the order of this series (which is in this case equal to 1). We will see in Lemma 3.13 that this is a general fact.

The following lemma shows that the non-zero essential elements of a set E relative to any positive integer are irreducible elements of E: Lemma 3.10. The essential elements of a set E Ă Q `with bounded denominators relative to a number p P N ˚are irreducible elements of E.

Proof. If esspE, pq " pǫ 0 , . . . , ǫ d q, then we have that ǫ 0 " min E, which is an irreducible element of E. Let us show that the property is also true for ǫ l , where l ě 1. If ǫ l was reducible, then it would be a non-trivial sum of elements of E, which would therefore be strictly less than ǫ l . By the definition of ǫ l , the terms of this sum would belong to the group Ztp, ǫ 0 , . . . , ǫ l´1 u. This would imply that ǫ l belongs also to this group, which contradicts Definition 3.6. Lemma 3.11. Assume that E Ă Q `has bounded denominators and that p P N ˚. Then we have the following equality of essential sequences: esspE, pq " esspIrrpEq, pq.

Proof. If esspE, pq " pǫ 0 , . . . , ǫ d q and esspIrrpEq, pq " pǫ 1 0 , . . . , ǫ 1 d 1 q, then Remark 3.3 and Definition 3.6 imply that: ǫ 0 " min E " min IrrpEq " ǫ 1 0 . Assume by induction that ǫ 0 " ǫ 1 0 , . . . , ǫ l´1 " ǫ 1 l´1 , for 1 ď l ă d. Then we get: ǫ 1 l " minpIrrpEqzZtp, ǫ 0 , . . . , ǫ l´1 uq ď ǫ l " minpEzZtp, ǫ 0 , . . . , ǫ l´1 uq, since ǫ l is an irreducible element of E by Lemma 3.10. The inclusion IrrpEq Ă E implies also that ǫ l ď ǫ 1 l , hence ǫ l " ǫ 1 l . This proves that ǫ l " ǫ 1 l for 0 ď l ď d and d ď d 1 . By Lemma 3.10 and the definition of the essential exponents, one has the inclusions IrrpEq Ă E Ă Ztp, ǫ 0 , . . . , ǫ d u, which imply that d 1 " d. Definition 3.12. If ψ P Krrx 1{n ss is a non-zero series and p P N ˚, then we will write: esspψ, pq :" esspSpψq, pq, and we will speak about the sequence of essential exponents of ψ relative to p.

The characteristic exponents of a Newton-Puiseux series are intimately related to its essential exponents relative to 1: Lemma 3.13. Let pα 1 , . . . , α g q be the sequence of characteristic exponents of a series ψ P Krrx 1{n ss. It may be obtained from the sequence pǫ 0 , ǫ 1 , ..., ǫ d q of essential exponents of ψ relative to 1 in the following way:

' If ǫ 0 R Z, then g " d `1 and α i " ǫ i´1 for all i P t1, ..., d `1u. ' If ǫ 0 P Z, then g " d and α i " ǫ i for all i P t1, ..., du.

Proof.

' Consider first the case in which ǫ 0 R Z. As the first characteristic exponent is the minimal nonintegral exponent in the support of ψ, we deduce that α 1 " ǫ 0 . Assume by induction that α i " ǫ i´1 for all i P t0, ..., lu. Definition 3.6 implies that ǫ l is the first exponent of Spψq which is strictly greater than ǫ l´1 and which cannot be written as a fraction whose denominator is the least common denominator of the previous exponents in the support of ψ. By Definition 2.7, we get that α l`1 " ǫ l .

' Consider now the case in which ǫ 0 P Z. By definition, ǫ 0 P N cannot be a characteristic exponent of ψ and α 1 " ǫ 1 . The result follows by induction, using the same argument as in the first case.

In the rest of this section we will be especially interested in entire series with non-zero constant term, that is, in invertible elements of the multiplicative monoid pKrrtss, ¨q. They form a multiplicative group, which we will denote by pKrrtss ˚, ¨q.

Note that the entire series of the form t φptq for φ P Krrtss ˚, that is, the entire series of order 1, form the (non-commutative) group under composition of series which admit a reciprocal (an inverse for composition). We denote by pt Krrtss ˚, ˝q this group. Division by t transforms it bijectively into pKrrtss ˚, ¨q, but is not a morphism of groups. What is essential for us is that the inversion for composition becomes an involution of the set Krrtss ˚which has special properties with respect to the terms whose exponents are irreducible (see Proposition 3.16 (2)). We use the following vocabulary for this involution: Definition 3.14. If φ P Krrtss ˚, then its dual is the unique entire series q φ P Krruss ˚such that u q φpuq and t φptq are reciprocal. Remark 3.15. If φ P Krrtss ˚, then setting u " t φptq defines a change of variable in the ring Krrtss. Notice that Krrtss " Krruss and by Definition 3.14 the following equivalence holds:

(3.1)
u " t φptq ô t " u q φpuq.

We use two variables t and u in our notations for a dual pair of series, in order to relate them easily from the notational point of view to the two sets of Newton-Puiseux series of an irreducible f px, yq P Krrx, yss, which depend on the two variables x and y (see Section 4).

The following proposition expresses the coefficients of the positive integral powers and of the dual of an entire series φ P Krrtss ˚in terms of those of φ. It may be deduced from the statement and the proof of Wall's [START_REF] Wall | Singular points of plane curves[END_REF]Lemma 3.5.4]. But as we could not find it formulated in the literature and as it lies at the core of our first proof of the Halphen-Stolz theorem, we give a detailed proof of it. Proposition 3.16. Let φ P Krrtss ˚and N P N ˚. Then:

(1) Irrpφ N q " Irrpφq. Moreover rφ N s 0 " rφs N 0 and rφ N s r " N rφs N ´1 0 rφs r , for all r P Irrpφqzt0u.

(2) Irrp q φq " Irrpφq. Moreover r q φs 0 " rφs ´1 0 and r q φs r " ´rφs ´r´2 0 rφs r , for all r P Irrpφqzt0u.

Proof. One has:

(3.2) φptq " ÿ jPSpφq rφs j t j .
The hypothesis φ P Krrtss ˚translates into 0 P Spφq, that is, rφs 0 ‰ 0.

(1) Consider first the case of φ N . By equation (3.2), we have:

(3.3) φ N ptq " ÿ j1,.
..,jN PSpφq rφs j1 ¨¨¨rφs jN t j1`¨¨¨`jN .

' Let us show first that Irrpφq Ă Irrpφ N q and that one has the stated equalities between coefficients. Consider r P Irrpφq. If r " 0, one has obviously r P Irrpφ N q and rφ N s 0 " rφs N 0 . Assume therefore that r ą 0. The only way to write r as a sum of N elements of Spφq, is that one of them be equal to r, and the other ones vanish. There are N different positions in the sum for the non-vanishing one, therefore:

rφ N s r " N rφs N ´1 0 rφs r ,
which is the desired formula.

In particular, rφ N s r ‰ 0, which shows that r P Spφ N q. If r was reducible in Spφ N q, it could be written as a non-trivial sum of elements of Spφ N q. By formula (3.3), it would also be a non-trivial sum of elements of Spφq, which would contradict the fact that it is an irreducible element of Spφq. Therefore r P Irrpφ N q. ' Let us show now the reverse inclusion Irrpφ N q Ă Irrpφq. Consider an element r P Irrpφ N q. By formula (3.3), we know that it may be written as a sum of N elements of Spφq. In particular, it may be written as a sum of irreducible elements of Spφq. By the previous point, we know that those elements are also irreducible in Spφ N q. Our hypothesis r P Irrpφ N q implies that there is only one non-zero term in this sum, which proves the desired membership r P Irrpφq.

(2) Consider now the case of q φ. Write the analogue of (3.2) for q φ:

(3.4) q φpuq " ÿ kPSp q φq r q φs k u k .

As t φptq P Krrtss and u q φpuq P Krruss are reciprocal series, one has by definition the identity:

t " pt φptqq q φpt φptqq, which, after division by t and combination with the expansion (3.4), gives:

(3.5) 1 " ÿ kPSp q φq r q φs k t k φptq k`1 .
Expand now the powers φptq k`1 using equation (3.3). We get:

1 " ÿ k P Sp q φq j1, ..., j k`1 P Spφq r q φs k rφs j1 ¨¨¨rφs j k`1 t k`j1`¨¨¨`j k`1 .

Therefore:

(3.6) ÿ k P Sp q φq j1, ..., j k`1 P Spφq k `j1 `¨¨¨`j k`1 " p r q φs k rφs j1 ¨¨¨rφs j k`1 " 0, for all p P N ˚.

' Let us show first that Irrpφq " Irrp q φq. By Lemma 3.5 (1), the irreducible elements of a set are determined by the semigroup it generates. In order to show that the sets Irrpφq and Irrp q φq coincide, it is therefore enough to prove that:

(3.7) N ˚pSpφqq " N ˚pSp q φqq.
The situation being symmetric between φ and q φ, we may prove only the inclusion:

(3.8) N ˚pSp q φqq Ă N ˚pSpφqq.
We will argue by contradiction, assuming that the previous inclusion is false. Consider then:

(3.9) r P N ˚pSp q φqq z N ˚pSpφqq, which is minimal with this property. As 0 P Spφq , we have r ą 0. Apply equation (3.6) to p " r. Consider a tuple:

(3.10) pk, j 1 , ..., j k`1 q P Sp q φq ˆSpφq k`1 such that:

(3.11) k `j1 `¨¨¨`j k`1 " r.
Let us show that this implies the equality k " r. Reasoning again by contradiction, assume that k ă r.

As k P Sp q φq Ă N ˚pSp q φqq, the minimality of r shows that k P N ˚pSpφqq. Combining condition (3.10) and equation (3.11), we deduce that r P N ˚pSpφqq, which contradicts the assumption (3.9).

Therefore, if both (3.10) and (3.11) are true, then k " r, which implies that j 1 " ¨¨¨" j r`1 " 0. Hence there is only one term in the sum of the left-hand side of equation (3.6) for p " r, and we get: r q φs r rφs r`1 0 " 0, which contradicts the assumption that both coefficients r q φs r and rφs 0 are non-zero (as they are associated to elements of the supports of φ and q φ). Our proof of the inclusion (3.8) is finished. Therefore, as explained above, we get the desired equality Irrp q φq " Irrpφq.

' Let us prove the identities relating the coefficients associated to the irreducible exponents of φ and q φ. Consider r P Irrpφq " Irrp q φq. Look again at the tuples pk, j 1 , ..., j k`1 q satisfying the conditions (3.10) and (3.11) above.

If k R t0, ru, then at least one of the numbers j 1 , ..., j k`1 would not vanish. Equation (3.11) gives a non-trivial decomposition of r inside Spφq Y Sp q φq Ă N ˚pSpφqq p3.7q

" N ˚pSp q φqq, which shows that r R IrrpN ˚Spφqq. This contradicts Lemma 3.5 [START_REF] Abhyankar | On the ramification of algebraic functions[END_REF].

Therefore, one has necessarily k " 0 or k " r. Both possibilities determine completely pj 1 , ..., j k`1 q through equation (3.11). Applying equation (3.6) to p " r, we get:

(3.12) r q φs 0 rφs r `r q φs r rφs r`1 0 " 0.

The equalities (3.12) and r q φs 0 " rφs ´1 0 imply the formula for r q φs r written in the statement of the proposition.

Combining Proposition 3.16 with Lemma 3.11, we get: Corollary 3.17. Let φ P Krrtss ˚and N, p P N ˚. Then the sequences of essential exponents of φ, φ N and q φ relative to p coincide.

In the next section we apply Proposition 3.16 and its Corollary 3.17 in order to relate the essential exponents relative to 1 and their coefficients for the Newton-Puiseux series of an irreducible series f px, yq P Krrx, yss.

Applications to inversion formulae for Newton-Puiseux series

Let f px, yq P Krrx, yss be an irreducible formal power series. One has therefore associated Newton-Puiseux series relative to both coordinates x and y. In this section we prove in two ways the Halphen-Stolz theorem (Corollary 4.5), which relates the coefficients of the terms with essential exponents relative to 1 in both series. The first proof, summarized in the flow-chart (4.17), applies directly the results of the previous section. The second one passes through a more general result, allowing to compute recursively all the coefficients of one series in terms of those of the other one (see Proposition 4.10). In turn, this proposition is a consequence of a version of the classical Lagrange inversion formula (see Theorem 4.9).

The first proof of the Halphen-Stolz theorem.

There is no natural bijection between the Newton-Puiseux series of a formal power series f px, yq relative to x and y, for the simple reason that their numbers are in general different. We want to explain first that if one takes adequate roots of them, then one gets two sets which are naturally in a bijective correspondence (see Proposition 4.2).

Let us denote by ηpx 1{n q P Krrx 1{n ss a Newton-Puiseux series of f px, yq with respect to x, where η P Krrtss. We assume that the representation of this Newton-Puiseux power series is primitive (see Definition 2.5). The series η is of the form: η " a ¨tm `higher order terms, with m ą 0 and a P K ˚. Let us choose an m-th root ã P K ˚of a. Then, we have a unique m-th root t ηptq P Krrtss of η:

(4.1)
ηptq " pt ηptqq m , such that the series η has constant term rηs 0 " ã ‰ 0. Therefore, we get a " 1, n " 4 and ψ " ηpx 1{4 q, where ηptq " t 6 `c¨t 7 " t 6 p1`c¨tq. This shows that m " 6 and if ã " 1 then ηptq " p1 `c ¨tq 1{6 . By (4.20) below we have the expansion ηptq " 1 `řkPN

˚`1{6 k ˘ck ¨tk .
Let us come back to the general case.

Denote by ξpuq P Krruss ˚the dual series of ηptq (see Definition 3.14). Hence, one has the following equivalence (see (3.1)):

(4.2)
u " t ηptq ô t " u ξpuq.

As ηpx 1{n q is a Newton-Puiseux-series of f px, yq relative to x, we have:

f px, ηpx 1{n qq " 0.
Replacing x by t n and using the equality (4.1), we get:

(4.3) f pt n , pt ηptqq m q " 0.
By the equivalence (4.2), we deduce:

(4.4) f ppu ξpuqq n , u m q " 0.

Consequently, if one defines:

(4.5) ξpuq " pu ξpuqq n (an equation which is analogous to (4.1)), then one sees that:

f pξpy 1{m q, yq " 0, that is, ξpy 1{m q is a Newton-Puiseux series of f px, yq with respect to the variable y.

In fact, one has the following proposition:

Proposition 4.2.

(1) The map x 1{n ηpx 1{n q ÝÑ y 1{m ξpy 1{m q induced by the duality involution on Krrtss ˚, gives a bijection from the set of m-th roots of the Newton-Puiseux series of f px, yq relative to x to the set of n-th roots of those relative to y.

(2) If f pt, uq :" f pt n , u m q P Krrt, uss then:

-the series of the form t ηptq P Krrtss are the Newton-Puiseux series of f relative to t.

-the series of the form u ξpuq P Krruss are the Newton-Puiseux series of f relative to u.

Proof.

(1) The two sets have both mn elements and the given map is injective because the map η Ñ ξ is an involution. Therefore the given map is bijective.

(2) Equation (4.3) shows that the Newton-Puiseux series of f relative to the variable t are exactly those of the form t ηptq. The situation is analogous for the series of the form u ξpuq.

The following lemma relates special sequences of essential exponents of the series ηptq and ηptq on one side, and of the series ξpuq and ξpuq on another side: Lemma 4.3. Denote: " esspη, nq " pm, ǫ 1 , ..., ǫ d q, esspξ, mq " pn, ǫ 1 1 , ..., ǫ 1 d 1 q. Then: " esspη, gcdpn, mqq " p0, ǫ 1 ´m, ..., ǫ d ´mq, essp ξ, gcdpn, mqq " p0, ǫ 1 1 ´n, ..., ǫ 1 d 1 ´nq. Proof. By symmetry, it is enough to treat the case of the series η. Since η P Krrtss ˚, Proposition 3.16 implies that Irrpηq " Irrpη m q. Then, by Lemma 3.11, for any integer p P N ˚one has: esspη, pq " esspIrrpηq, pq " esspIrrpη m q, pq " esspη m , pq.

Thus, it is enough to prove that p0, ǫ 1 ´m, ..., ǫ d ´mq is the sequence of essential exponents of ηm relative to gcdpn, mq. By formula (4.1), we have that ηm " t ´mη, therefore: Spη m q " Spηq ´m. Using Definition 3.6, we see that we have to prove that:

' minpSpηq ´mq " 0. ' For all k P t1, ..., du: ǫ k ´m " min ppSpηq ´mqzZt gcd pn, mq, 0, ǫ 1 ´m, ..., ǫ k´1 ´muq. ' Spηq ´m Ă Zt gcd pn, mq, 0, ǫ 1 ´m, ..., ǫ d ´mu. But all these facts are immediate from the definition of the essential exponents ǫ i , because:

Zt gcd pn, mq, 0, ǫ 1 ´m, ..., ǫ k´1 ´mu " Ztn, m, ǫ 1 , ..., ǫ k´1 u, for all 1 ď k ď d, an equality which is immediate to check by double inclusion.

We are ready to deduce an inversion formula, expressing the sequence of essential exponents of ξ relative to m and the associated coefficients in terms of the sequence of essential exponents of η relative to n and their associated coefficients. We chose to inverse also the order of presentation, by starting from any pair of dual series pη, ξq and any pair of positive integers pm, nq, and by associating to them the series pη, ξq by the formulae (4.1) and (4.5). In this way, we emphasize only univalued maps, in contrast to their reciprocals, which involve taking roots. Theorem 4.4. Let η P Krrtss ˚and ξ P Krruss ˚be dual of each other and m, n P N ˚. Let ã be the constant term of η. Denote:

" ηptq " pt ηptqq m , ξpuq " pu ξpuqq n , and: " esspη, nq " pm, ǫ 1 , ..., ǫ d q, esspξ, mq " pn, ǫ 1 1 , ..., ǫ 1 d 1 q. Then one has the following inversion formulae for exponents and coefficients:

(4.6) d 1 " d, (4.7) ǫ 1 k `m " ǫ k `n, for all k P t1, ..., du, (4.8)
rξs n " ã´n and rξs ǫ 1 k " ´n m ã´n´ǫ k rηs ǫ k , for all k P t1, ..., du.

Proof. The entire series η and ξ being dual in the sense of Definition 3.14, Corollary 3.17 shows that they have the same sequences of essential exponents relative to gcdpn, mq. Then, Lemma 4.3 allows us to deduce the desired formulae (4.6) and (4.7) relating the two sequences pǫ k q k and pǫ 1 k q k . Let us pass to the proof of the inversion formula (4.8) for the coefficients. Equation (4.5) implies that ξ " u n p ξq n . Therefore: The same proposition, combined with the equivalent form ǫ 1 k ´n " ǫ k ´m of the equality (4.7), implies that:

(4.11) r ξs ǫ 1 k ´n " ´rηs ´ǫk `m´2 0 rηs ǫ k ´m " ´ã ´ǫk `m´2 rηs ǫ k ´m.
Combining the equalities (4.9), (4.10) and (4.11), we obtain:

(4.12) rξs ǫ 1 k " ´n ã´ǫ k `m´n´1 rηs ǫ k ´m. Now, from the analogues of equations (4.9) and (4.10) for η, we get:

(4.13) rηs ǫ k ´m " 1 m rηs ´m`1 0 rη m s ǫ k ´m " 1 m rηs ´m`1 0 rηs ǫ k " 1 m ã´m`1 rηs ǫ k .
Combining formulae (4.12) and (4.13), we deduce the inversion formula for the coefficients rξs ǫ 1 k , for k P t1, ..., du.

Dividing by n all the terms of the sequence pm, ǫ 1 , ..., ǫ d q, one gets the sequence of essential exponents of η relative to 1 (see Remark 3.7). Similarly, dividing by m all the terms of the sequence pn, ǫ 1 1 , ..., ǫ 1 d q, one gets the sequence of essential exponents of ξ relative to 1. Theorem 4.4 translates therefore in the following inversion formula for the Newton-Puiseux series of f px, yq relative to x and to y, which is the theorem of Halphen-Stolz presented in the introduction: Let ηpx 1{n q and ξpy 1{m q be Newton-Puiseux series of an irreducible formal power series f px, yq P Krrx, yss relative to x and y respectively. As before, we assume that ηptq " pt ηptqq m and ξpuq " pu ξpuqq n , where ηptq, ξpuq are dual series and rηs 0 " ã. Denote: " esspηpx 1{n q, 1q " pm{n, e 1 , . . . , e d q, esspξpy 1{m q, 1q " pn{m, e 1 1 , . . . , e 1 d 1 q. Then one has the following inversion formulae for exponents and coefficients:

(4.14) d 1 " d.
(4.15) mp1 `e1 k q " np1 `ek q for all k P t1, ..., du.

(4.16) rξpy 1{m qs n{m " ã´n and rξpy 1{m qs e 1 k " ´n m ã´p1`e k qn rηpx 1{n qs e k for all k P t1, ..., du.

In the case in which ã " 1, the inversion formula for the coefficients stated in Corollary 4.5 may be written in a more symmetric way, easier to remember: Corollary 4.6. Assume moreover that the constant coefficient ã of η is equal to 1. Then: rξpy 1{m qs n{m " 1 " rηpx 1{n qs m{n and mrξpy 1{m qs e 1 k `nrηpx 1{n qs e k " 0 for all k P t1, ..., du.

Summary of the previous arguments. In order to understand better the line of reasoning we followed till now, the reader may find helpful the following flow-chart, in which f px, yq P Krrx, yss is irreducible:

(4.17) ' From the irreducible series f px, yq P Krrx, yss, one gets symmetrically two sets of Newton-Puiseux series tηpx 1{n qu and tξpy 1{m qu. The first one has n and the second one m elements. ' Follow now two analogous sequences of transformations of those sets, indicated in the diagram horizontally. We describe them only for the upper line of the diagram. ' The change of variables x " t n , indicated above the corresponding doubly-arrowed horizontal segment, puts the set tηpx 1{n qu in bijection with the set of entire series tηptqu. ' Lemma 3.7, mentioned below the same arrow, allows to pass from esspηpx 1{n q, 1q to esspηptq, nq.

esspηpx 1{n q,1q
The corresponding coefficients are unchanged. ' One extracts in all possible ways the m-th roots of the series ηptq. Then one divides the result by t, arriving at a set tηptqu with mn elements. The composition of the two operations is expressed by the formula η " pt ηq m , written above the corresponding arrow. ' Combining Proposition 3.16 [START_REF] Abhyankar | On the ramification of algebraic functions[END_REF] with Lemma 4.3, one passes from the sequence esspηptq, nq to esspηptq, gcdpm, nqq and one relates also the corresponding coefficients. ' There is a canonical bijection between the two sets tηptqu and t ξpuqu, indicated by the left vertical double-arrowed segment. This bijection associates two series ηptq and ξpuq whenever ηptq and ξpuq are dual of each other, which may be expressed by the two equivalent equalities marked at the right of the vertical segment.

' Proposition 3.16 (2), indicated to the left of the same segment, shows that the two sequences are equal, and allows to relate the corresponding coefficients. Note that this proposition allows in fact to relate the coefficients corresponding to all the irreducible exponents of the two dual series, not only those which are essential relative to gcdpm, nq. This is understandable if one thinks that, reading now the diagram from right to left, one may start from any pair pηptq, ξpuqq of dual series and only afterwards choose the pair of positive integers pm, nq, independently of the choice of the two dual series. One arrives then at the series f px, yq by taking either the minimal polynomial of ηpx 1{n q or that of ξpy 1{m q, and multiplying it with an invertible element of the ring Krrx, yss.

Remark 4.7. Using Lemma 3.13, the Halphen-Stolz inversion theorem (Corollary 4.5) may also be expressed in terms of the characteristic exponents of η and ξ. That lemma shows that the sequences of characteristic exponents of η and of ξ do not have necessarily the same lengths, which has as consequence the fact that the elements of the two series which are related have not necessarily the same position in both sequences. For this reason, it is easier to express the inversion formulae as we have done in Theorem 4.4 and in Corollary 4.5, in terms of the essential exponents.

Remark 4.8. The part of Corollary 4.5 concerning the exponents is usually expressed nowadays in terms of the characteristic exponents and is sometimes attributed to Abhyankar's paper [START_REF] Abhyankar | Inversion and invariance of characteristic pairs[END_REF] of 1967 or to Zariski's paper [START_REF] Zariski | Studies in equisingularity III. Saturation of local rings and equisingularity[END_REF] of 1968. In fact, it was already stated precisely in terms of the sequences of essential exponents relative to 1 (called there "exposants caractéristiques" from their second term on) by Halphen [20, page 91] in 1876. But Halphen stated also the previous formulae (of course, with different notations) for the inversion of the corresponding coefficients. He did not prove those formulae, and as far as we know, the unique proof was provided by Stolz [36, page 133] in 1879. We searched new proofs because we were not fully convinced by Stolz' arguments and because we wanted to extend the theorem to higher dimensions.

4.2.

The second proof of the Halphen-Stolz theorem.

Let us pass now to our second proof of Theorem 4.4. Corollary 4.5 concerns only the terms of the two Newton-Puiseux series whose exponents are essential relative to 1. We explain now a way to get formulae for all the coefficients of ξ as rational fractions of those of η.

We recall first a form of Lagrange's inversion formula which, given two reciprocal entire series X and Y , allows to express the coefficients of the integral powers of X in terms of those of Y . Several proofs of it may be found in [START_REF] Stanley | Enumerative combinatorics II[END_REF]Theorem 5.4.2], and historical explanations in [35, pages 67-68]. Let us mention only that the founding result for this kind of formulae was stated by Lagrange in [START_REF] Lagrange | Nouvelle méthode pour résoudre les équations littérales par le moyen des séries[END_REF]Par. 16]. Theorem 4.9 (Lagrange inversion formula). Let Xpuq P u Krruss ˚and Y ptq P t Krrtss ˚be two reciprocal series. For any p, q P Z, one has: p ¨rXpuq q s p " q ¨rY ptq ´ps ´q.

Note that the lifting to a negative integral power produces a meromorphic series which has a finite number of terms with negative exponents.

Let us apply Theorem 4.9 in our context. Recall that ã P K ˚is the constant term of η, hence by formula (4.1), we may write: (4.18) ηptq " ãm t m ˜1 `ÿ kąm c k t k´m ¸.

Therefore: Using formula (4.19), one gets the following consequence of Theorem 4.9, which allows to compute the coefficients of the Newton-Puiseux series ξ of the irreducible power series f px, yq P Krrx, yss relative to y in terms of those η relative to x. Note that one gets rational fractions whose numerators are polynomials with rational coefficients in the coefficients c k and whose denominators are positive integral powers of ã: Proposition 4.10. Assume that:

(4.19) ηptq " ã ˜1 `ÿ kąm c k t k´m
ηpx 1{n q " ãm x m{n ˜1 `ÿ kąm c k x k´m n ¸.
Then one has the following formula for the coefficients rξs q m of the corresponding Newton-Puiseux series ξpy 1{m q P Krry 1{m ss, for all integer q ě n:

rξs q m " n q ã´q » -1 `ÿ iě1 ˆ´q{m i ˙˜ÿ sąm c s x s´m n ¸ifi fl ´1`q n .
Proof. By Theorem 4.9 applied after replacing the pair pp, qq by pq, nq, we have: q ¨ru n ξpuq n s q " n ¨rt ´q ηptq ´q s ´n.

We get: (4.23) q ¨rξpuqs q " q ¨ru n ξpuq n s q " nã ´q » -¨1

`ÿ iě1 ˆ´q{m i ˙˜ÿ sąm c s t s´m ¸i'fi fl q´n .
It is enough now to divide by q, to replace u by y 1{m and t by x 1{n in order to get the desired formula.

As a corollary, we obtain:

Second proof of Theorem 4.4. Recall the notation esspη, nq " pm, ǫ 1 , . . . , ǫ d q. We set then:

ǫ0 " n, ǫ1 " ǫ 1 ´m `n, . . . , ǫd " ǫ d ´m `n.
We prove first by induction on the integer q ě n that if (4.24) ǫk ď q ă ǫk`1 for some k P t0, . . . , du, then the terms of the sequence esspξ, mq which are lower than or equal to q are precisely ǫ0 " n, ǫ1 , . . . , ǫk .

Here the case k " d in (4.24) means simply that ǫd ď q.

If q " n, we get from (4.23) that rξs n " ã´n ‰ 0 is the dominant term of the series ξ, hence the assertion (4.24) holds by Definition 3.12. Assume that (4.24) holds for some q ą n. We distinguish two cases: ' Assume that ǫk ă q, so ǫk ď q ´1. Then, by the induction hypothesis applied to q ´1, we have that the terms of the sequence esspξ, mq which are lower than or equal to q ´1 are precisely ǫ0 " n, ǫ1 , . . . , ǫk . If rξpuqs q " 0, then there is nothing to prove. Assume that rξpuqs q ‰ 0. Since q ´n ă ǫ k`1 ´m by (4.24), the exponent of a term appearing in the polynomial p ř măsăǫ k`1 c s t s´m q i must belong to the group Ztn, m, ǫ 1 , . . . , ǫ k u, by the definition of the essential exponents esspη, nq. We deduce from this and the right hand side of the equality (4.23) that q must belong to the subgroup Ztn, m, ǫ 1 , . . . , ǫ k u. Since by definition we have the equality:

(4.25)
Ztn, m, ǫ 1 , . . . , ǫ k u " Ztn, m, ǫ1 , . . . , ǫk u, q cannot be an essential exponent of ξ with respect to n (see Definition 3.12).

' Assume that ǫk " q. Then, by the induction hypothesis applied to q ´1, we have that the terms of the sequence esspξ, mq which are lower or equal to q ´1 are precisely ǫ0 " n, ǫ1 , . . . , ǫk´1 . We have to prove that the coefficient rξpuqs q does not vanish. Notice that there is a term with exponent equal to q ´n appearing in the polynomial p ř măsďǫ k c s t s´m q i if and only if i " 1 and then this term is equal to c ǫ k t q´n . Indeed, arguing as in the previous case, we see that any other term would provide an expansion of ǫ k " q ´n `m in the group Ztn, m, ǫ 1 , . . . , ǫ k´1 u, contradicting the definition of the essential exponent ǫ k . Notice that c ǫ k " ã´m rηs ǫ k , by (4.18). It follows from (4.23) that:

q ¨rξs q " nã ´ǫk `m´n ˆ´q m ˙ã ´mrηs ǫ k " ´q n m ã´ǫ k ´nrηs ǫ k ,
thus rξs q " ´n m ã´ǫ k ´nrηs ǫ k is nonzero. This finishes the proof of the assertion. Theorem 4.4 is proved, since we have also proved the inversion formula (4.8) for the coefficients. l Example 4.11. Let us consider again the Newton-Puiseux series ηpx 1{4 q of Example 4.1. Denote by ξpy 1{6 q a Newton-Puiseux series corresponding by inversion to η, through the bijection described in Proposition 4.2. Applying Proposition 4.10, we get:

rξpy 1{6 qs p{6 " 4 p « 1 `ÿ iě1 ˆ´p{6 i ˙pcx 1 4 q i ff ´1`p 4 " 4 p ˆ´p{6 p ´4˙c p´4 .
That is:

ξpy 1{6 q " ÿ pě4 4 p ˆ´p{6 p ´4˙c p´4 y p{6 .
The first two exponents in the support Spξpy 1{6 qq are therefore 4{6 " 2{3 and 5{6, which shows that they constitute the characteristic sequence of ξ. The corresponding terms of ξ are, according to the previous formula: 1 ¨y2{3 , p´2 3 cq ¨y5{6 . One may verify then immediately the correcteness of the formulae stated in the Halphen-Stolz inversion theorem (Corollary 4.5).

Remark 4.12. In order to compute recursively the coefficients of ξpyq starting from those of ηpxq, one could also use the method explained by Borodzik [START_REF] Borodzik | Puiseux expansion of a cuspidal singularity[END_REF].

Remark 4. [START_REF] Fischer | Plane algebraic curves[END_REF]. We believe that one can use Abhyankar's [3, First Inversion Theorem, page 111] in order to obtain a third proof of the Halphen-Stolz inversion theorem. The approach of that paper seems to be similar in spirit to our first approach.

Generalization to an arbitrary number of variables

In this section we generalize our first proof of the Halphen-Stolz theorem to an arbitrary number of variables. We formulate the needed generalizations of the definitions and propositions used in that proof. We only sketch their proofs, insisting in the differences with respect to the one-variable case. Finally, we explain how our result generalizes Lipman's inversion theorem for the characteristic exponents of quasi-ordinary branches.

Throughout the section, we consider a fixed number h P N ˚and we work with the Q-vector space Q h and various free subgroups of it of rank h, which we will call briefly lattices of Q h . We denote by pν 1 , ν 2 , . . . , ν h q the canonical basis of Q h . 5.1. Irreducible exponents of subsets of Q h with bounded denominators.

The notions of set with bounded denominators (Definition 3.1) and of its irreducible elements (Definition 3.2) extend immediately from subsets of Q `to subsets of Q h `. If E is such a set, it generates again a semigroup N ˚E Ă Q h `and a group Z E Ă Q h . Lemma 3.5 (1) remains true in this setting:

Lemma 5.1. If E Ă Q h
`is a set with bounded denominators, then IrrpEq " IrrpN ˚Eq and this set is the minimal generating set of the semigroup N ˚E, relative to the inclusion partial order between its generating sets.

Notice that point (2) of Lemma 3.5 is not necessarily true for h ě 2, as shown by the following standard example:

Example 5.2. Take E " pN ˚q2 . Then IrrpEq " pN ˚ˆt1uq Y pt1u ˆN˚q . Therefore IrrpEq is infinite.

When h ě 2, we will need also to use special order relations on the group pQ h , `q: Definition 5.3. An additive order on Q h is a partial order relation ĺ on Q h satisfying:

(1) ĺ is a total order;

(2) if α, β, γ P Z h and α ĺ β, then α `γ ĺ β `γ. The additive order ĺ dominates a set θ Ă Q h if any non empty subset of θ with bounded denominators has a minimum relative to ĺ. Remark 5.4. If ĺ is an additive order of Q h , then there exist an integer s P r1, hs, linear forms u 1 , . . . , u s P pR h q ˚, and an increasing injective group morphism: pQ h , ĺq Ñ pR h , ď lex q, v Ñ pu 1 pvq, . . . , u s pvqq, where ď lex denotes the lexicographical order (see [START_REF] Robbiano | On the theory of graded structures[END_REF]Theorem 2.5]). The lexicographical order is additive and dominates Z h `. More generally, if ĺ dominates Z h `, then ĺ defines a well-order on Z h `, hence Definition 5.3 is a generalization of the notion of term order explained in [8, Chap. `with bounded denominators. Let M be a lattice of pQ h , `q and ĺ be an additive order on Q h dominating its subset Q h `.

Then the sequence esspE, M, ĺq :" pesspE, M, ĺq l q l of essential elements of E relative to M is defined inductively by: ' esspE, M, ĺq 0 :" min E. ' If l ě 1, then the term esspE, M, ĺq l is defined if and only if E is not included in the group M `ZtesspE, M, ĺq 0 , ..., esspE, M, ĺq l´1 u. In this case: esspE, M, ĺq l :" min pEz pM `ZtesspE, M, ĺq 0 , . . . , esspE, M, ĺq l´1 uqq .

One gets Definition 3.6 by taking h " 1, M " p Z and ĺ to be the unique additive order on (Q, +) which dominates Q `, that is, the usual order. Indeed, then the sequence esspE, p Z, ĺq defined according to Definition 5.5 is precisely the sequence esspE, pq defined according to Definition 3.6.

Lemma 3.8 about the finiteness of the sequences esspE, M q holds also in our larger context: Lemma 5.6. Assume that the subset E Ă Q h `has bounded denominators, that M is a lattice of Q h and that ĺ is an additive order dominating Q h `. Then the sequence of essential exponents esspE, M, ĺq of E relative to M is finite.

Proof. For every integer l ě 0 for which esspE, M, ĺq l is defined, let us denote by M l the abelian group M `ZtesspE, M, ĺq 0 , . . . , esspE, M, ĺq l u. Since pM l q l is an increasing sequence of abelian groups, the union Ť l M l is also an abelian group. The hypothesis that E has bounded denominators implies that this group Ť l M l is a lattice of Q h . Any ascending chain of subgroups of a free abelian group of finite rank being stationary, the sequence pM l q l must be finite. Therefore, the sequence esspE, M, ĺq is also finite.

If q P GLph, Qq and if ĺ is a additive order on Q h , we denote by ĺ q the additive order defined by: α ĺ q β ô qpαq ĺ qpβq.

By using this notion, Lemma 3.7 extends immediately into: Lemma 5.7. Assume that E Ă Q h `has bounded denominators and that M is a lattice of Q h . Take q P GLph, Qq such that qpQ h `q Ă Q h `and let ĺ be a additive order dominating Q h `. Then: q pesspE, M, ĺ q qq " ess pqpEq, qpM q, ĺq .

Lemmas 3.10 and 3.11 also extend immediately to our more general context: Lemma 5.8. The essential elements of a set E Ă Q h `with bounded denominators relative to any lattice M of Q h and an additive order ĺ dominating Q h `are irreducible elements of E. Lemma 5.9. Let E Ă Q h `be a set with bounded denominators, M be a lattice of Q h and ĺ be an additive order dominating Q h `. Then we have the following equality of essential sequences: esspE, M, ĺq " esspIrrpEq, M, ĺq. is invertible for composition, as its linearization pt 1 , t 2 , ..., t h q Ñ pα t 1 , t 2 , ..., t h q is invertible in GLph, Kq.

One has the following generalization of the duality of series in Krrtss ˚, introduced in Definition 3.14:

Definition 5.10. If φ P Krrt 1 , t 2 , ..., t h ss ˚, then its dual relative to the first variable is the unique entire series q φ P Krru 1 , t 2 , ..., t h , ss ˚such that the following maps are reciprocal: pu 1 , t 2 , ..., t h q Ñ pu 1 q φpu 1 , t 2 , ..., t h q, t 2 , ..., t h q, pt 1 , t 2 , ..., t h q Ñ pt 1 φpt 1 , t 2 , ..., t h q, t 2 , ..., t h q.

Remark 5.11. Note that the previous definition depends in an essential way on the choice of the first variable t 1 , but that it is symmetric in the other variables. If φ P Krrt 1 , t 2 , ..., t h ss ˚, then setting u 1 " t 1 φpt 1 , t 2 , ..., t h q defines a change of variables in the ring Krrt 1 , t 2 , ..., t h ss. Notice that Krrt 1 , t 2 , ..., t h ss " Krru 1 , t 2 , ..., t h ss and by Definition 5.10 one has the equivalence:

(5.2) u 1 " t 1 φpt 1 , t 2 , ..., t h q ô t 1 " u 1 q φpu 1 , t 2 , ..., t h q.

The following proposition generalizes Proposition 3.16 to the case of an arbitrary number of variables:

Proposition 5.12. Let φ P Krrt 1 , ..., t h ss ˚and N P N ˚. Then:

(1) Irrpφ N q " Irrpφq. Moreover rφ N s 0 " rφs N 0 and rφ N s r " N rφs N ´1 0 rφs r , for all r P Irrpφqzt0u.

(2) Irrp q φq " Irrpφq. Moreover r q φs 0 " rφs ´1 0 and r q φs r " ´rφs ´r1´2 0 rφs r , for all r P Irrpφqzt0u.

Proof. In what follows, if k " pk 1 , ..., k h q P N h , we will write simply:

t k :" t k1 1 ¨¨¨t k h h .
One has the following analogue of equation (3.2): (5.3) φptq " ÿ jPSpφq rφs j t j .

The hypothesis φ P Krrt 1 , ..., t h ss ˚translates into 0 P Spφq, that is, rφs 0 ‰ 0.

(1) Consider first the case of φ N . By equation ( 5.3), we get the exact analogue of the expansion (3.3):

(5.4) φ N ptq " ÿ j1,...,jN PSpφq rφs j1 ¨¨¨rφs jN t j1`¨¨¨`jN .

Then the proof is identical to that of the one-variable case.

(2) Consider now the case of q φ. Write the analogue of the expansion (5.3) for the series q φ:

(5.5) q φpuq " ÿ kPSp q φq r q φs k u k , where:

u k :" u k1 1 t k2 2 ¨¨¨t k h
h for all k " pk 1 , . . . , k h q P N h . By Definition 5.10, if q φpu 1 , t 2 , ..., t h q is the dual with respect to t 1 of the series φpt 1 , t 2 , ..., t h q, then one has the identity: t 1 " pt 1 φpt 1 , t 2 , ..., t h qq ¨q φpt 1 φpt 1 , ..., t h q, t 2 , ..., t h q which, after division by t 1 and combination with the expansion (5.5), gives:

1 " ÿ kPSp q φq r q φs k t k φptq k1`1 .

Expand now the powers φptq k1`1 using equation (5.4). We get:

1 " ÿ k P Sp q φq j1, ..., j k 1 `1 P Spφq r q φs k rφs j1 ¨¨¨rφs j k 1 `1 t k`j1`¨¨¨`j k 1 `1 .

Therefore:

(5.6) ÿ k P Sp q φq j1, ..., j k 1 `1 P Spφq k `j1 `¨¨¨`j k 1 `1 " p r q φs k rφs j1 ¨¨¨rφs j k 1 `1 " 0, for all p P N h zt0u. ' Let us show first that Irrpφq " Irrp q φq. Using Lemma 5.1, we reason as in the one variable case. We must take into account that for h ě 2, the element r P N ˚pSp q φqq z N ˚pSpφqq chosen to be minimal with this property (for the componentwise partial order) is not necessarily unique. Let us choose r to be the smallest element with this property, relative to an additive order ĺ dominating Q h `. Then the proof of the assertion follows exactly by the same argument as for h " 1.

' We prove the identities relating the coefficients associated to the irreducible exponents of φ and q φ. If r P Irrpφq " Irrp q φq and if: k `j1 `¨¨¨`j k1`1 " r for k P Sp q φq and j 1 , . . . j k1`1 P Spφq, then we obtain, by the same argument as in the one variable case, that k " r or k " 0. We deduce the following analogues of equation (3.12):

(5.7) r q φs 0 rφs r `r q φs r rφs r1`1 0 " 0 from which one gets the stated equality between coefficients of terms with irreducible exponents.

Combining Proposition 5.12 with Lemma 5.9, we obtain the following extension of Corollary 3.17:

Corollary 5.13. Let φ P Krrt 1 , t 2 , . . . , t h ss ˚, N P N ˚and M be any lattice of Q h . Then the sequences of essential exponents of φ, φ N and q φ relative to M coincide.

5.3.

Newton-Puiseux series in several variables.

We will consider the following analogue of the ring of Newton-Puiseux series in one variable:

Krrx 1{N 1 , x 1{N 2 , ..., x 1{N h ss :" ď niPN ˚, 1ďiďh Krrx 1{n1 1 , x 1{n2 2
, ..., x 1{n h h ss.

We say that its elements are Newton-Puiseux series in the variables x 1 , ..., x h . The support Spηq of such a series η is a subset with bounded denominators of Q h `.

Definition 5.14. Assume that f px 1 , y 1 , x 2 , . . . , x h q P Krrx 1 , y 1 , x 2 , . . . , x h ss has vanishing constant term. A Newton-Puiseux series of f relative to px 1 , x 2 , . . . , x h q is a series: ss is a Newton-Puiseux series, then the field extension Kppx 1 , . . . , x h qq Ă Kppx 1 , . . . , x h qqrψs is finite and its Galois group G is isomorphic to the quotient of G n1 ˆ¨¨¨ˆG n h by its subgroup formed by those elements which leave ψ fixed. If n " |G| and ψ 1 " ψ, ψ 2 , . . . , ψ n are the different conjugates of ψ under the action of the group G, then the polynomial:

ψ P Krrx 1{N 1 , x 1{N
f " n ź j"1 py ´ψj q P Krrx 1{n1 1 , . . .

, x

1{n h h ssrys is invariant under the action of G n1 ˆ¨¨¨ˆG n h on its coefficients. It follows that f must belong to Krrx 1 , . . . , x h ssrys and that ψ is a Newton-Puiseux series relative to f . Remark 5.17. Let f P Krrx 1 , . . . , x h ssry 1 s be an irreducible polynomial such that its discriminant ∆ y1 f is the product of a monomial and of a unit in the ring Krrx 1 , . . . , x h ss. Then, by the Jung-Abhyankar theorem, all the roots of f are Newton-Puiseux series in the variables x 1 , . . . , x h (see [START_REF] Abhyankar | On the ramification of algebraic functions[END_REF]). Let us mention that the roots obtained in this way have special properties, for instance, the Newton-Puiseux series x 3{2 1

`x5{2

2 cannot be a root of the polynomial f (see Lemma 5.24). Notice also that if the discriminant of f is not of this form, the roots may not be expressible as Newton-Puiseux series in the variables x 1 , . . . , x h . An example of this last phenomenon is the polynomial f

" x 3 1 `x3 2 `y2 1 .
Then one has the following inversion formulae for exponents and coefficients, where we denote by e k,1 the first coordinate of e k P Q h `(that is, the coefficient of ν 1 in the expansion e k " ř h i"1 e k,i ν i ): (5.18) d 1 " d.

(5.19) m 1 pν 1 `e1 k q " n 1 pν 1 `ek q for all k P t1, ..., du. , . . .qs e k for all k P t1, . . . , du.

In the case in which ã " 1, the inversion formula for the coefficients stated in Corollary 5.21 may be written in a more symmetric way, easier to remember, and analogous to Corollary 4.6: Corollary 5.22. Assume moreover that the constant coefficient ã of η is equal to 1. Then:

rξs n1{m1 " 1 " rηs m1{n1 ,
and m 1 rξs ǫ 1 k `n1 rηs ǫ k " 0 for all k P t1, ..., du. In order to summarize our reasoning, let us draw the analogue of the flow-chart (4.17) in which f px 1 , y 1 , x 2 , . . . , x h q P Krrx 1 , y 1 , x 2 , ..., x h ss is an irreducible series:

(5.21) Among the Newton-Puiseux series in several variables, the quasi-ordinary ones form a distinguished subclass, having many special properties. We compare both classes in this section with the help of additive orders and toric modifications. In particular, we get that Lipman's inversion theorem for quasi-ordinary series can be seen as a particular case of our generalized inversion theorem (Corollary 5.21). Definition 5.23. A series ψ P Krrx 1{N 1 , . . . , x 1{N h ss is quasi-ordinary if ψ is a Newton-Puiseux series relative to an irreducible polynomial f P Krrx 1 , . . . , x h ssry 1 s, such that the discriminant, ∆ y1 pf q P Krrx 1 , . . . , x h ss of f with respect to y 1 , is the product of a monomial and of a unit in the ring Krrx 1 , . . . , x h ss.

If the discriminant ∆ y1 pf q of f P Krrx 1 , . . . , x d ssry 1 s is a monomial times a unit then, by the Jung-Abhyankar theorem (see [START_REF] Abhyankar | On the ramification of algebraic functions[END_REF] and [START_REF] Jung | Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Veränderlichen x,y in der Umgebung einer stelle x " a, y " b[END_REF]), f factors in the ring Krrx 1{N 1 , . . . , x 1{N h ssry 1 s as a product of polynomials of degree 1 in the variable y 1 . If y 1 ´ψ, y 1 ´ψ1 are two different factors of f in this ring, then ψ ´ψ1 divides the discriminant ∆ y1 pf q, hence ψ ´ψ1 is the product of a monomial times a unit in the ring Krrx α k " pα k,1 , . . . , α k,h q, for k P t1, . . . , gu, the characteristic exponents of the quasi-ordinary series ψ. Lipman showed that the characteristic exponents determine many features of the geometry of the germ of hypersurface defined by f (for precise definitions and related results, see for instance [START_REF] Lipman | Quasi-ordinary singularities of embedded surfaces[END_REF][START_REF] Lipman | Quasi-ordinary singularities of surfaces in C 3[END_REF][START_REF] Lipman | Topological invariants of quasi-ordinary singularities[END_REF][START_REF] Gau | Embedded topological classification of quasi-ordinary singularities[END_REF]). He also proved the following combinatorial characterization of quasi-ordinary power series (see [START_REF] Lipman | Quasi-ordinary singularities of embedded surfaces[END_REF]Proposition 1.5] and [14, Proposition 1.3]):

Lemma 5.24. Denote by ď the coordinate-wise order on Q h . A series ψ P Krrx 1{N 1 , . . . , x 1{N h ss is quasiordinary if and only if there exist an integer n ě 1 and elements λ 1 , . . . , λ r P Spψq such that:

(1) The support Spψq is included in 1 n Z h `.

(2) Every λ P Spψq belongs to the group Z h `řλjďλ Zλ j .

(3) λ i ď λ i`1 , for every i P t1, . . . , r ´1u.

(4) λ i does not belong to the group Z h `řjďi´1 Zλ j , for every i P t1, . . . , ru. If such elements exist, then they are the characteristic exponents of ψ.

The following lemma is an analogue of Lemma 3.13. Its proof is a consequence of the definitions of essential exponents and of Lemma 5.24. It shows how to recover the characteristic exponents of a quasi-ordinary series from a sequence of essential exponents relative to the lattice Z h and any additive order ĺ dominating Q h `.

Lemma 5.25. Let ψ P Krrx 1{N 1 , . . . , x 1{N h ss be a quasi-ordinary series with characteristic exponents α 1 , . . . , α g . Let us denote by pe 0 , . . . , e d q the sequence of essential elements of the support Spψq relative to the lattice Z h and a fixed additive order ĺ dominating Q h `. Then: ' If e 0 R Z h , then g " d `1 and α k " e k´1 for k P t1, . . . , d `1u. ' If e 0 P Z h , then g " d and α k " e k for k P t1, . . . , du.

Remark 5.26. Lipman proved an inversion theorem for the characteristic exponents of a quasi-ordinary series ψ, when ψ is x 1 -dominant. This result appeared in Lipman's PhD Thesis [24, Lemma 2.3 and table 4.4], see also [START_REF] Lipman | Quasi-ordinary singularities of surfaces in C 3[END_REF], while its proof was published later in [START_REF] Lipman | Embedded topological classification of quasi-ordinary Singularities[END_REF]. This proof is written in the two variable case but it extends naturally to more variables. See also [START_REF] González Pérez | The semigroup of a quasi-ordinary hypersurface[END_REF]Proposition 5.5]. Thanks to Corollary 5.21 and Lemma 5.25, we see that Lipman's inversion theorem for quasi-ordinary series is a particular case of the part concerning exponents of our inversion theorem (Corollary 5.21) for x 1 -dominant Newton-Puiseux series in several variables.

We end this paper with some remarks relating geometrically the Newton-Puiseux series with the quasiordinary series by using methods of toric geometry. They are inspired by the second-named author's proof of [START_REF] González Pérez | Singularités quasi-ordinaires toriques et polyèdre de Newton du discriminant[END_REF]Théorème 3].

Let f P Krrx 1 , . . . , x h ssry 1 s be a reduced polynomial such that f p0, . . . , 0q " 0. Assume that f is not quasi-ordinary. Then, the discriminant ∆ y1 pf q P Krrx 1 , . . . , x h ss is not of the form a monomial times a unit. It follows that the dual fan associated to the Newton polyhedron of ∆ y1 pf q defines a nontrivial subdivision of the positive quadrant pR h q _ `of the vector space pR h q _ of real weights of monomials x k1

1 ¨¨¨x k h h . Let Σ be a regular subdivision of this dual fan. One has an associated toric modification X Σ Ñ K h , which is obtained by patching the monomial maps associated to the h-dimensional cones of Σ. See for instance [START_REF] Goldin | Resolving singularities of plane analytic branches with one toric morphism[END_REF] or [START_REF] González Pérez | Singularités quasi-ordinaires toriques et polyèdre de Newton du discriminant[END_REF] for the basic definitions used in these methods of toric geometry.

Let σ P Σ be a h-dimensional cone of the fan Σ. It is spanned by the forms γ 1 , . . . , γ h , which are the primitive lattice vectors of the lattice pZ h q _ lying on the edges of the cone σ. By the definition of the dual fan, the following property holds: Lemma 5.27. All the forms γ 1 , . . . , γ h reach their minimum value on the support of the discriminant ∆ y1 pf q at the same vertex λ 0 of its Newton polyhedron.

We consider the coordinates pγ s,1 , . . . , γ s,h q of the vectors γ s , s P t1, . . . , hu, with respect to the dual basis of ν 1 , . . . , ν h . Let q σ P GLph, Qq be the linear map defined, with respect to the canonical basis ν 1 , . . . , ν h of Q h , by the matrix whose rows are pγ s,1 , . . . , γ s,h q, for every s P t1, . . . , hu.

The monomial map (5.22) Krx 1 , . . . , x h s Ñ Krv 1 , . . . , v h s, x λ Ñ v qσ pλq

4 .

 4 -Puiseux series and their characteristic exponents 3 3. A calculus for the irreducible terms of invertible entire series 6 Applications to inversion formulae for Newton-Puiseux series 11 4.1. The first proof of the Halphen-Stolz theorem 11 4.2. The second proof of the Halphen-Stolz theorem 15 5. Generalization to an arbitrary number of variables 17 5.1. Irreducible exponents of subsets of Q h with bounded denominators 18 5.2. On the notions of dual and reciprocal series in several variables 19 5.3. Newton-Puiseux series in several variables 21 5.4. The generalized Halphen-Stolz inversion theorem 22 5.5. The special case of quasi-ordinary series 24

Example 4 . 1 .

 41 Start from a branch C with Newton-Puiseux series: ψ " x 3{2 `c ¨x7{4 , where c P K ˚.

(4. 9 ) rξs ǫ 1 k " rp ξq n s ǫ 1 rp ξq n s ǫ 1 k´n " n r ξs n´1 0 r ξs ǫ 1 k

 911101 ´n " n ã´n`1 r ξs ǫ 1 k ´n.

Corollary 4 . 5 (

 45 The Halphen-Stolz inversion theorem).

¸1{m ,

 ¸1{m in which the right-hand-side can be computed using the generalized binomial expansion: ´1q ¨¨¨pr ´k `1q k! , for all r P R.

(4. 22 ) q ¨ru n ξpuq n s q

 22

Definition 5 . 5 .

 55 2.2]. Definition 5.3 allows to generalize the notion of sequence of essential elements relative to an integer p (Definition 3.6) in the following way: Let us consider a set E Ă Q h

1 , 1 ,

 11 . . .qs n1{m1 " ã´n1 and rξpy 1{m1

u 1 " t 1 η õ t 1 " u 1 ξ f px 1 1 ,y1"u1 m 1 / 5 . 5 .

 1111155 .. q,n1Zν1`¨¨¨q tηpt 1 , t 2 , . . . qu esspηpt1,... q,gcdpm1,n1qZν1`¨¨¨q tηpt 1 , . . . qu ˆpProp.5.12p1q, , y 1 , x 2 , . . . , x h q 1{m 1 ... q,Z h q o o pLm.5.7q / tξpu 1 , t 2 , . . . qu esspξpu1,... q,m1Zν1`¨¨¨q t ξpu 1 , . . . qu essp ξpu1,... q,gcdpm1,n1qZν1`¨¨¨q ˆpProp.5.12p1q, The special case of quasi-ordinary series.

1{N 1 ,,

 1 . . . , x 1{N h ss. The monomials obtained in this way:x for k P t1, . . . , gu, are called the characteristic monomials, and the tuples:

  5.2. On the notions of dual and reciprocal series in several variables. Consider now the ring Krrt 1 , t 2 , ..., t h ss, and its subset: Krrt 1 , t 2 , ..., t h ss ˚" tφ P Krrt 1 , t 2 , ..., t h ss : φp0, ..., 0q ‰ 0u consisting of the series with non-zero constant term. It is the group of multiplicatively invertible elements of the ring Krrt 1 , t 2 , ..., t h ss. If φ P Krrt 1 , t 2 , ..., t h ss ˚has constant term α ‰ 0, then the map: Krrt 1 , t 2 , ..., t h ss ÝÑ Krrt 1 , t 2 , ..., t h ss pt 1 , t 2 , ..., t h q ÝÑ pt 1 φpt 1 , t 2 , ..., t h q, t 2 , ..., t h q

	(5.1)

  px 1 , ψ, x 2 , . . . , x h q " 0. The series ψ is called x 1 -dominating if it is of the form: are not x 1 -dominating. Instead, the second one is x 2 -dominating. 1{N h ss, then there is a series f P Krrx 1 , y 1 , x 2 , . . . , x h ss such that f px 1 , ψ, x 2 , . . . , x h q " 0. Indeed, one can get such an f in the ring Krrx 1 , . . . , x h ssry 1 s (cf. Remark 2.4 in the 1-variable case): Remark 5.16. Recall that for any n P N ˚, we denote by G n the subgroup of pK ˚, ¨q consisting of the n-roots of unity. Consider n 1 , . . . , n h P N ˚. Let Kppx The field extension Kppx 1 , . . . , x h qq Ă Kppx

							2 , . . . , x 1{N h ss
	such that f a ¨xλ 1 p1 `higher order termsq,
							1{N 1 , x 1{N 2 , . . . , x 1{N h ss of the form:
	(5.8)		ψ " ηpx 1{n1 1	, x 1{n2 2	1{n h , . . . , x h
	x 1{2 1	`x1{2 2	and x 1{3 2 `x1 x 2{3 2		3{2 1	`x7{4 1 x 1{2 2 ´2x 2 1 x 1{3 3	is x 1 -dominating. But the series:
	If ψ P Krrx 1{N 1 , . . . , x
	1{n1 1 ss. 1{n1 , . . . , x 1{n h h 1 , . . . , x qq be the fraction field of 1{n h qq is finite and Galois. h Its Galois group is isomorphic to G n1 ˆ¨¨¨ˆG n h , acting on Kppx Krrx 1{n1 1 , . . . , x 1{n h h 1{n1 1 , . . . , x 1{n h qq by: h ´pρ 1 , . . . , ρ h q, x a1{n1 1 ¨¨¨x a h {n h h ¯Ñ ρ a1 1 ¨¨¨ρ a h h ¨xa1{n1 1 ¨¨¨x a h {n h h .
	If ψ P Krrx 1{n1 1	1{n h , . . . , x h

where λ P Q ˚, and a P K ˚. A representation of ψ P Krrx q with η P Krrt 1 , t 2 , . . . , t h ss is called primitive if it is primitive in each variable separately in the sense of Definition 2.5.

Example 5.15. The Newton-Puiseux series x

The generalized Halphen-Stolz inversion theorem.

In this subsection we assume that ψ is a x 1 -dominating Newton-Puiseux power series with primitive representation (see Definition 5.14):

, ¨¨¨, x 1{n h h q.

Then the dominating term a ¨xλ 1 of ψ satisfies:

with m 1 P N ˚. The series η is therefore of the form:

ηpt 1 , t 2 , ..., t h q " a ¨tm1 1 p1 `higher order termsq, with m 1 ą 0 and a P K ˚. Let us choose an m 1 -th root ã P K ˚of a. Then, we have a unique m 1 -th root t 1 ηpt 1 , t 2 , ..., t h q P Krrt 1 , t 2 , ..., t h ss of η:

(5.9) ηpt 1 , t 2 , ..., t h q " pt 1 ηpt 1 , t 2 , ..., t h qq m1 , with constant term rηs 0 " ã ‰ 0.

Example 5.18. Start from the Newton-Puiseux series:

Let us come back to the general case. Denote by ξpu 1 , t 2 , ..., t h q P Krru 1 , t 2 , ..., t h ss ˚the dual series of ηpt 1 , t 2 , ..., t h q with respect to t 1 (see Definition 5.10). Hence, one has the equivalence (see (5.2)):

(5.10)

We know that there exists a series f px 1 , y 1 , x 2 , . . . , x h , q P Krrx 1 , y 1 , x 2 , . . . , x h ss such that:

, ..., x 1{n h h q, x 2 , ..., x h q " 0.

Replacing each x i by t ni i and using the equality (5.9), we get: (5.11) f pt n1 1 , pt 1 ηpt 1 , t 2 , ..., t h qq m1 , t n2 2 , ..., t n h h q " 0. By doing the second change of variable of formula (5.10), we deduce that:

(5.12)

f ppu 1 ξpu 1 , t 2 , ..., t h qq n1 , u m1 1 , t n2 2 , ..., t n h h q " 0. Consequently, if one defines:

(5.13) ξpu 1 , t 2 , ..., t h q " pu 1 ξpu 1 , t 2 , ..., t h qq n1

(an equation which is analogous to (5.9)), then one sees that:

, ..., x

, ..., x 1{n h h q is a Newton-Puiseux series of f px 1 , y 1 , x 2 , ..., x h q with respect to the variables py 1 , x 2 , ..., x h q (see Definition 5.14).

Recall that we denote by pν 1 , ν 2 , . . . , ν h q the canonical basis of Q h . The following lemma is a multivariable analogue of Lemma 4.3, formulated using Definition 5.3: Lemma 5.19. Let ĺ be an additive order of Q h dominating Q h `. Denote:

Proof. By symmetry, we may treat only the case of the series η. As η P Krrt 1 , t 2 , ..., t h ss ˚, Proposition 5.12 implies that Irrpηq " Irrpη m q. Combining this with Lemma 5.9, we see that for any lattice M of Q h one has: esspη, M, ĺq " esspIrrpηq, M, ĺq " esspIrrpη m q, M, ĺq " esspη m , M, ĺq. Thus, it is enough to prove that p0, ǫ 1 ´m1 ν 1 , ..., ǫ d ´m1 ν 1 q is the sequence of essential exponents of ηm1 relative to gcdpn 1 , m 1 qZν 1 `Zν 2 `¨¨¨`Zν h and the chosen additive order. By formula (5.9), we get ηm1 " t ´m1 1 η. Therefore Spη m1 q " Spηq ´m1 ν 1 . Using Definition 5.5, we see that we are done if we prove that:

' minpSpηq ´m1 ν 1 q " 0.

' For all k P t1, ..., du:

, where the minimum is taken with respect to the additive order ĺ. But all these facts are immediate from the definition of the essential exponents ǫ i , because:

for all 1 ď k ď d, an equality which may be proved immediately by double inclusion.

Our extension of Theorem 4.4 to the case of an arbitrary number of variables follows then exactly as in the one variable case: Theorem 5.20. Let η P Krrt 1 , t 2 , ..., t h ss ˚and ξ P Krru 1 , t 2 , ..., t h ss ˚be dual relative to the first coordinate and consider m 1 , n 1 P N ˚. Let ã be the constant term of η. Denote by pν 1 , ν 2 , ..., ν h q the canonical basis of the free abelian group Z h . Introduce the t 1 -dominating and u 1 -dominating series:

" ηpt 1 , t 2 , ..., t h q " pt 1 ηpt 1 , t 2 , ..., t h qq m1 , ξpu 1 , t 2 , ..., t h q " pu 1 ξpu 1 , t 2 , ..., t h qq n1 , and their sequences of essential exponents relative to an additive order ĺ dominating Q h ě0 : " esspη, n 1 Zν 1 `Zν 2 `¨¨¨`Zν h , ĺq " pm 1 ν 1 , ǫ 1 , ..., ǫ d q, esspξ, m 1 Zν 1 `Zν 2 `¨¨¨`Zν h , ĺq " pn 1 ν 1 , ǫ 1 1 , ..., ǫ 1 d 1 q. Then one has the following inversion formulae for exponents and coefficients, where we denote by ǫ k,1 the first coordinate of ǫ k P Q h `(that is, the coefficient of ν 1 in the expansion ǫ k "

)

, ..., x

1{n h h q be Newton-Puiseux series of f px 1 , y 1 , x 2 , ..., x h q relative to px 1 , x 2 , ..., x h q and py 1 , x 2 , ..., x h q respectively. As before, we assume that ηpt 1 , t 2 , ..., t h q " pt 1 ηpt 1 , t 2 , ..., t h qq m1 and ξpu 1 , t 2 , ..., t h q " pu 1 ξpu 1 , t 2 , ..., t h qq n1 , where ηpt 1 , t 2 , ..., t h q and ξpu 1 , t 2 , ..., t h q are dual relative to the first coordinate and rηs 0 " ã. Let ĺ be a fixed additive order dominating Q h `.

Denote

, ..., x

, ..., x

then the pull-back of F on K h σ ˆK is defined by:

h ss is a Newton-Puiseux series of f . Then, by definition

h ss is a Newton-Puiseux series of the pull-back f σ of f . In addition, qpλ 0 q belongs to the support of the discriminant ∆ y1 pf σ q. By Lemma 5.27, if qpλq belongs to the support of ∆ y1 pf σ q, then qpλ 0 q ď qpλq (for the coordinate-wise order). This implies that the Newton-Puiseux series ψ σ is quasi-ordinary, since the discriminant ∆ y1 pf σ q is of the form v qσ pλ0q times a unit. Using Lemma 5.7 and the fact that σ is a regular cone (which implies that q σ pZ h q " Z h ), we obtain also the relation: (5.23) q σ pesspψ, Z h , ĺ qσ qq " esspψ σ , Z h , ĺq between the essential exponents of ψ and the essential exponents of the quasi-ordinary series ψ σ .

Example 5.28. By Lemma 5.24, the series ψ " x

is not quasi-ordinary. If we consider the chart of the blowing up of 0 P K 2 given by x 1 " v 1 v 2 and x 2 " v 2 , whose associated cone we denote by σ, then we obtain the series:

. By Lemma 5.24, the series ψ σ is quasi-ordinary. It has essential exponents p0, 1{4q and p3{2, 3{2q with respect to the lattice Z 2 and any additive order ĺ of Q 2 dominating Q 2 `. By Lemma 5.25, these pairs are also the characteristic exponents of ψ σ . By (5.23), the pairs p0, 1{4q and p3{2, 0q are the essential exponents of ψ with respect to the order ĺ qσ .

Remark 5.29. Tornero studied in [START_REF] Tornero | On Kummer extensions of the power series field[END_REF] a notion of distinguished exponents of the Newton-Puiseux series ψ P Krrx 1{N 1 , . . . , x 1{N h ss with respect to a fixed additive order ĺ 1 of N d . In Example 5.28, one can check that the distinguished exponents of ψ relative to the additive order ĺ qσ correspond to the characteristic exponents of the quasi-ordinary series ψ σ . One can prove that this is a general phenomenon. By the previous discussion, it is enough to show that given a fixed additive order ĺ 1 , there exists a unique h-dimensional cone σ P Σ and a unique additive order ĺ dominating Q h `such that the orders ĺ 1 and ĺ qσ coincide. Indeed, the cone σ is the unique h-dimensional cone of Σ such that the additive order ĺ 1 dominates σ _ X Q h , where σ _ is the dual cone of σ (the existence of σ is a consequence of the properties of the Zariski-Riemann space of the fan Σ, see [START_REF] Ewald | Completion of real fans and Zariski-Riemann spaces[END_REF] and [START_REF] González Pérez | Toric geometry and the Semple-Nash modification[END_REF]Section 3.5]).