Milnor Fiber of a Plane Curve

Milnor Fiber of a Plane Curve

Pierrette Cassou-Noguès and Michel Raibaut

Pour Antonio, en témoignage de notre amitié

Abstract In this article we give an expression of the motivic Milnor fiber at the origin of a polynomial in two variables with coefficients in an algebraically closed field. The expression is given in terms of some motives associated to the faces of the Newton polygons appearing in the Newton algorithm. In the complex setting, we deduce a computation of the Euler characteristic of the Milnor fiber in terms of the area of the surfaces under the Newton polygons encountered in the Newton algorithm which generalizes the Milnor number computation by Kouchnirenko in the isolated case.

1 Introduction

Letkbe an algebraically closed field of characteristic zero. Letf be a regular map defined on a smoothk-varietyX. Using the motivic integration theory introduced by Kontsevich in [20], Denef and Loeser defined in [8,11], the motivic Milnor fiber of f at a pointxofX, denoted byS_f,xas an element ofM_k^μˆ, a modified Grothendieck ring of varieties overkendowed with an action of the group of roots of unityμ. Theyˆ proved that the motiveS_f,x is a “motivic” incarnation of the topological Milnor fiber of f at x denoted by F_x and endowed with its monodromy actionT_x. For instance, whenkis the field of complex numbers, they proved in [8, Theorem 4.2.1,

P. Cassou-Noguès ()

Institut de Mathématiques de Bordeaux, (UMR 5251), Université de Bordeaux, Talence Cedex, France

e-mail:Pierrette.Cassou-Nogues@math.u-bordeaux.fr M. Raibaut

Université Grenoble Alpes, Université Savoie Mont Blanc, CNRS, LAMA, 73000, Chambéry, France

e-mail:Michel.Raibaut@univ-smb.fr

© Springer Nature Switzerland AG 2018 145

Corollay 4.3.1] that the motiveSf,xrealizes on usual invariants of(Fx, Tx)as the Euler characteristic, the monodromy zeta function or Steenbrink’s spectrum.

In [15] Guibert computed the motivic Milnor fiber at the origin of a polynomial which is non degenerate with respect to its Newton polyhedron. The computation is given in terms of the faces of the Newton polyhedron. For more general point of views we refer to the memoir of Artal-Bartolo, Cassou-Nogues, Luengo and Melle-Hernandez [2], the composition of a non-degenerate polynomial with regular functions with separated variables of Guibert, Loeser and Merle in [16] and the logarithmic approach of Bultot and Nicaise in [3,4].

In [15] Guibert computed also the motivic fiber at the origin of a function in two variables in terms of the Puiseux pairs of its branches and their orders of contact. This approach is generalized in [18], where the authors compute the motivic Milnor fiber at the origin of a composition of a polynomial with two functions with the same variables but transversality assumptions. More recently using resolution of singularites Lê Quy Thuong computed in [24] the motivic Milnor fiber in an inductive and combinatoric way using the extended simplified resolution graph. An other recursive approach is given by González-Villa, Kennedy and McEwan in [14].

The quasi-ordinary case is treated in the article of González-Villa and González- Pérez [13].

Inspired by the works of the first author and Veys in the case of an ideal of k[[x, y]]in [5,6], the aim of this article is to give in Theorem2an expression of the motivic Milnor fiber at the origin of a polynomial ink[x, y]in terms of some motives associated to the faces of the Newton polygons appearing in the Newton algorithm. The Newton algorithm is an algorithm which allows to compute the tree of the resolution of a germ at the origin in terms of Newton polygons.

We present the strategy of the proof of the theorem and the structure of the article.

LetGm be the multiplicative group ofk. We will work with a Grothendieck ring M_G^G_m^m of the category Var^G_G^m

m ofGm-algebraic varieties endowed with a monomial Gm-action (see Sect.2for details). This ring is isomorphic to the ringM_k^μˆ (see [17,

§2.4]).

We denote byL(A²_k)the arc space of the affine planeA²_kwhosek-rational points are formal series ink[[t]]². The multiplicative groupGmacts canonically onL(A²_k) byλ.ϕ(t)equal toϕ(λt)for anyλinGmand any arcϕ inL(A²_k). For a non zero elementψink[[t]], we denote by ord(ψ)theorderofψand by ac(ψ)itsangular componentequal to its first non-zero coefficient with the convention that ac(0)is zero.

Letf be a polynomial ink[x, y]withf (0,0)=0. Themotivic zeta functionof f at the origin is the formal series

Zf(T )

(0,0)=

n≥1

mes(Xn(f ))Tⁿ

inM_G^G_m^m[[T]], where for anyn≥1, mes(X_n(f ))is themotivic measureof the arc space

X_n(f )= {ϕ ∈L(A²_k)|ϕ(0)=(0,0), ordf (ϕ)=n}

endowed with the arrow “angular component” ac(f )toGmand the standard action ofGmon arcs. Denef and Loeser show in [8,11] that this zeta function is rational by giving a formula of

Z_f(T )

(0,0)in terms of a resolution off. It admits a limit whenT goes to∞and themotivic Milnor fiberoff at the origin is the motive of M_G^G_m^mdefined as

S_f

(0,0)= − lim

T→∞

Z_f(T )

(0,0).

Following the strategy of [6] we describe now how to compute the motivic Milnor fiber

Sf

(0,0) using the Newton algorithm and taking account ofGm-actions (see Remark7). Assume thatf is written asf (x, y)=

(a,b)∈N²c_a,bx^ay^b. The Newton diagram off at the origin, denoted byNf, is the convex hull of the set of points {(a, b)+R²₊|c_a,b = 0}. We denote bym the usual support function ofNf (see Proposition2). Letγbe a face ofNf andf_γ be theface polynomialoff associated toγ and defined as

f_γ(x, y)=

(a,b)∈γ

c_a,bx^ay^b=cx^aγy^bγ

1≤i≤r

(y^p−μ_ix^q)^νi,

withc inGm,(p, q) ∈ N² coprime, μ_i inGm all different and called roots of f_γ andν_i inN^∗. The set of rootsμ_i is denoted byR_γ. The last equality needs the assumptionk algebraically closed. The main remark is that for any arcϕ = (x(t), y(t))with ordx(t)and ordy(t)inNwe have

m(ordx(t),ordy(t))≤ordf (ϕ)

with equality if and only iff_γ(acx(t),acy(t))is non zero, whereγ is the face of Nf whose dual cone Cγ contains the couple(ordx(t),ordy(t)). This implies a decomposition of the motivic zeta function

Zf(T )

(0,0)=

γ∈N(f )

Zγ(T )=

γ∈N(f )

Z_γ⁼(T )+Z^<_γ(T )

in terms of faces of the Newton polygon and a dichotomy based on the fact that the face polynomial vanishes or doesn’t vanish (see Sect.4.1and Proposition4).

The case of arcs such that the face polynomial evaluated on their angular compo- nents does not vanish is done in Sect.4.3.1. The other case, necessarily for one dimensional faces, is treated in Remark12. The main tool is the Newton map. For

instance, considerγa one dimensional face ofN (f )andμone of the roots of the face polynomialfγ. The faceγ is supported by a line with normal vector(p, q)in N²with gcd(p, q)=1. Let(p, q)be inN²such thatpp−qq =1. A Newton map ofγ andμis the map

σ_(p,q,μ): k[x, y] −→k[x₁, y₁]

f (x, y) → f (μ^qx₁^p, x₁^q(y₁+μ^p))=f_σ(x₁, y₁). Then in Proposition7we obtained the refined decomposition

Z^<_γ(T )=

μ∈R_γ

Z_{fσ (p,q,μ),ωp,q}(T )

(0,0)

in terms of roots and Newton transforms. Hereωp,qis the differential form induced by the Newton map and motivic zeta functions with differential form are recalled in Sect.2. The main step is Proposition6comparing a measure of arcs forf and for fσ. Finally, the motivic zeta function can be computed inductively using the Newton algorithm which consists in considering the Newton polygon offσ and applying the same process (see Lemma2). It is proven (see Lemma4) that after a finite number of steps, we end up with a monomial orx^k(y+μx^q+g(x, y))^ν multiplied by a unit ink[[x, y]]withkinN,g(x, y)=

a+bq>qca,bx^ay^bink[x, y]. These base cases are studied in Sect.4.3.3. After the proof of Theorem2, we conclude Sect.4by two interesting examples. Cauwergs’ example in [7] shows that two functions which have the same topological type can have different motivic Milnor fibers. Schrauwen, Steenbrink and Stevens example in [26] shows, on the other hand, that two functions with different topological type can have same motivic Milnor fiber.

In the last sections, we apply the previous results to the computation of invariants off using what is known for quasihomogeneous polynomials. We use in particular computations of the monodromy zeta function by Martin-Moralés in [23]. In Sect.5 we deduce from Theorem 2 a generalization of Kouchnirenko’s theorem which computes the Milnor number, in the isolated case, in terms of the area of the surfaces under the Newton polygons appearing in the Newton algorithm. Furthermore, in Sect.3we recover the formula for the monodromy Zeta function given by Eisenbud and Neumann in [12] and that of Varchenko in [27].

2 Motivic Milnor Fibers

Below we explain some definitions and properties that will be used throughout the paper. We refer to [10,17,21,22] and [16] for further discussion.

2.1 Motivic Setting

2.1.1 Grothendieck Rings

Letkbe a field of characteristic 0 and denote byGmits multiplicative group. We call k-variety, a separated reduced scheme of finite type overk. We denote byV ar_kthe category ofk-varieties and byV ar_Gm the category ofGm-varieties, where objects are morphismsX→GminV ar_k. As usual we denote by M_Gmthe localization of the Grothendieck ring of varieties over Gmwith respect to the relative line. We will use also theGm-equivariant variant M_G^G_m^mintroduced in [17, §2] or [16, §2], which is generated by classes of objects Y →Gmendowed with a monomialGm-action.

In this context the class of the projection from A¹_k×Gmto Gmendowed with the trivial action is denoted byL.

2.1.2 Rational Series

Let Abe one of the rings Z[L,L⁻¹] or M_G^G_m^m. We denote byA[[T]]sr the A- submodule ofA[[T]]generated by 1 and finite products of termsp_e,i(T )defined as L^eTⁱ/(1−L^eTⁱ)witheinZandiinN>0. There is a uniqueA-linear morphism limT→∞ : A[[T]]sr → Asuch that for any subset(e_i, j_i)_i∈I ofZ×N>0withI finite or empty, limT→∞(

i∈Ip_ei,ji(T ))is equal to(−1)^|I|.

We will use the following lemma similar to [5, §3] or [15, 2.1.5] and [17, 2.9].

Lemma 1 Letϕandηbe twoZ-linear forms defined onZ²withϕ(N²)andη(N²) included inN. Let C be a rational polyhedral convex cone ofR²\ {(0,0)}. We assume that for anyn ≥ 1, the setC_n defined asϕ⁻¹(n)∩C ∩N² is finite. We consider the formal series inZ"

L,L⁻¹# [[T]].

S_ϕ,η,C(T )=

n≥1

(k,l)∈Cn

L^−η(k,l)Tⁿ.

• IfCis equal toR>0ω₁+R>0ω₂whereω₁andω₂are two non colinear primitive vectors inN²withϕ(ω₁)=0andϕ(ω₂)=0then, if

P =(]0,1]ω₁+]0,1]ω₂)∩N² we have

S_ϕ,η,C(T )=

(k0,l0)∈P

L^−η(k0,l0)T^ϕ(k0,l0)

(1−L^−η(ω1)T^ϕ(ω1))(1−L^−η(ω2)T^ϕ(ω2)) (1) andlimT→∞S_ϕ,η,C(T )=1.

• IfCis equal toR>0ωwhereωis a primitive vector inN²withϕ(ω)=0then we have

Sϕ,η,C(T )= L^−η(ω)T^ϕ(ω)

1−L^−η(ω)T^ϕ(ω) and lim

T→∞Sϕ,η,C(T )= −1. (2) Proof Assume thatC is equal toR>0ω₁+R>0ω₂ whereω₁andω₂ are two non colinear primitive vectors inN²withϕ(ω₁)=0 andϕ(ω₂)=0. LetPbe defined as(]0,1]ω₁+]0,1]ω₂)∩N². For any(k, l)∈Cthere is a unique(k₀, l₀)∈P, there is a unique(α, β)inN²such that

(k, l)=(k₀, l₀)+αω₁+βω₂. As the setP is finite, we have

S_ϕ,η,C(T )=

(k0,l0)∈P

L^−η(k0,l0)T^ϕ(k0,l0)S_k0,l0(T )

where S_k0,l0(T )=

n≥1

α≥0, β≥0 αϕ(ω1)+βϕ(ω2)=n−ϕ(k0, l0)

L^−η(ω1)T^ϕ(ω1) _α

L^−η(ω2)T^ϕ(ω2) _β

=

n≥ϕ(k0,l0)

α≥0, β≥0 αϕ(ω1)+βϕ(ω2)=n−ϕ(k0, l0)

L^−η(ω1)T^ϕ(ω1) _α

L^−η(ω2)T^ϕ(ω2) _β

= m≥0

α≥0, β≥0 αϕ(ω1)+βϕ(ω2)=m

L^−η(ω1)T^ϕ(ω1) _α

L^−η(ω2)T^ϕ(ω2) _β

which implies the equality (1). Asϕ(1,0)andϕ(0,1)are non negative integers, for any(k₀, l₀)inP,ϕ(k₀, l₀)≤ϕ(ω₁)+ϕ(ω₂)with equality only in the case

(k₀, l₀)=ω₁+ω₂

which implies the result on the limit in (1). The proof of (2) is similar.

2.2 Arcs

2.2.1 Arc Spaces

LetXbe ak-variety. We denote byLn(X)thespace ofn-jetsofX. This set is ak- scheme of finite type and itsK-rational points are morphisms fromSpec K[t]/tⁿ⁺¹ toX, for any extensionKofk. There are canonical morphisms formLn+1(X)to Ln(X)induced by the truncation modulotⁿ⁺¹. These morphisms areA^d_k-bundles

whenXis smooth with pure dimensiond. Thearc spaceofX, denoted byL(X), is the projective limit of this system. This set is ak-scheme and we denote byπnfrom L(X)toLn(X)the canonical morphisms. For more details we refer for instance to [9,22].

2.2.2 Origin, Order, Angular Component and Action

For a non zero elementϕ in K[[t]] or in K[t]/tⁿ⁺¹, we denote by ord(ϕ) the valuation ofϕ and by ac(ϕ)its first non-zero coefficient. By convention ac(0)is zero. The scalar ac(ϕ) is called theangular componentofϕ. The multiplicative group Gm acts canonically on Ln(X) and on L(X) by λ.ϕ(t) := ϕ(λt). We consider the applicationoriginwhich maps an arcϕtoϕ(0)=ϕ modt.

2.3 The Motivic Milnor Fiber

Letf be a polynomial ink[x, y], withf (0,0)=0. For anyn≥1, X_n(f )=$

ϕ ∈L(A²_k)|ϕ(0)=(0,0), ordf (ϕ)=n

%

is a scheme endowed with the arrow “angular component” ac(f )to Gmand the standard action ofGmon arcs. In particular, for anym≥nthe image ofXn(f )by the truncation mapπ_mis a variety in M_G^G_m^m denoted byX^(m)_n (f ). In particular, by smoothness of A²_kwe have the equality

&

X_n^(m)(f )

'L^−2m=&

X⁽ⁿ⁾_n (f )

'L⁻²ⁿ∈M_G^G_m^m

this element is the motivic measure of X_n(f ) denoted by mes (X_n(f )) and introduced by Kontsevich in [20].

Denef and Loeser defined in [8] themotivic zeta functionoff at the origin as Zf(T )

(0,0)=

n≥1

mes(Xn(f ))Tⁿ

in M_G^G_m^m[[T]]. They show that the motivic zeta function is rational by giving a formula forZf(T )in terms of a resolution off, see for instance [8,11]. It admits a limit whenT goes to∞and by definition themotivic Milnor fiberoff at(0,0)is

Sf

(0,0)= − lim

T→∞

Zf(T )

(0,0)∈M_G^G_m^m.

Denef and Loeser proved that this motive contains usual invariants of the Milnor fiber off at(0,0)see for instance [8, Theorem 4.2.1, Corollary 4.3.1] or [11,19].

We will apply these theorems in Sects.5and6to compute the Euler characteristic and the monodromy zeta function of the Milnor fiber in terms of the Newton polygons off.

2.4 Motivic Zeta Function and Differential Form

Letf be a polynomial ink[x, y]withf (0,0)=0. Letωbe the differential form ω=x^ν−1dx∧dywithν ≥ 1. Ifx does not dividef then we assumeν =1. For any(n, l)in(N^∗)², we define

X_n,l(f, ω)=$

ϕ ∈L(A²_k) |ϕ(0)=(0,0),ordf (ϕ(t))=n,ordω(ϕ(t))=l

%

endowed with the arrow “angular component” ac(f )toGmand the standard action ofGmon arcs. We consider the following motivic zeta function

Z_f,ω(T )

(0,0)=

n≥1

⎛

⎝

l≥1

mes

X_n,l(f, ω) L^−l

⎞

⎠Tⁿ∈M_G^G_m^m[[T]]. (3)

The assumption onωensures that the sum overlis finite. This zeta function is for instance studied in [2,5,7,28]. It is rational by Denef-Loeser standard arguments, and admits a limit whenT goes to∞, and we denote

Sf,ω

(0,0)= − lim

T→∞

Zf,ω(T )

(0,0). Proposition 1 The motives

Sf

(0,0)and Sf,ω

(0,0)are equal.

Proof By taking the limit whenT goes to infinity, this equality follows immediately from the expression of the rational form of the motivic zeta functions

Zf,ω(T )

(0,0)

and Z_f(T )

(0,0)on a log-resolution of(A², f⁻¹(0)∪ {x =0})adapted tox =0.

See for instance [7, §1.4].

3 Newton Algorithm

Let k be an algebraically closed field of characteristic zero, with multiplicative group denoted byGm.

Definition 1 Letf be an element ofk[x, y]written asf (x, y)=

(a,b)∈Z²ca,b

x^ay^b.

We define thesupportoff by Suppf =$

(a, b)∈Z²|c_a,b=0

% .

In this section (Lemma2), we define the local Newton algorithm for a polynomial f ink[x, y].

3.1 Newton Polygons

Definition 2 For any set E inN×N, denote by Δ(E) the smallest convex set containing

E+(R>0)²=$

a+b, a∈E, b∈(R>0)²

% .

A subsetΔofR²is calledNewton diagramif there exists a setEinN×N, such thatΔis equal toΔ(E). The smallest setE0ofN×Nsuch thatΔis equal toΔ(E0) is called the set of verticesofΔ.

Remark 1 The set of vertices of a Newton diagram is finite.

Definition 3 LetE0be the set{v0,· · · , vd}withvi =(ai, bi)inN×Nsatisfying ai−1 < ai andbi−1 > bi for anyiin{1, . . . , d}. For suchi, we denote bySi the segment[vi−1, vi]and bylS_i the line supporting the segmentSi. We call the set

N(Δ)= {S_i}i∈{1,...,d}∪{v_i}i∈{0,...,d}

theNewton polygonofΔ. The integerh(Δ)=b₀−b_dis called theheightofΔ.

Letf be an element ofk[x, y]equal to f (x, y)=

(a,b)∈N×N

ca,bx^ay^b.

Definition 4 The Newton diagram of f is the Newton diagram Δ(f ) equal to Δ(Suppf ). TheNewton polygon at the originoff is the Newton polygonN (f ) equal toN (Δ(f )). Theheight of f denoted byh(f )is the heighth(Δ(f )).

Definition 5 Let l be a line inR². Theinitial partoff with respect tol is the quasi-homogeneous polynomial

in(f, l)=

(a,b)∈l∩N(f ))

c_a,bx^ay^b.

Notation 1 In the following, for a faceγ ofN (f )contained in a linel, instead of writingin(f, l)we will simply writefγ.

Remark 2 Letγ be a one-dimensional face ofN (f ). We have the equality f_γ =x^aγy^bγF_γ(x^q, y^p),

where(a_γ, b_γ)belongs toN×N,pandqare coprime positive integers and F_γ(x, y)=c

1≤i≤r

(y−μ_ix)^νi,

withcink^∗,μ_i ink^∗(all different) andν_i inN^∗. We callf_γtheface polynomialof f associated toγ.

Definition 6 We say thatf isnon degeneratewith respect to its Newton polygon N (f )if for each one dimensional faceγinN (f ), the polynomialF_γ has simple roots.

Remark 3 The polynomialf is non degenerate with respect toN(f )if and only if for any one-dimensional faceγ, the initial partf_γ has no critical point on the torus (k^∗)².

3.2 Newton Algorithm

Definition 7 (Newton map) Let(p, q)be inN²with gcd(p, q)=1. Let(p, q) be inN²such thatpp−qq=1. Letμbe inGm. We define the application

σ_(p,q,μ): k[x, y] −→ k[x₁, y₁] f (x, y) → f

μ^qx₁^p, x₁^q(y₁+μ^p)

We say thatσ_(p,q,μ)is aNewton map.

Remark 4 The Newton mapσ_(p,q,μ)depends on(p, q)but this doesn’t affect our results. Indeed, if(p+lq, q+lp)is another pair, then

f

μ^q+lpx₁^p, x₁^q(y₁+μ^p+lq) =f

μ^q(x₁μ^l)^p, (x₁μ^l)^q(y₁μ^−lq +μ^p)

. Furthermore, there is exactly one choice of (p, q) satisfying pp −qq = 1 and moreoverp ≤ q andq < p. In the sequel we will always assume these inequalities. This will make procedures canonical.

In the well-known following proposition we introduce notations used in the rest of the paper.

Proposition 2 LetEbe a set of some(m, n) fornandminN. Let(p, q)be in N²withgcd(p, q) = 1. We consider the linear forml(p,q) which maps(a, b)on ap+bq.

1. The minimum ofl(p,q)|Δ(E), denoted bym(p, q), is obtained on a face of Δ(E) denoted byγ (p, q). In particular,l(p,q)is constant onγ (p, q).

2. For any faceγofΔ(E), we denote byCγthe interior, in its own generated vector space inR², of the positive cone generated by the set{(p, q)∈N²|γ (p, q)= γ}. This set is a rational polyhedral cone, convex and relatively open.

For a one dimensional faceγ, we denote bynγ the normal vector toγ with integral non negative coordinates and the smallest norm. With that notation we have : 3. For a one dimensional faceγ,C_γ =R>0nγ.

4. A zero dimensional faceγ is an intersection of two one dimensional facesγ₁and γ₂, and C_γ =R>0nγ1+R>0nγ2.

5. The set of conesC_γ is a fan, called the dual fan ofΔ(E).

Lemma 2 (Newton algorithm) Let(p, q)be inN²withgcd(p, q)=1. Letμbe inGm.

Let f be a non zero element in k[x, y] and f₁ be its Newton transform σ_(p,q,μ)(f )in k[x₁, y₁].

1. If there does not exist a faceSofN (f )of dimension1whose supporting line has equationpa+qb=N, for someN, then

f1(x1, y1)=x₁^m(p,q)u(x1, y1) withu(x₁, y₁)ink[x₁, y₁]andu(0,0)=0.

2. If there exists a face S of N (f ) of dimension1 whose supporting line has equationpa+qb=N, and ifF_S(1, μ)=0, thenm(p, q)=N and

f₁(x₁, y₁)=x₁^Nu(x₁, y₁) withu(x₁, y₁)ink[x₁, y₁]andu(0,0)=0.

3. If there exists a face S of N (f ) of dimension1 whose supporting line has equationpa+qb=N, and ifF_S(1, μ)=0, thenm(p, q)=N and

f₁(x₁, y₁)=x₁^Ng₁(x₁, y₁)

withg₁(x₁, y₁)ink[x₁, y₁]andg₁(0,0) = 0, g1(0, y1) = dy₁^ν + · · ·,where d=0andνis the multiplicity ofμas root ofF_S(1, Y )=0. In particular in that case,ν≥h(σ_(p,q,μ)(f )).

Proof We consider the linear forml(p,q)which maps(a, b)onap+bq. Writing f (x, y)=

(a,b)∈supp(f )ca,bx^ay^bwe obtain f1(x1, y1)=

(a,b)∈supp(f )

c_a,bμ^qa(y1+μ^p)^b

x^l₁^(p,q)(a,b).

Letm(p, q)be the minimum min(a,b)∈supp(f )l(p,q)(a, b). By convexity and defini- tion of the Newton diagram off, this minimum is reached on a zero dimensional or one dimensional faceγ. Then, we can write

f₁(x₁, y₁)=x₁^m(p,q)

(a,b)∈γc_a,bμ^qa

y₁+μ^p b

+x₁^m(p,q)

(a,b) /∈γ

c_a,bμ^qa(y₁+μ^p)^b

x^l(p,q)(a,b)−m(p,q)

1 .

Remark that for any(a, b)not inγ,l(p,q)(a, b)−m(p, q) >0.

1. If there does not exist a faceSofN (f )of dimension 1 whose supporting line has directionpa+qb=0, then the faceγ is a vertex and the results follows.

2. If there exists a one dimensional faceS ofN (f )whose supporting line has equationpa+qb=N, thenγis equal toS. In particular we haveN =m(p, q) and

x₁^m(p,q)

(a,b)∈γ

ca,bμ^qa(y1+μ^p)^b=in(f, γ )

μ^qx₁^p, x₁^q(y1+μ^p)

,

and the result follows by computations.

Remark 5 Iff1(x1, y1)is equal tox₁ⁿ¹y₁^m1u(x1, y1), where(n1, m1)belongs toN² andu ∈ k[x1, y1]is a unit in k[[x1, y1]], we say for short thatf1is a monomial times a unit.

Remark 6 From this lemma, we see that there are a finite number of(p, q, μ)such thatσ(p,q,μ)(f )is eventually not a monomial times a unit in k[[x1, y1]]. These triples are given by the equations of the faces of the Newton polygon and the roots of the corresponding face polynomials.

Notation 2 Ifσ =σ(p,q,μ)is a Newton map, we denote byfσ =σ(p,q,μ)(f ).

Lemma 3 (Lemma 2.11 [5]) If the height offσ is equal to the height off, then the Newton polygon off has a unique faceS withfS = x^ky^l(y−μx^q)^ν, with (k, l, ν)inN×N²andq inN^∗.

LetΣn = (σ1,· · · , σn)whereσi is a Newton map for all i, we definefΣ_n by induction:fΣ₁ =fσ₁,fΣ_i =(fΣ_i−1)σ_i.

Definition 8 If the height in the Newton process remains constant, we say that the Newton algorithm stabilizes.

Lemma 4 Letf be ink[x, y]. If the Newton algorithm off stabilizes with height νthen

f (x, y)=U (x, y)x^k(y+x^q+g(x, y))^ν, with(k, q) ∈ N², g(x, y) ∈ k[x, y], g(x, y) =

a+bq>qc_a,bx^ay^b, U (x, y) ∈ k[[x, y]]andU (0,0)=0.

Proof Assume that the Newton algorithm off is stabilized in heightν. We can write

f (x, y)=

i>ν

yⁱa_i(x)+y^νa_ν(x)+

j <ν

y^ja_j(x)

witha_ν(x)= 1+ · · ·, forj < ν,a_j(x)= _j!(ν^ν₋^!_{j )!}μ^ν−jx^{q(ν−j )}+ · · · where· · · means higher order terms. Letf₁(x, y)be the(ν−1)th derivative off with respect toy.

f₁(x, y)=

i>ν

i!

(i−ν)!y^i−ν+1a_i(x)+ν!ya_ν(x)+(ν−1)!a_ν−1(x) f1(x, y)=ν!(y+μx^q+g(x, y))

with g(x, y) ∈ k[x, y], g(x, y) =

a+bq>qc_a,bx^ay^b. The hypothesis that the Newton algorithm is stabilized in height ν, implies that at each step the face polynomial is of the formx^l(y+ ˜μx^r)^ν for somel,μ, r˜ and it implies thatf has a Puiseux series of the formy₀=

c_ixⁱ withi >0 as its unique root of multiplicity ν. This point follows from Puiseux theory, see for instance [29, 2.1]. Then, there is a formal seriesu(x, y)ink[[x, y]]withu(0,0)=0 such that

f (x, y)=u(x, y)(y−y₀)^ν. Theny₀is also the unique root of the polynomialf₁, and

f₁(x, y)=ν!(y+μx^q+g(x, y))=u₁(x, y)(y−y₀) withu₁(x, y)∈k[[x, y]], u₁(0,0)=0 . Then we obtain

f (x, y)=(ν!)^ν u(x, y)

u1(x, y)^ν(y+μx^q+g(x, y))^ν =U (x, y)(y+μx^q+g(x, y))^ν withU (x, y)∈k[[x, y]], U (0,0)=0.

Theorem 1 For allf (x, y)in k[x, y], there exists a natural integern0, such that for any sequence of Newton mapsΣn =(σ1,· · ·, σn)withn≥n0,fΣ_n is either a

monomial times a unit or there exists an integerνinNsuch thatfΣ_nis of the form U (x, y)x^k(y+μx^q+g(x, y))^ν wherekis an integer inN,g(x, y)is a polynomial ink[x, y]equal to

a+bq>qca,bx^ay^b,μbelongs toC^∗andU (x, y)is ink[[x, y]]

withU (0,0)=0.

Proof From Lemma2, we first observe that the number of Newton mapsσ, such thatfσ is not a monomial times a unit is finite, bounded by the sum on all facesS of the number of roots ofFS.

We argue by induction on the height off. Ifh(f )=0,f is a monomial times a unit andn0=0. Consider the case whereh(f ) >0. In that case,N (f )has at least one face of dimension 1. Choose one,S, and a root ofFS with multiplicityν ≥1.

Letap+bq =Nbe the equation of the supporting line ofS. Then f_σ(x₁, y₁)=x₁^Ng₁(x₁, y₁),

withNinN,g₁(x₁, y₁)ink[x₁, y₁]and the height ofΔ(f₁)is equal toν ≤ h(f ).

Either we haveh(f_σ) = 0, orh(f ) ≥ h(f_σ) ≥ 1. We continue the process and either the height vanishes or it stabilizes at a positive value.

Example 1 Letf be the polynomial

f (x, y)=(x³−y²)²x²+y⁷.

The Newton polygon has two compact faces of dimension one. Then, we consider v₀ =(0,7), v1 = (2,4), v2 =(8,0). The faceγ₁ = [v₀, v₁]has supporting line with equation 3α+2β =14 and polynomial facey⁴(x²+y³). The faceγ₂= [v₁, v₂] has supporting line 2α+3β = 16 and face polynomialx²(x³−y²)². Relatively to the faceγ₁, the Newton map givesx = x₁³, y = x₁²(y₁−1)and the Newton transform is

f₁¹(x₁, y₁)=x₁¹⁴(3y1−2x₁⁵+g(x₁, y₁)).

The Newton algorithm stabilizes in height 1. Relatively to the faceγ₂, the Newton map givesx =x₁², y=x₁³(y₁+1)and the Newton transform is

f₁²(x₁, y₁)=x₁¹⁶(4y₁²+x₁⁵+g₁(x₁, y₁)).

There is a second stepx₁= −x₂², y=x₂⁵(y₂+1/2)and the Newton transform is f₂²(x₂, y₂)=x₂⁴²(−512y2+664x₂¹⁰+g₂(x₂, y₂)).

The Newton algorithm stabilizes in height 1.

4 Motivic Milnor Fibers and Newton Algorithm

Notation 1 Letkbe an algebraically closed field of characteristic zero. Letf be a polynomial ink[x, y]withf (0,0)=0. Letωbe the differential formω=x^ν−1dx∧ dywithν ≥1. Ifxdoes not dividef then we assumeν=1.

In this section, using the Newton algorithm Sect.3.1and the strategy of the first author and Veys in [6] we express the motivic zeta function(Z_f,ω(T ))_(0,0)and the motivic Milnor fiber

S_f

(0,0)in terms of the Newton polygons off and its Newton transforms.

Remark 7 Compared to [6], we work in this article relatively to the standard action on arcs, varieties involved are, then, endowed with an action of the multiplicative groupGm. Applying the Newton algorithm as in [6] we have to take into account these actions and use the construction of the Grothendieck ringM_G^G_m^m to identify classes[X→Gm, σ]and[X→Gm, σ]for different actionsσandσofGm. This identification allows factorization in zeta functions, see for instance Proposition5 and its proof.

Theorem 2 Let f be a polynomial in k[x, y] with f (0,0) = 0, let ω be the differential form x^ν−1dx ∧ dy, with ν = 1 if x does not divide f. Denote by γ_h =(a_h, b_h)andγ_v =(a_v, b_v)the zero dimensional faces contained respectively in the horizontal and vertical faces of N(f ). The motivic zeta function off in (0,0)relatively to the differential formωis equal to

Zf,ω(T )

(0,0) =

(a,b)∈{γ_h,γ_v}[x^ay^b:G^rm→Gm, σ_Gm]R_xay^b,ω(T ) +

γ∈N(f )\{γ_h,γ_v}"

fγ :G²m\(fγ =0)→Gm, σγ

#Rγ ,ω(T ) +

γ∈N(f ),dimγ=1

μ∈Rγ

Z_{fσ (p,q,μ),ωp,q}(T )

(0,0),

(4) where

• r=1ifa=0orb=0andr=2ifaandbare non zero,

• σ (p, q, μ)are the associated Newton transforms andωp,qis the differential form ωp,q(v, w)=v^(νp+q−1)dv∧dw,

• for a faceγ,Rγ ,ωare rational fonctions in formulas(12),(13),(14)and(15)of Proposition3and formulas(19)and(20)of Proposition5,

• For a face(a, b)in{γh, γv}the actionσ_GmofGmis defined in Proposition3. For a faceγnot contained in{γh, γv}the actionσγ ofGmis defined in Proposition5.

Furthermore, the motivic Milnor fiber off in(0,0)is : S_f

(0,0)=

(a,b)∈{γh,γv}(−1)^r+1[x^ay^b:G^rm→Gm, σ_Gm] +

γ∈N(f )\{γh,γv}(−1)^{dim(γ )−1}"

fγ :G²m\(fγ =0)→Gm, σγ

# +

γ∈N(f ),dimγ=1

μ∈R_γ

Sf_{σ (p,q,μ)}

(0,0).

(5)