Zeta Functions for Analytic Mappings, Log-principalization of Ideals, and Newton Polyhedra

In this paper we provide a geometric description of the possible poles of the Igusa local zeta function associated to an analytic mapping and a locally constant function, in terms of a log-principalizaton of an ideal naturally attached to the mapping. Typically our new method provides a much shorter list of possible poles compared with the previous methods. We determine the largest real part of the poles of the Igusa zeta function, and then as a corollary, we obtain an asymptotic estimation for the number of solutions of an arbitrary system of polynomial congruences in terms of the log-canonical threshold of the subscheme given by the ideal attached to the mapping. We associate to an analytic mapping a Newton polyhedron and a new notion of non-degeneracy with respect to it. By constructing a log-principalization, we give an explicit list for the possible poles of the Igusa zeta function associated to a non-degenerate mapping.


Introduction
Let K be a p−adic field, i.e. [K : Q p ] < ∞. Let R K be the valuation ring of K, P K the maximal ideal of R K , and K = R K /P K the residue field of K. The cardinality of the residue field of K is denoted by q, thus K = F q . For z ∈ K, ord (z) ∈ Z ∪ {+∞} denotes the valuation of z, and |z| K = q −ord(z) its absolute value. The absolute value |·| K can be extended to K l by defining z K = max 1≤i≤l |z i | K , for z = (z 1 , . . . , z l ) ∈ K l . Let f 1 , . . . , f l be polynomials in K [x 1 , . . . , x n ], or, more generally, K−analytic functions on an open set U ⊂ K n . We consider the mapping f = (f 1 , . . . , f l ) : K n → K l , respectively, U → K l . Let Φ : K n → C be a Schwartz-Bruhat function (with support in U in the second case). The Igusa local zeta function associated to the above data is defined as for s ∈ C with Re(s) > 0, where | dx | is the Haar measure on K n normalized in such a way that R n K has measure 1. We write Z(s, f ), Z 0 (s, f ) and Z W (s, f ) when Φ is the characteristic function of R n K , P n K , and an open compact subset W of K n , respectively.
The function Z Φ (s, f ) admits a meromorphic continuation to the complex plane as a rational function of q −s . Igusa established this result in the hypersurface case using Hironaka's resolution theorem [16,Theorem 8.2.1]. In the case l ≥ 1 the rationality of Z Φ (s, f ) was established by Meuser in [24], however, as mentioned in the review MR 83g:12015 of [24], a trick by Serre allows to deduce the general case from the hypersurface case. Denef gave a completely different proof of the rationality of Z Φ (s, f ), l ≥ 1, using p-adic cell decomposition [4]. The mentioned results do not give any information about the poles of Z Φ (s, f ) in the case l > 1. In [37] the second author showed that a list of possible poles of Z Φ (s, f ), l ≥ 1, can be computed from an embedded resolution of singularities of the divisor ∪ l i=1 f −1 i (0) by using toroidal geometry. In the special case in which f is a non-degenerate homogeneous polynomial mapping the possible poles of Z Φ (s, f ) are given in [38].
In this paper we provide a geometric description of the possible poles of Z Φ (s, f ), l ≥ 1, in terms of a log-principalization of the K [x] −ideal I f = (f 1 , . . . , f l ) (see Theorem 2.4). At this point it is important to mention that the main result in [37] gives an algorithm to compute a list of possible poles of Z Φ (s, f ), l ≥ 1, in terms of an embedded resolution of singularities of the divisor ∪ l i=1 f −1 i (0), while Theorem 2.4 gives a list of candidates to poles in terms of a log-principalization of the ideal I f . Typically our new method provides a much shorter list of possible poles (see Example 2.5). It is important to mention that in the case l = 1 the problem of determining the poles of the meromorphic continuation of Z Φ (s, f ) in Re(s) < 0 has been studied extensively (see e.g. [3], [14], [28], [23], [32], [34]). The relevance of this problem is due to the existence of several conjectures relating the poles of Z Φ (s, f ) with the structure of the singular locus of f . In the case of polynomials in two variables, as a consequence of the works of Igusa, Strauss, Meuser and the first author, there is a complete solution of this problem [14], [27], [23], [33]. For general polynomials the problem of determination of the poles of Z Φ (s, f ) is still open. There exists a generic class of polynomials named non-degenerate with respect to its Newton polyhedron for which it is possible to give a small set of candidates for the poles of Z Φ (s, f ). The poles of the local zeta functions attached to nondegenerate polynomials can be described in terms of Newton polyhedra. The case of two variables was studied by Lichtin and Meuser [21]. In [5], Denef gave a procedure based on monomial changes of variables to determine a small set of candidates for the poles of Z Φ (s, f ) in terms of the Newton polyhedron of f . This result was obtained by the second author, using an approach based on the p−adic stationary phase formula and Néron p−desingularization, for polynomials with coefficients in a non-archimedean local field of arbitrary characteristic [36], (see also [7], [29]).
In the case l = 1, among the conjectures relating the poles of Igusa's zeta function with topology and singularity theory, we mention here a conjecture of Igusa that proposes that the real parts of the poles of the Igusa zeta function of f are roots of the Bernstein polynomial of f (see e.g. [3], [16], and references therein). It seems reasonable to believe that such relations between poles and singularity theory extend to the case l > 1. Indeed, recently it was proved that the above-mentioned conjecture of Igusa is valid in the case in which I f is a monomial ideal [13].
In the case l = 1, the largest real part of the poles of the Igusa zeta function has been extensively studied both in the archimedean and non-archimedean cases [7], [21], [31], [36]. In the case l ≥ 1 we show that the largest real part −λ (I f ) of the poles of the Igusa zeta function attached to f can easily be determined from a logprincipalization of the ideal I f (see Theorem 2.7). As a consequence of this result we obtain an asymptotic estimation for the number of solutions of an arbitrary system of polynomial congruences in terms of the log-canonical threshold of a logprincipalization (see Corollary 2.9, and the comments that follow). At this point we have to mention that in the case l = 1 Loeser found lower and upper bounds for λ (I f ) in terms of certain geometric invariants introduced by Teissier [22, Theorem 2.6 and Proposition 3.1.1], [30]. In this form he derived a geometric bound for the number of solutions of a polynomial congruence involving one polynomial.
If f is a polynomial mapping with coefficients in a number field F , then for every maximal ideal P of the ring of algebraic integers of F , we can consider Z(s, f , K), l ≥ 1, where K is the completion of F with respect to P . We give an explicit formula for Z(s, f , K), l ≥ 1, that is valid for almost all P (see Theorem 2.10). The proof of this formula follows by adapting the argument given by Denef for the case l = 1 [6].
One can also associate to a sheaf of ideals I on a smooth algebraic variety (over a field of characteristic zero) a motivic zeta function (see Definition 2.16). By using a log-principalization of I we give a similar explicit formula for it (see Theorem 2.17). The proof is a reasonably straightforward generalization of the one given by Denef and Loeser in [8]. By specializing to Euler characteristics one obtains the topological zeta function associated to I.
We attach to an analytic mapping f = (f 1 , . . . , f l ) a Newton polyhedron Γ (f ) and a new notion of non-degeneracy with respect to Γ (f ). The novelty of this notion resides in the fact that it depends on one Newton polyhedron, and Khovanskii's non-degeneracy notion depends on the Newton polyhedra of f 1 , . . . , f l (see [18], [26]). By constructing a log-principalization, we give an explicit list for the possible poles of Z Φ (s, f ), l ≥ 1, in the case in which f is non degenerate with respect to Γ (f ) (see Theorem 3.11). This theorem provides a generalization to the case l ≥ 1 of a well-known result that describes the poles of the local zeta function associated to a non-degenerate polynomial in terms of the corresponding Newton polyhedron [5], [21], [36]. This result was originally established by Varchenko [31] for local zeta functions over R. If f is non-degenerate with respect to Γ (f ), then λ (I f ) can be computed from Γ (f ) in the classical way (see Corollary 3.12).
By using our notion of non-degeneracy and toroidal geometry we give an explicit formula for Z(s, f ) and Z 0 (s, f ), l ≥ 1. This formula generalizes one given by Denef and Hoornaert in the case l = 1 [7,Theorem 4.2], and one given by the second author for the local zeta function of a monomial mapping [36,Theorem 6.1].
The authors wish to thank the referee for his/her constructive remarks about the paper. The first author would like to thank Robert Lazarsfeld for suggestions and several inspiring conversations on poles of zeta functions, and Orlando Villamayor for very useful information concerning principalization in the analytic setting.
2. The Igusa local zeta function of a polynomial mapping 2.1. Log-principalization and poles of the Igusa local zeta function. We state the two versions of log-principalization of ideals that we will use in this paper. The first is the 'classical' algebraic formulation, see for example [11], [12], [35]. The second is in the context of p−adic analytic functions. It follows from the results in [11], see 5.11 in that paper (noticing that 'Property D' there is valid in the p−adic analytic setting).
Theorem 2.1 (Hironaka). Let X 0 be a smooth algebraic variety over a field of characteristic zero, and I a sheaf of ideals on X 0 . There exists a log-principalization of I, that is a sequence has only simple normal crossings and C i has simple normal crossings with E i , and (2) the total transform (σ r ) * (I) is the ideal of a simple normal crossings divisor E # . If the subscheme determined by I has no components of codimension one, then E # is a natural combination of the irreducible components of the divisor E r .
Remark 2.2. We use notations like (σ r ) * (I) as in [35]. However, other authors use the notation IO X for the same object, for example in [11]. As many other authors we use the term 'log-principalization'. The terms 'principalization' and 'monomialization' are also used for the same purpose by other authors. . . , f l ) is not trivial. Then there exists a log-principalization σ : X K → U of I f , that is, (1) X K is an n−dimensional K−analytic manifold, σ is a proper K−analytic map which is a composition of a finite number of blow-ups in closed submanifolds, and which is an isomorphism outside of the common zero set Z K of f 1 , . . . , f l ; (2) σ −1 (Z K ) = ∪ i∈T E i , where the E i are closed submanifolds of X K of codimension one, each equipped with a pair of positive integers (N i , v i ) satisfying the following. At every point b of X K there exist local coordinates (y 1 , . . . , y n ) on X K around b such that, if E 1 , . . . , E p are the E i containing b, we have on some neighborhood of b that E i is given by y i = 0 for i = 1, . . . , p, where ε (y), η (y) are units in the local ring of X K at b.
The (N i , v i ), i ∈ T , are called the numerical data of σ.
Let K be a p−adic field. Let f 1 , . . . , f l be polynomials over K or K−analytic functions on U ⊂ K n . We set I f to be the K−analytic ideal generated by the f i ; we suppose it is not trivial. Let Φ : K n → C or U → C be a Schwartz-Bruhat function, that is, a locally constant function with compact support. We associate to f = (f 1 , . . . , f l ) and Φ the Igusa zeta function Z Φ (s, f ) as in the introduction. The following theorem yields a new proof of its meromorphic continuation, but especially it gives a list of its possible poles in terms of the numerical data of a log-principalization.
Theorem 2.4. The local zeta function Z Φ (s, f ) admits a meromorphic continuation to the complex plane as a rational function of q −s . Furthermore, the poles have the form where the (N i , v i ) are the numerical data of a log-principalization σ : Proof. We pick a log-principalization σ of I f as in Theorem 2.3 and we use all notations that were introduced there. At every point b ∈ X K we can take a chart (V, φ V ) with coordinates (y 1 , . . . , y n ), which may be schrinked later when necessary. Let g(y) be a generator of σ * (I f ) = σ * (f 1 , . . . , f l ) in V . Then where ε(y) and η(y) are units of the local ring of X K at b. Furthermore, since σ * (I f ) is locally generated by g(y) we have f * i (y) = g(y) f i (y), for i = 1, . . . , l, y ∈ V , where each f i (y) is an analytic function on V . And, since g(y) ∈ σ * (I f ), we also have g(y) = l i=1 a i (y)f * i (y), with a i (y) an analytic function on V for each i; therefore Then there exists at least one index i 0 such that f i0 (b) = 0, hence we may assume that f i0 (y) = 0 on V and that We may further suppose that ) is compact open in X K , hence we can express it as a finite disjoint union of compact open sets B α such that each B α is contained in some V above. Since Φ is locally constant we may assume (after sub- The conclusion is now obtained by computing the integral in the previous expression like in the case l = 1 (see [16,Lemma 8.2.1]).
. . , f M ). Let Φ be a Schwartz-Bruhat function whose support is contained in a sufficiently small neighborhood of the origin. A log-principalization of the ideal I f (over a neighborhood of the origin) is obtained by blowing-up the origin of K 2 . There is only one exceptional curve E = P 1 (K) whose numerical datum is (a, 2), and therefore the possible poles of Z Φ (s, f ) have real part −2 a . In [37] an algorithm for computing a list of candidates for the poles of Z Φ (s, f ) in terms of the numerical data of an embedded resolution of the divisor ∪ M j=1 f −1 j (0) was given. Since the f j (x, y) are arbitrary polynomials for 3 ≤ j ≤ M , the mentioned algorithm gives in general a very long list of possible poles.
The following lemma is known by the experts, however we did not find a suitable reference for it; for the sake of completeness we include its proof here.
. . = f l (x 0 ) = 0, by using the continuity of f K , there exist infinitely many i such that there exists Since U is open we have for all those i that the Haar measure of the set is not a Laurent polynomial in q −s .
(2) The second part follows directly from the first one.
. . , f l ) : U −→ K l be an analytic mapping defined on a compact open neighborhood of the origin U such that f (0) = 0. We take a log-principalization σ : Proof. The proof will be achieved by establishing that q λ is the radius of convergence R of Z U (s, f ) considered as a function in q −s . Certainly R≥ q λ , since (by Theorem 2.4) the candidate poles closest to the origin have modulus q λ . We shall show that R≤ q λ by proving a lower bound for the coefficients of Z U (q −s , f ), considered as power series in q −s : Take a generic point b on a component E r with vr Nr = λ, and a small enough chart B (⊂ X K ) around b with coordinates (y 1 , . . . , y n ) such that on B, where |ε| K and |η| K are constant (and nonzero) on B. After an eventual K−analytic coordinate change, we may assume furthermore that B = R n K . Claim. For j big enough and divisible by N r we have where C is a positive constant.
By the above claim we have Therefore, since Z U (q −s , f ) is a rational function of q −s , we conclude that uq λ is a pole of Z U (q −s , f ), for some complex N r -th root of the unity u.
Proof of the claim. By the p−adic change of variables formula [16, Proposition where Jac σ is the Jacobian determinant of σ. With the same reasoning as in the proof of Theorem 2.
Note that on this subset of B we have for some positive constant C.
Remark 2.8. (1) In [15] Igusa showed in the case l = 1 that −λ (I f ) is a pole of Z U (s, f ) for a suitable compact open set U containing the origin. The argument uses Langlands' description of residues in terms of principal value integrals [20]. Furthermore, this argument is valid for archimedean and non-archimedean local zeta functions (see also [2, Théorème 5, part 3a, page 186], [31]).
This well-known important invariant (see e.g. [19], [25]) is defined analogously as λ(I f ) but in a geometric setting, i.e. working over an algebraic closure of K.
In order to obtain a log-principalization in this context maybe more exceptional components are needed, and then the inequality above could be strict.  [24,Theorem 2]). In the proof of the previous theorem was established that q λ is the radius of convergence R of Z (s, f ) considered as a function in q −s . By using this fact, and the above-mentioned relation between P (t, f ) and Z(s, f ), we obtain the following corollary. Corollary 2.9. With the above notation, . . , f l ). Let d be the maximal order of the poles of P (t, f ) with modulus q λ(I f ) . As a consequence of the above corollary and of the rationality of P (t, f ) we have that N j (f ) ≤ Cj d−1 q (n−λ(I f ))j for j big enough, where C is a positive constant. And by Remark 2.8 (2), we have then that N j (f ) ≤ Cj d−1 q (n−lct(I f ))j for j big enough.

2.3.
Denef 's explicit formula. For polynomials f 1 , . . . , f l over a number field F , we can consider local zeta functions Z W (s, f,K) for all (non-archimedean) completions K of F . When l = 1, Denef presented in [6, Theorem 3.1] an explicit formula, which is valid simultaneously for almost all these zeta functions. His arguments extend to the several polynomials case, by replacing resolution by logprincipalization (as in Theorem 2.1).
are the irreducible components of the simple normal crossings divisor given by the principal ideal σ * (I f ). For every maximal ideal P of the ring of integers of F , we consider the completion K of F with respect to P . Denote the valuation ring and the residue field of K by R and K = F q respectively. Then for almost all completions K (i.e. for all except a finite number) we have where W ⊂ R n is a union of cosets mod (P ) n , and Here · denotes the reduction mod P , for which we refer to [6, Sect. 2].
. . , f M as in Example 2.5 as being defined over a number field F . Then the formula of Theorem 2.10 for W = (P ) 2 yields Example 2.12. Let K = Q p , f 1 (x, y) = x, f 2 (x, y) = x + p 0 y, where p 0 is a fixed prime number, and let f = (f 1 , f 2 ). A direct calculation shows that A log-principalization for the ideal I f is attained by blowing-up the origin. One easily verifies that the expression for p = p 0 is the one given by Theorem 2.10.
As a consequence of Theorem 2.4 (or [4], [24] ) Z W (s, f ) can be written as where P (T ) and Q(T ) are polynomials in T = q −s with rational coefficients. We The proof follows from the explicit formula (Theorem 2.10) by analogous arguments as in [6] (or [16]) where the case l = 1 is treated. We should mention that by using model-theoretic arguments Denef already showed the above result (see [6,Theorem 5.2,and Example 5.4]). So in this paper we give a geometric proof of this fact.
Note that for the case p = p 0 in Example 2.12 it is not true that deg Z(s, f , Q p ) = −1, though f 1 , f 2 are homogeneous of degree 1.
We shall present an alternative formula to compute this example in Section 4, where only one candidate pole will appear.
Note that the support of I f has two 1−dimensional components C and C ′ with a singularity at the origin of K 3 .
We first blow up the origin yielding the exceptional surface E 1 ∼ = P 2 with (N 1 , v 1 ) = (2, 3). The strict transform of C and C ′ and E 1 have one common point. Next we blow up this point obtaining the new exceptional surface E 2 ∼ = P 2 with (N 2 , v 2 ) = (3,5). At this stage (the strict transforms of) C and C ′ are disjoint and both meet E 2 in one point of the intersection of E 2 with (the strict transform of) E 1 . Now we blow up the curve E 1 ∩ E 2 ; the new exceptional component E 3 is a ruled surface over that curve and (N 3 , v 3 ) = (6,8). We have that E 3 ∩ E 1 and E 3 ∩ E 2 are disjoint sections of E 3 , and C and C ′ intersect E 3 transversely outside E 3 ∩E 1 and E 3 ∩E 2 . Finally we blow up C and C ′ , yielding the last two exceptional surfaces E 4 and E ′ 4 with numerical data (1,2). The formula of Theorem 2.10 yields where N q −s = (q 2 − q − 1)q −10−7s + (q 2 + q − 1)q −8−6s − (q + 1)q −7−5s +q −4−4s − q −4−3s + (q + 1)q −2−2s .

2.4.
Motivic and topological zeta functions. The analogue of the original explicit formula of Denef plays an important role in the study of the motivic zeta function associated to one regular function [8]. One can associate more generally a motivic zeta function to any sheaf of ideals on a smooth variety, and obtain a similar formula for it in terms of a log-principalization using the argument of [8].
We just formulate the more general definition and formula, referring to e.g. [9], [32] for the notion of jets and Grothendieck ring.
Definition 2.16. Let Y be a smooth algebraic variety of dimension n over over a field F of characteristic zero, and I a sheaf of ideals on Y . Let W be a subvariety of Y . Denote for i ∈ N by X i,W the variety of i−jets γ on Y with origin in W for which ord t (γ * I) = i. The motivic zeta function associated to I (and W ) is the formal power series where [·] denotes the class of a variety in the Grothendieck ring of algebraic varieties over F , and L = A 1 .
Theorem 2.17. Let σ : X → Y be a log-principalization of I. With the analogous notation E i , N i , v i , (i ∈ T ) as before, and also E • In particular Z W (I, T ) is rational in T .
Specializing to topological Euler characteristics, denoted by χ (·), as in [8, (2.3)] or [32, (6.6)] we obtain the expression which is then independent of the chosen log-principalization. (When the base field is not the complex numbers, we consider χ(·) inétale Q-cohomology as in [8].) It can be taken as a definition for the topological zeta function associated to I (and W ), generalizing the original one of Denef and Loeser associated to one polynomial [10].

Newton polyhedra and non-degeneracy conditions
3.1. Newton polyhedra. We set R + := {x ∈ R | x 0}. Let G be a nonempty subset of N n . The Newton polyhedron Γ = Γ (G) associated to G is the convex hull in R n + of the set ∪ m∈G m + R n + . For instance classically one associates a Newton polyhedron (at the origin) to g(x) = m c m x m (x = (x 1 , . . . , x n ), g(0) = 0), being a nonconstant polynomial function over K or K−analytic function in a neighborhood of the origin, where G =supp(g) := {m ∈ N n | c m = 0}. Further we will associate more generally a Newton polyhedron to an analytic mapping.
We fix a Newton polyhedron Γ as above. We first collect some notions and results about Newton polyhedra that will be used in the next sections. Let ·, · denote the usual inner product of R n , and identify the dual space of R n with R n itself by means of it.
For a ∈ R n + , we define We define an equivalence relation in R n + by taking a ∼ a ′ ⇔ F (a) = F (a ′ ). The equivalence classes of ∼ are sets of the form where τ is a face of Γ. We recall that the cone strictly spanned by the vectors a 1 , . . . , a r ∈ R n + \{0} is the set ∆ = {λ 1 a 1 + ... + λ r a r | λ i ∈ R + , λ i > 0}. If a 1 , . . . , a r are linearly independent over R, ∆ is called a simplicial cone. If a 1 , . . . , a r ∈ Z n , we say ∆ is a rational cone. If {a 1 , . . . , a r } is a subset of a basis of the Z-module Z n , we call ∆ a simple cone.
A precise description of the geometry of the equivalence classes modulo ∼ is as follows. Each facet (i.e. a face of codimension one) γ of Γ has a unique vector a(γ) = (a γ,1 , . . . , a γ,n ) ∈ N n \ {0}, whose nonzero coordinates are relatively prime, which is perpendicular to γ. We denote by D(Γ) the set of such vectors. The equivalence classes are rational cones of the form where τ runs through the set of faces of Γ, and γ i , i = 1, . . . , r are the facets containing τ . We note that ∆ τ = {0} if and only if τ = Γ. The family {∆ τ } τ , with τ running over the proper faces of Γ, is a partition of R n + \{0}; we call this partition a polyhedral subdivision of R n + subordinated to Γ. We call ∆ τ τ , the family formed by the topological closures of the ∆ τ , a fan subordinated to Γ.
Each cone ∆ τ can be partitioned into a finite number of simplicial cones ∆ τ ,i . In addition, the subdivision can be chosen such that each ∆ τ ,i is spanned by part of D(Γ) . Thus from the above considerations we have the following partition of R n + \{0}: where τ runs over the proper faces of Γ, and each ∆ τ ,i is a simplicial cone contained in ∆ τ . We will say that {∆ τ ,i } is a simplicial polyhedral subdivision of R n + subordinated to Γ; and that ∆ τ ,i is a simplicial fan subordinated to Γ. By adding new rays , each simplicial cone can be partitioned further into a finite number of simple cones. In this way we obtain a simple polyhedral subdivision of R n + subordinated to Γ; and a simple fan subordinated to Γ (see e.g. [17]).

3.2.
The Newton polyhedron associated to an analytic mapping. Let f = (f 1 , . . . , f l ), f (0) = 0, be a nonconstant polynomial mapping, or more generally, an analytic mapping defined on a neighborhood U ⊆ K n of the origin. In this paper we associate to f a Newton polyhedron Γ (f ) := Γ ∪ l i=1 supp (f i ) , and a non-degeneracy condition to f and Γ (f ).
(2) If for a given face γ at least one f i,γ is a monomial, then the rank condition on γ is satisfied. This is in particular true if γ is a point. Example 3.3. Let f (x, y) = x 3 − xy, y . The mapping f is strongly nondegenerate at the origin with respect to Γ(f ), and also strongly non-degenerate with respect to Γ(f ).
Example 3.4. Let f (x, y, z) = x 2 , y 2 , z 2 , xy, xz, yz . Then f is strongly nondegenerate at the origin with respect to Γ (f ), and also strongly non-degenerate with respect to Γ (f ), 3.2.1. Monomial mappings. Any monomial mapping is strongly non-degenerate at the origin with respect to its Newton polyhedron. If f 0 is a fixed monomial mapping with Newton polyhedron Γ (f 0 ), and f = f 0 +g is a deformation of f 0 such that all the monomials in g have exponents in the interior of Γ (f 0 ), then f is strongly nondegenerate at the origin with respect to Γ (f ) = Γ (f 0 ). This type of mapping was introduced by the second author in [37, Definition 6.1]. Furthermore, the corresponding local zeta function can be computed by using a simple polyhedral subdivision subordinated to Γ (f 0 ) [37, Theorem 6.1].

3.2.2.
Saia's non-degeneracy condition. In [28] Saia introduced the following notion of non-degeneracy for ideals. Let I = (f 1 , . . . f l ) be a polynomial ideal. I is non-degenerate with respect to Γ (I) (where Γ (I) = Γ ∪ l i=1 supp (f i ) ), if for every compact face τ of Γ (I), the system of equations f 1,τ (z) = 0, . . . f l,τ (z) = 0 does not have a solution in the torus (K × ) n . Thus Saia's notion of non-degeneracy is a particular case of our notion of non-degeneracy. Saia's notion of non-degeneracy plays an important role in the study of the integral closure of ideals.

3.2.3.
Khovanskii's non-degeneracy condition. Now we discuss the relation between our notion of non-degeneracy and Khovanskii's notion of non-degeneracy of an analytic mapping with respect to several Newton polyhedra ( [18], see also [26]). Given a positive vector a (i.e. a ∈ (N\ {0}) n ), and an analytic mapping g, we set g a (x) := g F (a) (x), where F (a) is the first meet locus of a with respect to Γ (g).
To make explicit the dependence between F (a) and Γ (g) we shall write F (a, Γ (g)) instead of F (a). . . = f l,a (z) = 0 it verifies that Here f j,a (z) = f j,F (a,Γ(fj )) (z) for every j.
The above definition is equivalent to the non-degeneracy notion given by Oka in [26], that is in turn a reformulation of the notion of non-degeneracy introduced by Khovanskii in [18]. The following is the relation between Khovanskii's non-degeneracy notion and the one introduced here. Proposition 3.7. Let f = (f 1 , . . . , f l ) : U −→ K l be an analytic mapping strongly non-degenerate at the origin with respect to Γ(f ). Then f is non-degenerate with respect to (Γ (f 1 ) , . . . , Γ (f l )).
Hence by using (3.2) the system of equations f j,a (x) = 0, j = 1, . . . , l, has no solutions in K × n , and so the condition on a in Definition 3.5 is satisfied. Now, we may assume that card(I) ≥ min {l, n}, and that f j,τ (x) = 0, j ∈ I, has solutions in K × n .
Since f is strongly non-degenerate with respect to Γ (f ), it follows that for any z ∈ z ∈ (K × ) n | f j,τ (z) = 0, j ∈ I . Then by (3.2), Therefore, f is non-degenerate in the sense of Khovanskii.

Newton polyhedra and log-principalizations.
Proposition 3.9. Let f = (f 1 , . . . , f l ) : U (⊆ K n ) −→ K l be a polynomial mapping (or more generally, an analytic mapping defined on U ) strongly non-degenerate at the origin with respect to Γ (f ). Let F f be a simple fan subordinated to Γ (f ). Let Y K be the toric manifold corresponding to F f , and let be the restriction of the corresponding toric map to the inverse image of U . Denote by Z the set of common zeroes of I f = (f 1 , . . . , f l ) in U ∩ (K × ) n . When U is taken small enough, either Z = ∅ or it is a submanifold of codimension l. In this last case we have l < n and we denote the closure of Z in U and Y K by Z U and Z Y , respectively.
(2) If Z = ∅, we have that Z Y is a closed submanifold of Y K , having normal crossings with the exceptional divisor of σ 0 . Let σ 1 : X K −→ Y K be the blowingup of Y K with center Z Y , and let σ = σ 0 • σ 1 : X K −→ U . Then the ideal σ * (I f ) is principal (and monomial) in a sufficiently small neighborhood of σ −1 {0}.
Proof. We first recall the construction of (Y K , σ 0 ) from a simple fan F f subordinated to Γ (f ) (see e.g. [2]). Let ∆ τ be an n−dimensional simple cone in F f such that F (a) = τ for any a ∈ ∆ τ . Then the face τ of Γ (f ) is necessarily a point. Let a 1 , . . . , a n be the generators of ∆ τ . Then in the chart of Y K corresponding to ∆ τ , the map σ 0 has the form . . , a n ]. Denote this chart by V τ . We slightly abuse notation here : since σ 0 only maps to U instead of to the whole of K n , at some charts it will not be defined everywhere on If supp(f i ) ∩ τ = ∅, then the minimum of all m, a j is attained at τ , and then . Now let us consider on V τ the points on σ −1 0 (0), different from the origin of V τ . We will study simultaneously points with exactly r zero coordinates (where 1 ≤ r ≤ n − 1); after permuting indices, we may assume that the first r coordinates are zero.
We have to study two cases. The first case occurs when there exists an index i such that f i ( p) = 0. In this case as before σ * 0 (I f ) is generated by The second case occurs when f i ( p) = 0, for all i = 1, . . . , l. We recall that, by the non-degeneracy condition, rank K (y) = min{l, n} for y ∈ (K × ) n ∩ {f 1,τ ′ (σ 0 (y)) = · · · = f l,τ ′ (σ 0 (y)) = 0}. Note that by (3.7) this condition on y is equivalent to y ∈ (K × ) n ∩ { f 1 (y) = · · · = f l (y) = 0} and that   6)). This last matrix having rank l implies that we can choose new coordinates y ′ = (y 1 , . . . , y r , y ′ r+1 , . . . , y ′ n ) in a neighborhood V p of p such that Since σ 0 is an isomorphism on (K × ) n , we have that {y ′ r+1 = · · · = y ′ r+l = 0} is the description in V p of Z Y ⊂ Y . (The local description (3.8) yields that Z is a submanifold of (K × ) n of codimension l.) Clearly Z Y is a submanifold of Y of codimension l, having normal crossings with the exceptional divisor of σ 0 . So, σ 1 being the blowing-up of Y in Z Y , we obtain by (3.8) that (σ 0 • σ 1 ) * (I f ) becomes principal.
Remark 3.10. If we replace in Proposition 3.9 the condition strongly non-degenerate at the origin with respect to Γ (f ) by the condition strongly non-degenerate with respect to Γ (f ), and U by K n , with a similar proof we obtain a global version of the proposition, that is, the conclusions (1) and (2) are valid without the condition in a sufficiently small neighborhood. In this case Z Y may have components that are disjoint with the exceptional divisor of σ 0 .
The numerical data of the log-principalizations constructed in Proposition 3.9 and Remark 3.10 can be computed directly from the explicit expressions for the generators of σ * 0 (I f ), σ * (I f ), and Lemma 8 in [2, page 201]. Then Theorem 2.4 yields that the poles of Z Φ (s, f ) belong to the set where the last set may be discarded if l ≥ n. This provides a generalization to the case l ≥ 1 of a well-known result that describes the poles of the local zeta function associated to a non-degenerate polynomial in terms of the corresponding Newton polyhedron [21], [5], [7], [36]. This result was originally established by Varchenko [31] for local zeta functions over R. As in the case l = 1, the list (3.9) is too big. More precisely, the set ∪ ξ∈Λ f P (ξ) is not necessary. This fact is established by analogous arguments as in [5] where the case l = 1 is studied. . . . , f l ) : U −→ K l be an analytic mapping strongly non-degenerate at the origin with respect to Γ (f ). If U is a sufficiently small neighborhood of the origin, and Φ is a Schwartz-Bruhat function whose support is contained in U , then the poles of Z Φ (s, f ) belong to the set ∪ ξ∈D(Γ(f )) P (ξ)∪ log q , k ∈ Z , where the last set may be discarded if l ≥ n.
(2) If f : K n −→ K l is a strongly non-degenerate polynomial mapping with respect to Γ (f ), then the poles of Z (s, f ) belong to the set The above result can be restated in a geometric form as follows. If s is a pole of Z Φ (s, f ), then Re(s) is −l, or Re(s) is of the form −1/t 0 , where (t 0 , . . . , t 0 ) is the intersection point of the diagonal {(t, . . . , t) ∈ R n } with the supporting hyperplane of a facet of Γ (f ).
By using Theorems 2.7 and 3.11 we obtain the following corollary.
(1) Let U be a sufficiently small neighborhood of the origin, and let f = (f 1 , . . . , f l ) : U −→ K l be an analytic mapping strongly non-degenerate at the origin with respect to Γ (f ). Let (t f , . . . , t f ) ∈ Q n be the intersection point of the diagonal {(t, . . . , t) ∈ R n } with the boundary of Γ (f ). If t f ≥ 1/l, then −1/t f is the largest real part of a pole of Z U (s, f ).
(2) Let f : K n −→ K l be a strongly non-degenerate polynomial mapping with respect to Γ (f ). If t f ≥ 1/l, then −1/t f is the largest real part of a pole of Z (s, f ).
The largest real part of the poles of Z (s, f ), l = 1, when f is non-degenerate with respect to its Newton polyhedron Γ (f ) and t f > 1 follows from observations made by Varchenko in [31] and was originally noted in the p−adic case in [21].
The case t f = 1 is treated in [7]. The case of t f < 1 is more difficult and is established in [7] with some additional conditions on Γ (f ) by using a difficult result on exponential sums. In [36] the second author established the case t f ≥ 1 when f is a non-degenerate polynomial with coefficients in a non-archimedean local field of arbitrary characteristic.

Explicit formulas and Newton polyhedra
In [7, Theorem 4.2] Denef and Hoornaert gave an explicit formula for Z(s, f ), l = 1, associated to a polynomial f in several variables over the p−adic numbers, when f is sufficiently non-degenerate with respect to its Newton polyhedron Γ (f ). This explicit formula can be generalized to the case l ≥ 1 by using the condition of non-degeneracy for polynomial mappings introduced in this paper.
Let as before K be a p−adic field with valuation ring R K , maximal ideal P K and residue field K = F q . For any polynomial g over R K we denote by g the polynomial over K obtained by reducing each coefficient of g modulo P K . . . = f l,τ (z) = 0. Analogously we call f strongly non-degenerate at the origin over K with respect to Γ(f ), if the same condition is satisfied but only for the compact faces τ of Γ(f ).  λ j a j | 0 ≤ λ j < 1 for j = 1, . . . , r ∆τ,i .
(2) With the same notations and only assuming that f is strongly non-degenerate at the origin over K we have The proof of the above result is analogous to the case l = 1 treated in [7,Theorem 4.2].
If we would use the natural simple polyhedral subdivision of the one above, introducing two new rays generated by (1, 1) and (1, 2), we would introduce the same superfluous (real) candidate poles −2 and − 3 2 as in Example 2.14. This is reasonable because the log-principalization of Proposition 3.9 associated to this simple fan is in fact the same as the one constructed in Example 2.14.
Remark 4.5. With the obvious analogous definitions for strongly non-degeneracy over C, we have the following. Suppose that f 1 , . . . f l are polynomials in n variables with coefficients in a number field F (⊆ C). Then we can consider f = (f 1 , . . . , f l ) as a map K n → K l for any non-archimedean completion K of F . If f is strongly non-degenerate at the origin over C with respect to Γ (f ), then f is strongly nondegenerate over K with respect to Γ (f ) for almost all the completions K of F . (And analogously for non-degeneracy at the origin.) This fact follows by applying the Weak Nullstellensatz.
Remark 4.6. By using our notion of non-degeneracy with respect to a Newton polyhedron it is also possible to give lists of candidate poles and explicit formulae for the motivic and topological zeta functions introduced in 2.4, associated to a polynomial ideal. These explicit formulas are reasonably straightforward generalizations of those in [1] and [10, Théorème 5.3 (i)]. For the topological zeta function one requires here strongly non-degeneracy with respect to all the faces of the "global" Newton polyhedron as in [10, (5.1)].