Local zeta function for curves, non degeneracy conditions and Newton polygons

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1 CADERNOS DE MATEMÁTICA 02, October (2001) ARTIGO NÚMERO SMA#119 Local zeta function for cures, non degeneracy conditions and Newton polygons Marcelo José Saia * Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Uniersidade de São Paulo - Campus de So Carlos, Caixa Postal 668, , São Carlos SP, Brazil mjsaia@icmc.sc.usp.br W. A. Zuniga-Galindo Laboratorio de Cómputo Especializado, Uniersidad Autónoma de Bucaramanga, A.A. 1642, Bucaramanga, Colombia Current address: Department of Mathematics and Computer Science, Barry Uniersity, N.E. Second Aenue Miami Shores, Florida 33161, USA wzuniga@bumanga.unab.edu.co This paper is dedicated to the description of the poles of the Igusa local zeta functions Z(s, f, ) when f(x, y) satisfies a new non degeneracy condition, that we hae called arithmetic non degeneracy. More precisely, we attach to each polynomial f(x, y), a collection of conex sets Γ A (f) = Γ f,1,.., Γ f,l0 called the arithmetic Newton polygon of f(x, y), and introduce the notion of arithmetic non degeneracy with respect to Γ A (f). The set of degenerate polynomials, with respect to a fixed geometric Newton polygon, contains an open subset, for the Zariski topology, formed by non degenerate polynomials with respect to some arithmetic Newton polygon.if L is a number field, our main result asserts that for almost all non-archimedean aluations of L, the poles of Z(s, f, ), with f(x, y) L[x, y], can be described explicitly in terms of the equations of the straight segments that conform the boundaries of the conex sets Γ f,1,.., Γ f,l0. Moreoer, our proof gies an effectie procedure to compute Z(s, f, ). October, 2001 ICMC-USP 1. INTRODUCTION Let L be a number field, a non-archimedean aluation of L, and L the completion of L with respect to. Let O the ring of integers of L, P the maximal ideal of O, and L = O / P the residue field of L. The cardinality of L is denoted by q, thus L = F q. * Partially supported by CNPq-Grant /92-6. Partially supported by COLCIENCIAS, Grant # Sob a superisão CPq/ICMC

2 252 M.J. SAIA AND W. A. ZUNIGA-GALINDO We fix a local parameter π for O. For z L, z := q (z) denotes its normalized absolute alue. Gien a non-constant polynomial g(x, y) L[x, y], and a non-archimedean aluation of L, the Igusa local zeta function Z(s, g, ) attached to g(x, y) and is defined as follows: Z(s, g, ) := g(x, y) s dxdy, O 2 where s C satisfying Re(s) > 0 and dxdy denotes a Haar measure of L 2 normalized so that the measure of O 2 is one. Igusa showed that the local zeta function Z(s, g, ) is a rational function of q s, for general polynomials with coefficients in a p adic field [5]. Igusa s local zeta functions are related to the number of solutions of congruences modulo p m and to exponential modulo p m. There are seeral conjectures and intriguing connections between Igusa s local zeta functions with topology and singularity theory [2]. This paper is dedicated to the description of the poles of the Igusa local zeta functions Z(s, f, ) when f(x, y) satisfies a new non degeneracy condition, that we hae called arithmetic non degeneracy. More precisely, we attach to each polynomial f(x, y), a collection of conex sets Γ A (f) = {Γ f,1,.., Γ f,l0 } called the arithmetic Newton polygon of f(x, y), and introduce the notion of arithmetic non degeneracy with respect to Γ A (f) (see section 4). The set of degenerate polynomials, with respect to a fixed geometric Newton polygon, contains an open subset, for the Zariski topology, formed by non degenerate polynomials with respect to some arithmetic Newton polygon. Our main result asserts that for almost all non-archimedean aluations of L, the poles of Z(s, f, ), with f(x, y) L[x, y], can be described explicitly in terms of the equations of the straight segments that conform the boundaries of the conex sets Γ f,1,.., Γ f,l0 (see theorems 6.1, 6.2). Moreoer, our proof gies an effectie procedure to compute Z(s, f, ). The description of the poles of the local zeta functions in terms of Newton polygons is a well known fact for non degenerate polynomials in the sense of Kouchnirenko [13], [10], [3], [4], [16], [17]. Howeer, in the degenerate case there are no preious results showing a relation between the poles of local zeta functions and Newton polygons. A second approach to the description of the poles of local zeta functions is based on resolution of singularities. Currently, as a consequence of the work of many people, among them, Veys [14], [15], Meuser [11], Igusa [7], Strauss [12], there is a completely description of the poles in terms of resolution of singularities for arbitrary polynomials in two ariables. As far as the authors understand the main result of this paper is complementary to those mentioned results. Indeed, our approach does not require the computation of a resolution of singularities, and it gies an effectie procedure to compute the corresponding local zeta function. Howeer, our approach is only alid for a certain class of polynomials, and not for all polynomials. The notion of non degeneracy due Kouchnirenko is useful in the computation of Milnor numbers, number of isolated solutions of polynomial equations, etc., [9]. The authors hope that the notion of arithmetic non degeneracy may be useful for other purposes, like those aboe mentioned, different from the description of the poles of local zeta functions. Sob a superisão da CPq/ICMC

3 LOCAL ZETA FUNCTIONS AND NEWTON POLYGONS 253 In this paper, we work with p adic fields of characteristic zero, howeer all results are alid in positie characteristic. The plan of this paper is as follows. In the section 2, we reiew some well-known results about geometric Newton polygons, non degeneracy in the sense of Kouchnirenko, and computation of certain p adic integrals attached to cones and non degenerate polynomials, in the sense of Kouchnirenko. In section 3, we compute explicitly the local zeta function of some degenerate polynomials, in the sense of Kouchnirenko. These examples constitute the basic models for the effectie computation of the local zeta functions of arithmetically non degenerate polynomials. In section 4, we introduce the notion of arithmetic Newton polygon and arithmetic non-degeneracy condition. In section 5, we compute the seeral p adic integrals that appears in the effectie computation of the local zeta function of an arithmetically non degenerate polynomial. In section 6, we proe the main result. 2. GEOMETRIC NEWTON POLYGONS In this section we reiew some well-known results and notions about Newton polygons, Kouchnirenko s non degeneracy condition, and the computation of certain p adic integrals attached to cones and non degenerate (in the sense of Kouchnirenko) polynomials. All these notions will be used along this paper. We set R + = {x R x 0}. Let f(x, y) = a i,j x i y j K[x, y], be a non-constant polynomial in two ariables with coefficients in a field K, satisfying f(0, 0) = 0. The set supp(f) = { (i, j) N 2 a i,j 0 } is called the support of f. The geometric Newton polygon Γ geom (f) of f is defined as the conex hull in R 2 + of the set { } (i,j) supp(f) (i, j) + R 2 +. By a proper face γ of Γ geom (f), we mean a non-empty conex set γ obtained by intersecting Γ geom (f) with an straight line H, such that Γ geom (f) is contained in one of two half-spaces determined by H. The straight line H is named the supporting line of γ. The faces are points (zero-dimensional faces) or straight segments (one-dimensional faces). The last ones are also called facets. Gien a face γ Γ geom (f) we set f γ (x, y) := a i,j x i y j. (i,j) (i,j) γ Definition 2.1. A non-constant polynomial f(x, y) satisfying f(0, 0) = 0 is named non degenerate with respect to Γ geom (f), in the sense Kouchnirenko, if it satisfies: 1. the origin of K 2 is a singular point of f(x, y); 2. for each proper face γ Γ geom (f), including Γ geom (f) itself, the system of equations does not hae solutions on K 2. f γ (x, y) = f γ x (x, y) = f γ (x, y) = 0, y Sob a superisão CPq/ICMC

4 254 M.J. SAIA AND W. A. ZUNIGA-GALINDO We set <, > for the usual inner product in R 2, and identify the dual ector space with R 2. For a R 2 +, we define m(a) := inf {< a, x >}. The first meet locus of a x Γ(f) R2 + {0} is defined by F (a) := {x Γ geom (f) < a, x >= m(a)}. The first meet locus F (a) of a is a proper face of Γ geom (f). Gien a face γ of Γ geom (f), the cone associated to γ is defined to be the set γ := { a (R + ) 2 {0} F (a) = γ }. We define an equialence relation on R 2 + {0} by a a if and only if F (a) = F (a ). The following two propositions describe the geometry of the equialences classes of (see e.g. [13], [3]). Proposition 2.1. Let γ be a proper face of Γ geom (f). If γ is a one-dimensional face and a is perpendicular ector on γ, then γ = {αa α R, α > 0}. (2.1) If γ is a zero-dimensional face and w 1, w 2 are the facets of Γ geom (f) which contain γ, and a 1, a 2 are ectors which are perpendicular on respectiely w 1, w 2, then γ = {α 1 a 1 + α 2 a 2 α i R, α i > 0}. (2.2) A set of the form (2.2) (respectiely (2.1)) is called a cone strictly positie spanned by the ectors {a 1, a 2 } (respectiely by {a}). Let be a strictly positie spanned by a ector set A, with A = {a 1, a 2 }, or A = {a}. If A is linearly independent oer R, the cone is called a simplicial cone. In this last case, if the elements of A are in Z 2, the cone is called a rational simplicial cone. If A can be completed to be a basis of Z module Z 2, the cone is named a simple cone. A ector a R 2 is called primitie if the components of a are positie integers whose greatest common diisor is one. For eery facet of Γ geom (f) there is a unique primitie ector in R 2 which is perpendicular to this facet. Let D be the set of all these ectors. Thus each equialence class under is a cone strictly positiely spanned by ectors of D. From all aboe, we hae that there exists a partition of R 2 R 2 = {(0, 0)} γ Γ geom (f) γ of the following form:, (2.3) where γ runs through all proper faces of Γ geom (f), and γ denotes the rational simplicial cone associated to the face γ Local zeta functions of non degenerate polynomials Let f(x, y) L [x, y] be a non-constant, non degenerate polynomial with respect to its geometric Newton polygon Γ geom (f). Along this subsection we shall suppose that Γ geom (f) Sob a superisão da CPq/ICMC

5 LOCAL ZETA FUNCTIONS AND NEWTON POLYGONS 255 has only a compact facet, whose supporting line is { } (x, y) R 2 ax + by = d 0. Then the polynomial f(x, y) admits a decomposition of the form f(x, y) = f 0 (x, y) + f 1 (x, y), where f 0 (x, y) is weighted homogeneous with weighted degree d 0, with respect to the weight (a, b), a, b coprime; and all monomials of f 1 (x, y) hae weighted degree higher than d 0. In this particular case, Γ geom (f) induces on R 2 the following partition: R 2 = {(0, 0)} 5 i, (2.4) with 1 := (0, 1)R +, 2 := (0, 1)R + +(a, b)r +, 3 := (a, b)r +, 4 := (a, b)r + +(1, 0)R + and 5 := (1, 0)R +. The polynomial f(x, y) is non degenerate with respect to i, i = 1, 2, 4, 5, i.e. if γ i, i = 1, 2, 4, 5, are the corresponding first meet locus, then the polynomials f γi, i = 1, 2, 4, 5, do not hae singular points on L 2, for i = 1, 2, 4, 5. and We define i=1 E i := {(x, y) O 2 ((x), (y)) i }, Z(s, f,, i ) := Z(s, f,, ) := E i f(x, y) s dxdy, i = 1, 2, 3, 4, 5, f(x, y) s dxdy. With all the aboe notation, and partition (2.4), it follows that Z(s, f, ) = Z(s, f,, ) + 5 Z(s, f,, i ). (2.5) i=1 The computation of the integral Z(s, f,, ) is well-known. It can be computed by using the p adic stationary phase formula, abbreiated SPF ([5], theorem ).We formally summarize, in the case of two ariables, this fact as follows. Lemma 2.1. Let E be a subset of F 2 q and S(f) E a subset consisting of all z E such that f(z) = f f x (z) = y (z) = 0. Let E, S(f) be the preimages of E, S(f) under O O /πo = Fq and N(f, E) the number of zeros of f(x) in E. Then it holds that E f(x, y) s dxdy = q 2 ( Card{E} N(f, E) ) Sob a superisão CPq/ICMC

6 256 M.J. SAIA AND W. A. ZUNIGA-GALINDO ( +q 2 N(f, E) Card{S(f)} (1 q 1 )q s ) 1 q 1 s + S f(x, y) s dxdy. By using an approach based on SPF, it is possible to compute explicitly seeral classes of local zeta functions ([5], [16], [17] and the references therein). In the case in which the polynomial f(x, y) does not hae singularities on L 2, by applying SPF repeatedly, it is possible to compute effectiely the corresponding local zeta function (see lemma 3.1 in [16], or lemma 2.4 in [17] ). We summarize formally this result as follows. Lemma 2.2. Let f(x, y) L [x, y] be a non-constant polynomial such that it does not hae singular points on L 2. Then with U(q s ) Q[q s ]. Z(s, f, ) = U(q s ) 1 q 1 s, The computation of the integrals Z(s, f,, i ), i = 1, 2, 4, 5, is also well-known. There are basically three different forms to study the poles of Z(s, f,, ), when f is non degenerate with respect. The first approach due to Lichtin and Meuser [10] uses toroidal resolution of singularities; this approach is based on Varchenko s approach to estimating oscillatory integrals with non degenerate phase [13]. The second approach due to Denef uses monomial changes of ariables [3] (see also [4]) and the last one approach is due to second author and it is based on SPF formula [16], [17]. We formally summarize, in the case of two ariables, the mentioned result. Lemma 2.3. Let f(x, y) L [x, y] be a non-constant polynomial, with f(0, 0) = 0, and Γ geom (f) its geometric Newton polygon. Γ geom (f) has three facets γ 1, γ 2, γ 3. We set y = d γ1 for the equation of the supporting line of γ 1 ; ax + by = d γ2, with a, b coprime, for the equation of the supporting line of γ 2 ; and x = d γ3 for the equation of the supporting line of γ 3. 1.If f γ1 (x, y) does not hae singularities on the torus L 2, then Z(s, f,, 1 ) = L 1 (q s ) (1 q 1 s )(1 q 1 dγ 1 s ), L 1(q s ) Q[q s ]. L 2 (q s ) 2. Z(s, f,, 2 ) = (1 q 1 s )(1 q 1 d γ s 1 )(1 q (a+b) d γ s 2 ), L 3(q s ) Q[q s ]. L 4 (q s ) 3. Z(s, f,, 4 ) = (1 q 1 s )(1 q 1 d γ s 3 )(1 q (a+b) d γ s 2 ), L 4(q s ) Q[q s ]. Sob a superisão da CPq/ICMC

7 LOCAL ZETA FUNCTIONS AND NEWTON POLYGONS If f γ3 (x, y) does not hae singularities on the torus L 2, then Z(s, f,, 5 ) = L 1 (q s ) (1 q 1 s )(1 q 1 dγ 3 s ), L 5(q s ) Q[q s ]. The core of this paper is the explicit computation of integrals of same type that Z(s, f,, 3 ), when f(x, y) is degenerate in the sense of Kouchnirenko, but satisfies a new non degeneracy condition. This will be done in sections 4, 5. Remark Gien a be a rational simplicial cone, there exists a partition of into simple cones ([8], p ). This fact can be used to obtain a partition of R 2 + in simple cones. In this case, it is possible to compute the integral Z(s, f,, ) in a simple form,when f is non degenerate, in the sense of Kouchnirenko. Howeer this procedure produces a larger list of candidates to poles of Z(s, f,, ). We shall illustrate this procedure in the next section when we shall compute explicitly the local zeta function of (y 3 x 2 ) 2 + x 4 y 4, and (y 3 x 2 ) 2 (y 3 ax 2 ) + x 4 y 4, a MODEL CASES In this section we shall compute explicitly the local zeta functions for some degenerate polynomials, in the sense of Kouchnirenko. These examples constitute the basic models for the effectie computation of the local zeta functions of arithmetically non degenerate polynomials The local zeta function of (y 3 x 2 ) 2 + x 4 y 4 We assume that the characteristic of the residue field of L is different of 2. The origin of L 2 is the only singular point of f(x, y).this polynomial is degenerate with respect to its geometric Newton polygon. By using a decomposition of R 2 + into rational simple cones, it is possible to reduce the computation of Z(s, f, ) to the computation of p adic integrals on rational simple cones, as it was explained in subsection 2.1. The following is the conic decomposition of R 2 + into rational simple cones, obtained from the geometric Newton polygon of f(x, y): R 2 = {(0, 0)} 9 j, (3.1) j=1 with Sob a superisão CPq/ICMC

8 258 M.J. SAIA AND W. A. ZUNIGA-GALINDO 1 := (0, 1)R +, E 1 = {(0, 1)n, n N \ {0}}, 2 := ((0, 1)R + + (1, 1)R + ), E 2 = {(0, 1)n + (1, 1)m, n, m N \ {0}}, 3 := (1, 1)R +, E 3 = {(1, 1)n, n N \ {0}}, 4 := ((1, 1)R + + (3, 2)R + ), E 4 = {(1, 1)n + (3, 2)m, n, m N \ {0}}, 5 =: (3, 2)R +, E 5 = {(3, 2)n, n N \ {0}}, 6 := ((3, 2)R + + (2, 1)R + ), E 6 = {(3, 2)n + (2, 1)m, n, m N \ {0}}, 7 := (2, 1)R +, E 7 = {(2, 1)n, n N \ {0}}, 8 := ((2, 1)R + + (1, 0)R + ), E 8 = {(2, 1)n + (1, 0)m, n, m N \ {0}}. 9 := (1, 0)R +, E 9 = {(1, 0)n, n N \ {0}}. The conic Decomposition of R 2 (3.2) By the considerations done in subsection 2.1, and with the notation introduced there; it is sufficient to compute the integrals Z(s, f,, ), and Z(s, f,, i ), i = 1,.., 9. Computation of Z(s, f,, ) Since f(x, y) does not hae singularities on L 2, the integral can be computed by using SPF. Computation of Z(s, f,, i ), i = 1, 2, 3, 4, 6, 7, 8, 9 These integrals correspond to the case in which f is non degenerate, in the sense of Kouchnirenko, with respect i, i = 1, 2, 3, 4, 6, 7, 8, 9. We compute explicitly the integrals corresponding to i, i = 1, 2, as follows. Z(s, f,, 1 ) = n=1 O π n O f(x, y) s dxdy = = q n n=1 (π 3n y 3 x 2 ) 2 + π 4n x 4 y 4 s dxdy = (1 q 1 ) 2 q n = q 1 (1 q 1 ). n=1 Z(s, f,, 2 ) = m=1 n=1 π m O π n+m O f(x, y) s dxdy = Sob a superisão da CPq/ICMC

9 m=1 n=1 LOCAL ZETA FUNCTIONS AND NEWTON POLYGONS 259 q 2m n 4ms (π 3n+m y 3 x 2 ) 2 +π 4m+4n x 4 y 4 s dxdy = (1 q 1 ) q 2 4s q (1 q 2 4s ). The other integrals can be computed in the same form. The following table summarizes these computations. Cone Z(s, f,, ) 1 q 1 (1 q 1 ) 2 (1 q 1 1 q 2 4s )q (1 q 2 4s ) 3 (1 q 1 ) 2 q 2 4s (1 q 2 4s ) 4 (1 q 1 ) 2 q 7 16s (1 q 2 4s )(1 q 5 12s ) (1 q 1 ) 2 q 8 18s (1 q 5 12s )(1 q 3 6s ) (1 q 1 ) 2 q 3 6s (1 q 3 6s ) (1 q 1 )q 4 6s (1 q 3 6s ) 9 q 1 (1 q 1 ) The Zeta functions on each cone (3.3) Computation of Z(s, f,, 5 ) (an integral on a degenerate face in the sense of Kouchnirenko) Z(s, f,, 5 ) = n=1 π 3n O π 2n O f(x, y) s dxdy = (3.4) = q 5n 12ns n=1 (y 3 x 2 ) 2 + π 8n x 4 y 4 s dxdy. In order to compute (3.4), we first compute the integral I(s, f (n), ) := (y 3 x 2 ) 2 + π 8n x 4 y 4 s dxdy, n 1. (3.5) Sob a superisão CPq/ICMC

10 260 M.J. SAIA AND W. A. ZUNIGA-GALINDO For that, we use the following change of ariables Φ : O 2 O 2 (x, y) (x 3 y, x 2 y). (3.6) The map Φ gies an analytic bijection of O 2 onto itself, and preseres the Haar measure because its Jacobian J Φ (x, y) = x 4 y satisfies J Φ (x, y) = 1, for eery x, y O. Thus, if we set with f (n) Φ(x, y) = x 12 y 4 f (n) (x, y), (3.7) f (n) (x, y) := (y 1) 2 + π 8n x 8 y 4, (3.8) then from (3.6), and (3.7), it follows that I(s, f (n), ) = f (n) (x, y) s dxdy. (3.9) In order to compute the integral I(s, f (n), ), n 1, we decompose O 2 as follows O 2 = O {y 0 + πo } O {1 + πo }, (3.10) y 0 1 mod π where y 0 runs through a set of representaties in O (3.8), it follows that of F q. From the partition (3.10), and I(s, f (n), ) = (q 2)q 1 (1 q 1 ) + O {1+πO } f (n) (x, y) s dxdy. (3.11) The integral in the right side of (3.11), admits the following expansion: I(s, f (n), ) = (q 2)q 1 (1 q 1 ) + j=1 q j f (n) (x, 1 + π j z) s dxdz, (3.12) where f (n) (x, 1 + π j z) = π 2j z 2 + π 8n x 8 (1 + π j z) 4. In the next step, we compute explicitly f (n) (x, 1 + π j z) s, for a fixed n 1. In this case, this can be done by a simple calculation, f q 2js, if 1 j 4n 1, (n) (x, 1 + π j z) s = q 8ns z 2 + x 8 (1 + π j z) 4 s, if j = 4n, q 8ns (1 q 1 ) 2, if j 4n + 1, (3.13) Sob a superisão da CPq/ICMC

11 LOCAL ZETA FUNCTIONS AND NEWTON POLYGONS 261 for any (x, z). We note that the explicit computation of the absolute alue of f (n) (x, 1 + π j z) depends on an explicit description of the set { (w, z) R 2 w min{2z, 8n} }, for a fixed n. We shall see in section 4 that this kind of set is the basis to construct the arithmetic Newton polygon associated to f(x). Now from (3.12), (3.13), and the identity expression for I(s, f n, ): B k=a z k = za z B+1 1 z, we obtain an explicit q 1 2s I(s, f (n), ) = (q 2)q 1 (1 q 1 ) + (1 q 1 ) 2 (1 q 1 2s ) + (3.14) where +q 1 (1 q 1 )q 4n 8ns (1 q 1 ) 2 q 4n 8ns (1 q 1 2s ) + q 4n 8ns I 1 (s), I 1 (s) := z 2 + x 8 (1 + π j z) 4 s dxdz. (3.15) Since the reduction modulo π of the polynomial z 2 + x 8 (1 + π j z) 4 does not hae singular points on F 2 q (the characteristic of the residue field is different of 2), SPF implies that I 1 (s) = U1(q s ) (1 q 1 s ),where U 1(q s ) is a polynomial in q s independent of j and n. Finally, from (3.4), (3.9) and (3.14), we obtain the following explicit expression for Z(s, f,, 5 ) : Z(s, f,, 5 ) = (q 2)(1 q 1 )q 6 12s (1 q 5 12s ) + (1 q 1 )q 10 20s (1 q 9 20s ) + (1 q 1 ) 2 q 6 14s (1 q 1 2s )(1 q 5 12s ) + (3.16) (1 q 1 ) 2 q 9 20s (1 q 1 2s )(1 q 9 20s ) + q 9 20s U 1 (q s ) (1 q 1 s )(1 q 9 20s ). From the explicit formulas for Z(s, f,, i ), i = 1, 2,.., 8, and Z(s, f,, ) (see (3.2), (3.16)), it follows that the real parts of the poles of Z(s, f, ) belong to the set { 1} { 5 12 } { 1 2, 9 }. (3.17) The local zeta function of (y 3 x 2 ) 2 (y 3 ax 2 ) + x 4 y 4 We set g(x, y) = (y 3 x 2 ) 2 (y 3 ax 2 ) + x 4 y 4 L [x, y], with a O, and a 1 mod π. We assume that the characteristic of the residue field of L is different from 2. The Sob a superisão CPq/ICMC

12 262 M.J. SAIA AND W. A. ZUNIGA-GALINDO polynomial g(x, y) is degenerate with respect to its geometric Newton polygon. The origin of L 2 is the only singular point of g(x, y). By using the same decomposition of R 2 + into rational simple cones as in the aboe example, it is possible to reduce the computation of Z(s, g, ) to the computation of the p adic integrals Z(s, g,, O 2 ), Z(s, g,, i ), i = 1,.., 8, 9. Computation of Z(s, g,, O 2 ) Since g(x, y) does not hae singularities on O 2, the integral can be computed by using SPF. Computation of Z(s, g,, i ), i = 1, 2, 3, 4, 6, 7, 8, 9 These integrals can be computed as the corresponding integrals in the aboe example. We summarize these computations bellow. Cone Z(s, g,, ) 1 q 1 (1 q 1 ) 2 (1 q 1 ) q 3 6s (1 q 2 6s ) 3 (1 q 1 ) 2 q 2 6s (1 q 2 6s ) 4 (1 q 1 ) 2 q 7 24s (1 q 2 6s )(1 q 5 18s ) (1 q 1 ) 2 q 8 27s (1 q 5 18s )(1 q 3 9s ) (1 q 1 ) 2 q 3 9s (1 q 3 9s ) (1 q 1 )q 4 9s (1 q 3 9s ) 9 q 1 (1 q 1 ) The Zeta functions on each cone (3.18) Computation of Z(s, f,, 5 ) (an integral on a degenerate face in the sense of Kouchnirenko)) Z(s, g,, 5 ) = n=1 π 3n O π 2n O g(x, y) s dxdy (3.19) = q 5n 18ns n=1 (y 3 x 2 ) 2 (y 3 ax 2 ) + π 2n x 4 y 4 s dxdy. Sob a superisão da CPq/ICMC

13 LOCAL ZETA FUNCTIONS AND NEWTON POLYGONS 263 In order to compute (3.19), we first compute the integral I(s, g (n), ) := (y 3 x 2 ) 2 (y 3 ax 2 ) + π 2n x 4 y 4 s dxdy, n 1. (3.20) By using the change of ariables Φ(x, y) defined in (3.6), we hae that g (n) Φ(x, y) = x 18 y 6 g (n) (x, y), (3.21) Since Φ(x, y) gies a bijection of O 2 from (3.20), and (3.21), that with g (n) (x, y) = (y 1) 2 (y a) + π 2n x 2 y 2. (3.22) I(s, g (n), ) = onto itself presering the Haar measure, it follows g (n) (x, y) s dxdy. (3.23) In order to compute the integral I(s, g (n), ), we decompose as follows: = ( O {y 0 + πo y 0 1, a mod π} ) ( O {1 + πo } ) ( O {a + πo } ). (3.24) From the aboe partition it follows that I(s, g (n), ) = (1 q 1 )q 1 (q 3)+ + q j j=1 O O g (n) (x, 1 + π j y) s dxdy + q j j=1 O O g (n) (x, a + π j y) s dxdy, (3.25) with g(n) (x, 1 + π j y) = π 2j γ 1 (x, y)y 2 + π 2n γ 2 (x, y)x 2, (3.26) where γ 1 (x, y) = 1 a + π j y, γ 2 (x, y) = (1 + π j y) 2, and g (n) (x, a + π j y) = π j δ 1 (x, y)y + π 2n δ 2 (x, y)x 2, (3.27) where δ 1 (x, y) = (a 1 + π j y) 2, δ 2 (x, y) = (a + π j y) 2. In the next step, we compute explicitly g (n) (x, 1 + π j y) s, and g (n) (x, a + π j y) s, for a fixed n 1. By some simple calculations, it holds that g q 2js, if 1 j n 1, (n) (x, 1 + π j y) s = q 2ns γ 1 (x, y)y 2 + γ 2 (x, y)x 2 s, if j = n, q 2ns, if j n + 1, (3.28) Sob a superisão CPq/ICMC

14 264 M.J. SAIA AND W. A. ZUNIGA-GALINDO for any (x, z) O 2, and g q js, if 1 j 2n 1, (n) (x, a + π j y) s = q 2ns δ 1 (x, y)y + δ 2 (x, y)x 2 s, if j = 2n, q 2ns, if j 2n + 1, (3.29) We note that the explicit computation of the absolute alues of g(n) (x, 1 + π j y), respectiely, g (n) (x, a+π j y),, for a fixed n, depends on an explicit description of the set { (w, z) R 2 w min{2z, 2n} }, respectiely, of the set { (w, z) R 2 w min{z, 2n} }. We shall see in section 4 that this kind of set is the basis to construct the arithmetic Newton polygon associated to f(x). Now, from (3.25),(3.28), (3.29) and the identity B z k = za z B+1 1 z, we obtain the following explicit expression for I(s, g (n), ) : k=a I(s, g (n), ) = q 1 (1 q 1 )(q 3) + ( 1 q 1) 2 ( 1 q 1) 2 q 1 2s + (3.30) 1 q 1 2s q n 2ns 1 q 1 2s + q n 2ns I 1 (s) + q 1 ( 1 q 1) q n 2ns + + ( 1 q 1) 2 q 1 s 1 q 1 s ( 1 q 1) 2 q 2n 2ns 1 q 1 s + q 2n 2ns I 2 (s) + q 1 ( 1 q 1) q 2n 2ns where I 1 (s) := γ 1 (x, y)y 2 + γ 2 (x, y)x 2 s dxdz, (3.31) I 2 (s) := δ 1 (x, y)y + δ 2 (x, y)x 2 s dxdz. Since the reduction modulo π of each of the polynomials γ 1 (x, y)y 2 +γ 2 (x, y)x 2, δ 1 (x, y)y +δ 2 (x, y)x 2 does not hae singular points on F 2 q (the characteristic of the residue field is different of 2), SPF implies that I 1 (s) = U 1(q s ) 1 q 1 s, I 2(s) = U 2(q s ), (3.32) 1 q 1 s where U 1 (q s ), and U 1 (q s ) are polynomials in q s, independent of j, and n. Finally, from (3.19), and (3.30), we obtain an explicit expression for Z(s, f,, 5 ) : Sob a superisão da CPq/ICMC

15 LOCAL ZETA FUNCTIONS AND NEWTON POLYGONS 265 Z(s, f,, 5 ) = (q 3)(1 q 1 )q 6 18s (1 q 5 18s ) + (1 q 1 ) 2 q 6 20s (1 q 1 2s )(1 q 5 18s ) + (1 q 1 ) 2 q 6 20s (1 q 1 2s )(1 q 6 20s ) + q 6 20s U 1 (q s ) (1 q 1 s )(1 q 6 20s ) + (1 q 1 )q 7 20s (1 q 6 20s ) (1 q 1 ) 2 q 6 19s + (1 q 1 s )(1 q 5 18s ) (1 q 1 ) 2 q 7 20s (1 q 1 s )(1 q 7 20s ) q 7 20s U 2 (q s ) + (1 q 1 s )(1 q 7 20s ) + (1 q 1 )q 8 20s (1 q 7 20s. (3.33) ) From the explicit formulas for Z(s, f,, i ), i = 1, 2,.., 8, and Z(s, f,, O 2 ) (see (3.18), (3.33)), it follows that the real parts of the poles of Z(s, f, ) belong to the set { 1} { 5 18 } { 1 2, 3 10 } { 7 }. (3.34) ARITHMETIC NEWTON POLYGONS AND NON DEGENERACY CONDITIONS In this section we introduce the notions of arithmetic Newton polygon and arithmetic non degeneracy. With these notions we construct an open subset of polynomials, for the Zariski topology, contained in the set of degenerate polynomials with a fixed geometric newton polygon Γ geom (f 0 ). We hae called this set of polynomials arithmetically non degenerate. The releance of this fact is that the poles of the local zeta function of an arithmetically non degenerate polynomial can be described in terms of the corresponding arithmetic Newton polygon (see section 6) Arithmetically non degenerate semiquasihomogeneous polynomials Let K be a field, and a, b two coprime positie integers. Along this paper, a quasihomogeneous polynomial g(x, y) K[x, y] with respect to the weight (a, b) will be a polynomial of type g(x, y) = cx u y l ( y a α i x b) e i, satisfying g(t a x, t b y) = t d g(x, y), (4.1) i=1 for eery t K. The integer d is the weighted degree of g(x, y) with respect to (a, b). The aboe definition of quasihomogeneity assumes implicitly that the polynomial g(x a, y b ) factorizes as a product of linear polynomials with coefficients in K. Sob a superisão CPq/ICMC

16 266 M.J. SAIA AND W. A. ZUNIGA-GALINDO We put f j (x, y) := c j x uj y j l j i=1 of f j (x, y) with respect to (a, b), i.e. d j := ab ( y a α i,j x b) e i,j, and call dj the weighted degree l j e i,j i=1 + au j + b j. (4.2) A semiquasihomogeneous polynomial f(x, y) with coefficients in K, in two ariables, is a polynomial of the form f(x, y) = l f j=0 f j (x, y), with d 0 < d 1 <.. < d lf. (4.3) The polynomial f 0 (x, y) is called the quasihomogeneous part of f(x, y). If f 0 (x, y) has singular points on the torus K 2, i.e., if some e i,0 > 1, i = 1, 2,.., l 0, the polynomial f(x, y) is degenerate, in the sense of Kouchnirenko, with respect to its geometric Newton polygon Γ geom (f). From now on, we shall suppose that f(x, y) is a degenerate polynomial in the sense of Kouchnirenko. Definition 4.1. Let f(x, y) K[x, y] be a semiquasihomogeneous polynomial, θ K a fixed root of f 0 (1, θ a ) = 0, and e j,θ the multiplicity of θ as root of the polynomial function f j (1, y a ). To each term f j (x, y) in (4.3), we associate an straight line of the form w j,θ (z) := (d j d 0 ) + e j,θ z, j = 0, 1,.., l f, and define the arithmetic Newton diagram N f,θ of f(x, y) at θ as N f,θ := {(z, w) R 2 + w min 0 j l f {w j,θ (z)}. (4.4) Let N f,θ denotes the topological boundary of N f,θ. The set Γ f,θ := N f,θ \ {(z, w) R 2 + w = 0}, will be called the arithmetic Newton polygon of f(x, y) at θ. Definition 4.2. Let f(x, y) K[x, y] be a non-constant semiquasihomogeneous polynomial. The arithmetic Newton diagram N A (f) of f(x, y) is defined to be the set N A (f) := {N f,θ θ K, f 0 (1, θ a ) = 0}, and the arithmetic Newton polygon Γ A (f) of f(x, y) is defined to be the set Γ A (f) := {Γ f,θ θ K, f 0 (1, θ a ) = 0}. Sob a superisão da CPq/ICMC

17 LOCAL ZETA FUNCTIONS AND NEWTON POLYGONS 267 We highlight that Γ A (f) (and also N A (f)) is a finite collection of conex sets; and no just a conex set as occurs in the case of Γ geom (f). If the polynomial f 0 (x, y) is interpreted as the weighted tangent cone of f(x, y), then for each tangent direction of the form (θ : 1), θ K, there exists an arithmetic Newton polygon Γ f,θ at θ, and all these Newton polygons constitute the arithmetic Newton polygon Γ A (f) of f(x, y). Let θ be a fixed root of f 0 (1, θ a ) = 0. Each arithmetic Newton polygon Γ f,θ is conformed by a finite number of straight segments. A point Q Γ f,θ will be called a ertex, if Q = (0, 0) or if it is the intersection point of two straight segments with different slopes. Let Q k, k = 0, 1,.., r +1, denote the ertices of Γ f,θ, with Q 0 := (0, 0), and Q r+1 is a point at infinity. In order to fix the notation, we shall suppose from now on that the straight segment between Q k 1 and Q k has an equation of the form w k,θ (z) = (D k d 0 ) + E k z, k = 1, 2,.., r + 1. (4.5) Then from the aboe notation, it follows that Q k = (τ k, (D k d 0 ) + E k τ k ), k = 1, 2,.., r, (4.6) with τ k := (D k+1 D k ) E k E k+1 > 0, k = 1, 2,.., r, (4.7) (see in the examples of the next section how to obtain these numbers for specific cases). We note that D k = d jk, and E k = e jk,θ, for some index j k. In particular, D 1 = d 0, E 1 = e 0,θ, and the first equation is w 1,θ (z) = E 1 z. We also remark that the last segment that determines Γ f,θ is a half-line with origin at Q r, i.e. a infinite segment from Q r to Q r+1. This half-line may be horizontal, i.e. E r+1 = 0. Gien a ertex Q of Γ f,θ the face function f Q (x, y) is defined to be the polynomial f Q (x, y) := w j,θ (Q)=0 f j (x, y), (4.8) where w j,θ (z) is the straight line corresponding to the term f j (x, y). Now we introduce a notion of non degeneracy with respect to an arithmetic Newton polygon. Definition 4.3. A semiquasihomogeneous polynomial f(x, y) is named arithmetically non degenerate with respect to Γ f,θ at θ, if it satisfies: 1. the origin of K 2 is a singular point of f(x, y); 2. the polynomial f(x, y) does not hae singular points on K 2 ; 3. the system of equations f Q (x, y) = f Q x (x, y) = f Q (x, y) = 0, does not hae solutions on K 2, for each ertex Q (0, 0) of Γ y f,θ. Sob a superisão CPq/ICMC

18 268 M.J. SAIA AND W. A. ZUNIGA-GALINDO Definition 4.4. A semiquasihomogeneous polynomial f(x, y) with coefficients in a field K, will be called arithmetically non degenerate with respect to Γ A (f), if it satisfies: 1. the origin of K 2 is a singular point of f(x, y); 2. the polynomial f(x, y) does not hae singular points on K 2 ; 3. the polynomial f(x, y) is arithmetically non degenerate with respect to Γ f,θ, for each θ K satisfying f 0 (1, θ a ) = Arithmetically non degenerate polynomials Let f(x, y) K[x, y] be a non-constant polynomial, and Γ geom (f) its geometric Newton polygon. Each facet γ of Γ geom (f) has a perpendicular primitie ector a γ = (a 1 (γ), a 2 (γ)) and the corresponding supporting line has an equation of the form < a, x >= d a (γ). The polynomial f(x, y) is a semiquasihomogeneous polynomial with respect to the weight a γ, for each facet γ of Γ geom (f). Definition 4.5. Let f(x, y) K[x, y] be a non-constant polynomial. The arithmetic Newton diagram N A (f) of f(x, y) is defined to be the set N A (f) := {N A (f γ ) γ a facet of Γ geom (f)}, and the arithmetic Newton polygon Γ A (f) of f(x, y) is defined to be the set Γ A (f) := {Γ A (f γ ) γ a facet of Γ geom (f)}. Definition 4.6. A polynomial f(x, y) K[x, y] is named arithmetically non degenerate with respect to its arithmetic Newton polygon, if it satisfies: 1. the origin of K 2 is a singular point of f(x, y); 2. the polynomial f(x, y) does not hae singular points on K 2 ; 3. for each facet γ of Γ geom (f), the semiquasihomogeneous polynomial f(x, y) with respect to the weight a γ, is arithmetically non degenerate with respect to Γ A (f γ ). As an immediate consequence of the definition of arithmetically non degenerate polynomial, we hae that for a gien fixed geometric Newton polygon Γ geom (f 0 ), the set of degenerate polynomials, in the sense of Kouchnirenko with respect to Γ geom (f 0 ), contains an open set consisting of arithmetically non degenerate polynomials Examples The following three examples illustrate the aboe definitions for specific polynomials, we assume that K is a field of characteristic zero. Sob a superisão da CPq/ICMC

19 LOCAL ZETA FUNCTIONS AND NEWTON POLYGONS 269 Example 4.1. We set f(x, y) = (y 3 x 2 ) 2 + x 4 y 4 K[x, y]. The polynomial f(x, y) is a degenerate, in the sense of Kouchnirenko, semiquasihomogeneous polynomial with respect to the weight (3, 2).The origin of K 2 is the only singular point of f(x, y). In this case the arithmetic Newton polygon of f(x, y) is equal to Γ f,1. The Newton polygon Γ f,1 is conformed by the following two straight segments: w 1,1 (z) = 2z, 0 z 4, and w 2,1 (z) = 8, z 4. Thus D 1 = d 0 = 12, E 1 = 2, D 2 = 20, τ 0 = 0, τ 1 = 4. The arithmetic N f,1 is showed in the next figure. Newton diagram The face functions are f (0,0) (x, y) = (y 3 x 2 ) 2, f (4,8) (x, y) = (y 3 x 2 ) 2 + x 4 y 4. Since f (4,8) (x, y) does not hae singular points on K 2, f(x, y) is arithmetically non degenerate. Example 4.2. Let g(x, y) = (y 3 x 2 ) 5 +(y 3 x 2 ) 3 x 6 y 3 +(y 3 x 2 ) 2 x 12 +x 24 K[x, y]. The polynomial g(x, y) is a degenerate, in the sense of Kouchnirenko, semiquasihomogeneous polynomial with respect to the weight (3, 2). The origin of K 2 is the only singular point of g(x, y). Following the definition 4.1 we obtain g(x, y) = g 0 (x, y) + g 1 (x, y) + g 2 (x, y) + g 3 (x, y), where g 0 (x, y) = (y 3 x 2 ) 5, with d 0 = 30, e 0,1 = 5 and w 0,1 = 5z; g 1 (x, y) = (y 3 x 2 ) 3 x 6 y 3, with d 1 = 42, e 1,1 = 3 and w 1,1 = 12+3z; g 2 (x, y) = (y 3 x 2 ) 2 x 12, with d 2 = 48, e 2,1 = 2 and w 2,1 = z; g 3 (x, y) = x 24, with d 3 = 72, e 3,1 = 0 and w 3,1 = 42. Since there is only one root θ = 1 of g 0 (1, θ a ) = 0, the arithmetic Newton polygon of g(x, y) is equal to Γ g,1, which is conformed by the following three straight segments: w 1,1 (z) = 5z, 0 z 6, w 2,1 (z) = 8 + 2z, 6 z 12, and w 3,1 (z) = 42, z 12. Thus D 1 = d 0 = 30, E 1 = 5, D 2 = 48, E 2 = 2, D 3 = 72, E 3 = 0, τ 0 = 0, τ 1 = 6, τ 2 = 12. We obsere that the segment w(z) = z has only the point (6, 30) in the arithmetic Newton diagram of N g,1, which is showed in the next figure. Sob a superisão CPq/ICMC

20 270 M.J. SAIA AND W. A. ZUNIGA-GALINDO The face functions are g (0,0) (x, y) = (y 3 x 2 ) 5, g (6,30) (x, y) = (y 3 x 2 ) 5 +(y 3 x 2 ) 3 x 6 y 3 +(y 3 x 2 ) 2 x 12, g (12,40) (x, y) = (y 3 x 2 ) 2 x 12 +x 24. Since g (6,30) (x, y) = ( y 3 x 2) 2 ( (y 3 x 2 ) 3 + (y 3 x 2 ) + x 12) has singular points on K 2, g(x, y) is arithmetically degenerate. Example 4.3. We set h(x, y) = (y 5 x 3 ) 4 (y 5 ax 3 )(y 5 bx 3 ) + x 20 K[x, y], with 1 a b. The polynomial h(x, y) is a degenerate, in the sense of Kouchnirenko, semiquasihomogeneous polynomial with respect the weight (5, 3). The origin of K 2 is a singular point of h(x, y), and there is no singularities on the torus K 2. In this case the arithmetic Newton polygon of h(x, y) is equal to {Γ h,1, Γ h,a, Γ h,b }. The Newton polygon Γ h,1 is conformed by the following two straight segments: w 1,1 (z) = 4z, 0 z 5 2, and w 2,1(z) = 10, z 5 2. The arithmetic Newton diagram N h,1 is showed in the next figure Arithmetic Newton Diagram of h(x, y) at 1. The face functions are h (0,0) (x, y) = (y 5 x 3 ) 4 (y 5 ax 3 )(y 5 bx 3 ) h ( 5 2,10) (x, y) = (y 5 x 3 ) 4 (y 5 ax 3 )(y 5 bx 3 ) + x 20. Since h ( 5 2,10) (x, y) has no singular points on K 2, f(x, y) is arithmetically non degenerate with respect to Γ h,1. The Newton polygon Γ h,a is conformed by the following two straight segments: w 1,a (z) = z, 0 z 10, and w 2,a (z) = 10, z 10. The arithmetic Newton diagram N h,a at a is showed in the next figure Sob a superisão da CPq/ICMC

21 LOCAL ZETA FUNCTIONS AND NEWTON POLYGONS 271 The face functions are h (0,0) (x, y) = (y 5 x 3 ) 4 (y 5 ax 3 )(y 5 bx 3 ), and h (10,10) (x, y) = (y 5 x 3 ) 4 (y 5 ax 3 )(y 5 bx 3 ) + x 20. Since h (10,10) (x, y) has no singular points on K 2, f(x, y) is arithmetically non degenerate with respect to Γ h,a. The Newton polygon Γ h,b is equal to Γ h,a. Thus the polynomial h(x, y) is arithmetically non degenerate with respect Γ A h. 5. THE INTEGRALS Z(S, F, V, 3 ) In this section, by using the notion of arithmetic non degeneracy introduced in the aboe section, we shall compute explicitly the degenerate integrals Z(s, f,, 3 ) := f(x, y) s dxdy, where f(x, y) L [x, y] is a semiquasihomogeneous E 3 polynomial with respect to the weight (a, b), a, b coprime, and f(x, y) is a degenerate polynomial, in the sense of Kouchnirenko, but it is arithmetically non degenerate; and the cone 3 is gien by partition (2.4) of R 2, i.e., 3 := (a, b)r +, and E 3 := {(x, y) O 2 ((x), (y)) 3 }. Definition 5.1. Gien a fixed semiquasihomogeneous polynomial f(x, y) non degenerate with respect to Γ A (f), we define for each Γ f,θ, with θ L, f 0 (1, θ a ) = 0, P(Γ f,θ ) := r θ i=1 { 1, (a + b) + τ i, (a + b) + τ i } E i D i+1 + E i+1 τ i D i + E i τ i and P(Γ A (f)) := {θ f 0 (1,θ a )=0} {E r+1 o} { 1 E r+1}, (5.1) P(Γ f,θ ). (5.2) The data D i, E i, τ i, for each i, are obtained from the equations of the straight segments conforming Γ f,θ. The main result of this section is the following. Theorem 5.1. Let f(x, y) = l f j=0 f j (x, y) O [x, y] be a semiquasihomogeneous polynomial, with respect to the weight (a, b), a, b coprime, with f j (x, y) as in (4.2), α i,j, c j O, for eery i, j, and α i,j α r,t mod π, when (i, j) (r, t). If f(x, y) is arithmetically non degenerate with respect to Γ A (f), then the real parts of the poles of Z(s, f,, 3 ) belong to the set { 1} { a + b } P(Γ A (f)). d 0 Sob a superisão CPq/ICMC

22 272 M.J. SAIA AND W. A. ZUNIGA-GALINDO The proof will be gien at the end of this section, and it will be accomplished by computing explicitly seeral p adic integrals. Moreoer the proof proides an effectie method to compute these integrals. The following two examples illustrate the theorem. Example 5.1. Let f(x, y) = (y 3 x 2 ) 2 + x 4 y 4 L [x, y]. We suppose that the characteristic of the residue field is different from 2. The polynomial f(x, y) is arithmetically non degenerate, semiquasihomogeneous polynomial with respect to the weight (3, 2) (see example (4.1). In this case, we hae the following data attached the arithmetic Newton polygon Γ A (f) = Γ f,1 : a = 3, b = 2, D 1 = d 0 = 12, τ 1 = 4, E 1 = 2, D 2 = 20. Thus according the aboe theorem, the real parts of the poles of the integral Z(s, f, ) belong to the set { 1} { 5 12 } { 1 2, 9 20 }. In subsection 3.1, we computed explicitly the local zeta function Z(s, f, ) (cf. (3.17)). Example 5.2. Let g(x, y) = (y 3 x 2 ) 2 (y 3 ax 2 ) + x 4 y 4 L [x, y], with a 1 mod π. We assume that the characteristic of the residue field of L is different from 2. The polynomial g(x, y) is quasihomogeneous with respect to the weight (3, 2). The origin of L 2 is the only singular point of g(x, y). The arithmetic Newton polygon Γ A (g) = {Γ g,1, Γ g,a }. The Newton polygon Γ g,1 is conformed by the following two straight segments: w 1,1 (z) = 2z, 0 z 1, and w 2,1 (z) = 2, z 1. Thus D 1 = d 0 = 18, E 1 = 2, D 2 = 20, E 2 = 0, τ 0 = 0, τ 1 = 1.The polynomial is arithmetically non degenerate with respect Γ g,1. According to the preious theorem the contribution of the arithmetic Newton polygon Γ g,1 to the set of real parts of Z(s, g, ) is { 1 2, 9 20 }. The Newton polygon Γ g,a is conformed by the following two straight segments: w 1,1 (z) = z, 0 z 2, and w 2,1 (z) = 2, z 2. Thus D 1 = d 0 = 18, E 1 = 1, D 2 = 20, E 2 = 0, τ 0 = 0, τ 1 = 2. The polynomial is arithmetically non degenerate with respect Γ g,a. According to the preious theorem the contribution of the arithmetic Newton polygon Γ g,a to the set of real parts of Z(s, g, ) is { 1, 7 20 }. Then according to the preious theorem the real parts of the poles of Z(s, g, ) belong to the set { 1} { 5 18 } { 1 2, 9 20 } { 7 20 }. In subsection 3.2, we computed explicitly the local zeta Z(s, g, ) (cf. (3.34)). Sob a superisão da CPq/ICMC

23 LOCAL ZETA FUNCTIONS AND NEWTON POLYGONS Computation of the integrals Z(s, f,, 3 ) The explicit computation of the integrals Z(s, f,, 3 ) is based on the explicit computation of seeral simple p adic integrals. In this subsection we present all these computations. Proposition 5.1. Let f(x, y) O [x, y] be a semiquasihomogeneous polynomial, with respect to the weight (a, b), a,b coprime, and f (m) (x, y) := π d0 f(π a x, π b y) = l f j=0 π (dj d0)m f j (x, y), (5.3) with f j (x, y) as in (4.2). Then there exists a measure-presering bijection Φ :, with (u, w) (Φ 1 (x, y), Φ 2 (x, y)), (5.4) such that f (m) Φ(x, y) = u N i w M i f (m) (u, w), with f (m) (u, w) not anishing identically on the hypersurface uw = 0, and with f (m) (u, w) = l f j=0 π (dj d0)m fj (u, w), (5.5) f j (u, w) = c j u A j w B j l j (w α i,j ) ei,j, or f j (u, w) = c j u L j w F j i=1 i=1 l j (1 α i,j u) ei,j. (5.6) Proof. We denote by Φ 1 the map Φ 1 (u, w) : O 2 O 2 with (u, w) (x, y), is such that x = u, y = uw, and by Φ 2 the map Φ 2 (u, w) : O 2 O 2 with (u, w) (x, y), is such that x = uw, y = w. Since det Φ 1(u, w) = 1, det Φ 2(u, w) = 1, and the maps Φ 1, Φ 2 gie a measurepresering bijection of O 2 to itself. We shall show that Φ is a finite composition of maps of the form Φ 1 or Φ 2. The proof will be accomplished by induction on min{a, b}. Case min{a, b} = 1 In this case, it is sufficient to take Φ as follows: Φ = { Φ1.. Φ 1, b times, if a = 1, Φ 2.. Φ 2, a times, if b = 1. Induction hypothesis Suppose by induction that the proposition is alid for all polynomials f (m) (x, y) of the form (5.3) satisfying 1 min{a, b} k, with k 2. Sob a superisão CPq/ICMC

24 274 M.J. SAIA AND W. A. ZUNIGA-GALINDO Case min{a, b} = k + 1, k 2 Let f (m) (x, y) be a polynomial of the form (5.3) satisfying min{a, b} = k +1, k 2, and a > b. By applying the Euclidean algorithm to a, b, we hae that a = q 1 b + r 1, 0 r 1 < b, for some q 1, r 1 N. Because a, b are coprime, necessarily 1 r 1 < b. We set Ψ = Φ 2.. Φ 2, q 1 times, i.e. x = uw q1, y = w, thus f (m) Ψ(x, y) = u Ai w Bi f (m) (u, w) with f (m) (u, w) not anishing identically on uw = 0, and f (m) (u, w) = l f j=0 π (d j d 0 )m f j (u, w), (5.7) with f j (u, w) = c j u Cj w Dj l j i=1 ( w r 1 α i,j u b) e i,j. Since min{r 1, b} = r 1, and 1 r 1 b 1 = k, by applying the induction hypothesis applied to the polynomial f (m) (u, ) in (5.7), there exists a map Θ, that is a finite composition of maps of the form Φ 1 or Φ 2, such that ( f (m) Ψ ) Θ = f (m) (Ψ Θ) = u Ni w Mi f (m) (u, ), and f (m) (u, ) has all announced properties in the proposition. Therefore, it is sufficient to take Φ = Ψ Θ. The case b > a is proed in an analogous form. Remark We shall use the aboe proposition, only in the case in which α i,j O, for eery i, j. In this case, we can suppose that f (m) (u, w) = l f j=0 π (d j d 0 )m fj (u, w), (5.8) with fj (u, w) = c j u A j w B j l j i=1 (w α i,j ) ei,j. (5.9) Because f j (u, w) = c j u Lj w Fj l j i=1 (1 α i,ju) e i,j can be transformed into the aboe case by a simple algebraic manipulation. We define I(s, f (m),, ) := f (m) (x, y) s dxdy. Proposition 5.2. Let f(x, y) O [x, y] be a semiquasihomogeneous polynomial, with respect to the weight (a, b), a, b coprime, and f (m) (x, y) = π d0 f(π a x, π b y) = l f j=0 π (dj d0)m f j (x, y), with f j (x, y) as in (4.2). (5.10) Sob a superisão da CPq/ICMC

25 LOCAL ZETA FUNCTIONS AND NEWTON POLYGONS 275 If α i,j, c j O, for eery i, j, and α i,j α r,t mod π, when (i, j) (r, t), then I(s, f (m),, ) = q 2 (q 1 l 0 )(q 1)+ + q k {θ O f 0 (1,θ a )=0} k=1 O O with l 0 :=Card({θ O f 0 (1, θ a ) = 0} and f (m) (x, θ + π k y) s dxdy, f (m) (x, θ + π k y) = l f j=0 where γ j (x, y) F q, for all (x, y) O O. π (dj d0)m+ke j,θ γ j (x, y)y e j,θ, (5.11) Proof. By proposition (5.1), it holds that I(s, f (m),, ) = f (m) (x, y) s dxdy = f (m) (x, y) s dxdy. (5.12) Now, we take the partition = A B, with A := O {z O y θ mod π, for any θ O, f 0 (1, θ a ) = 0 t}, B := O Thus I(s, f (m),, O 2 ) = A {θ O f 0(1,θ a )=0} f (m) (x, y) s dxdy + {z O y θ mod π}. B f (m) (x, y) s dxdy. (5.13) Since f (m) (x, z + πy) = 1, when z θ mod π (cf. remark 5.1, and α i,j α r,t mod π, when (i, j) (r, t) ), it holds that A f (m) (x, y) s dxdy = q 2 (q 1 Card({θ O f 0 (1, θ a ) = 0}))(q 1). (5.14) Now, the results follows from (5.13) and (5.14). Gien a real number x, [x] denotes the greatest integer less or equal to x. Proposition 5.3. With the hypothesis of proposition 5.2. If f(x, y) is arithmetically non degenerate with respect to Γ f,θ, then Sob a superisão CPq/ICMC

26 276 M.J. SAIA AND W. A. ZUNIGA-GALINDO J(s, m, θ) := q k k=1 O O f (m) (x, θ + π k y) s dxdy = (5.15) r 1 (1 q 1 ) 2 q (1+sE i+1) 1 q (1+sE i+1) q (D i+1 d 0 )ms (1+sE i+1)[mτ i ] + i=0 r 1 (1 q 1 ) q (1+sE i+1) q (D i+1 d 0 )ms (1+sE i+1)[mτ i+1] + i=0 +(1 q 1 ) 2 q (1+sEr+1) 1 q (1+sE r+1) q (Dr+1 d 0)ms (1+sE r+1)[mτ r] + + r q (D i d 0 )ms (1+sE i )[mτ i ] I i (s), i=1 where τ i, i = 0, 1,.., r, are the abscissas of the ertices of Γ f,αl,0, and I i (s) = L j(q s ) 1 q 1 s, L i (q s ) Q[q s ], i = 1, 2,.., r. Moreoer the I i (s) do not dependent on k. Proof. The proof is based on an explicit computation of the adic absolute alue of f (m) (x, θ + π k y), when x, y O. In order to accomplish it, we attach to f (m) (x, θ + π k y) the conex set N f g defined as follows. We associate to each term (m) (x,θ+π k y) π (d j d 0 )m+ke j,θ γ j (x, y)y e j,θ (5.16) of f (m) (x, θ + π k y) (see (5.11) ) an straight line of the form w j,θ ( z) := (d j d 0 )m + e j,θ z, j = 0, 1,.., l f, and associate to f (m) (x, θ + π k y) the conex set N f (m) (x,θ+π k y) = {( z, w) R 2 + w min 0 j l f { w j,θ ( z)}}. (5.17) Now, we set Ϝ m : R 2 R 2 with u = ũ (u, w) (ũ, w), m, z = ũ m. (see (4.4)). The homomorphism Ϝ f (m) (x,θ+π k y) m sends the topologi- Then Ϝ m (N f,θ ) = N g cal boundary of N f,θ into the topological boundary of N f g (m) (x,θ+π j y) N g f (m) (x,θ+π k y) are. Thus the ertices of Sob a superisão da CPq/ICMC

27 LOCAL ZETA FUNCTIONS AND NEWTON POLYGONS 277 Ϝ m (Q i ) := {.(mτ i, (D i d 0 )m + me i τ i ), i = 1, 2,.., r,(0, 0), i = 0; (5.18) where the τ i are the abscissas of the ertices of Γ f (m),θ (cf. (4.6), (4.7)). Now f (m) (x, θ + π k y) notation introduced at (4.5), (4.6), (4.7)): If mτ i < j < mτ i+1, i = 0, 1,.., r 1, then can be computed from N f g, as follows (we use the (m) (x,θ+π k y) f (m) (x, θ + π k y) = q (Di+1 d0)m Ei+1j, (x, y) O O. (5.19) If j > mτ r, then f (m) (x, θ + π k y) = q (D r+1 d 0 )m E r+1 j, (x, y) O O. (5.20) with If j = mτ i, i = 1, 2,.., r, then f (m) (x, θ + π k y) = q (D i d 0 )m E i j f (m) Ϝ m (Q i ) (x, y), (x, y) O O, (5.21) f (m) Ϝm(Q i )(x, y) = ew i,θ (Ϝ m (Q i ))=0 γ i (x, y)y e i,θ ; here w i,θ ( z) is the straight line corresponding to the term π (d i d 0 )m+ke i,θ γ i (x, y)y e i,θ. Then by using (5.19),(5.20), and (5.21), it follows that J(s, m, θ) = +(1 q 1 ) 2 with I i (s) := k=1 q k r 1 = (1 q 1 ) 2 j=[mτ r ]+1 i=0 O O q (D i+1 d 0 )ms q (D r+1 d 0 )ms (1+sE r+1 )j + f (m) Ϝ m(q i) (x, y) dxdy. f (m) (x, θ + π k y) s dxdy = (5.22) [mτ i+1] 1 j=[mτ i ]+1 q (1+sE i+1)j + r q (D i d 0 )ms (1+sE i )[mτ i ] I i (s), Since f Qi Φ(π ma x, π mb (θ + π k y)) = π md0 x M y N (m) f Ϝ m (Qi) (x, θ + πk y), and f Qi does not hae singularities on L 2, and Φ is a K-analytic isomorphism on L 2 (cf. proposition 5.1); it follows that f (m) Ϝ (x, θ m(q i) +πk y) does not hae singularities on L 2. Therefore by lemma i=1 Sob a superisão CPq/ICMC

28 278 M.J. SAIA AND W. A. ZUNIGA-GALINDO 2.2, it follows that I i (s) = L i(q s ) 1 q 1 s, L i(q s ) Q[q s ], i = 1, 2,.., r. (5.23) Finally, the result follows from (5.22), by using the algebraic identity B k=a z k = za z B+1. (5.24) 1 z Lemma 5.1. Let f(x, y) O [x, y] be a semiquasihomogeneous polynomial, with respect to the weight (a, b), a, b coprime, and f (m) (x, y)=π d0 f(π a x, π b y) = l f j=0 π (dj d0)m f j (x, y), with f j (x, y) as in (4.2), and α i,j, c j O, for eery i, j, and α i,j α r,t mod π, when (i, j) (r, t). If f(x, y) is arithmetically non degenerate with respect to Γ A (f), then I(s, f (m),, ) = q 2 (q 1 l 0 )(q 1) + {θ O f 0 (1,θ a )=0} where the functions J(s, m, θ) were explicitly computed in proposition 5.3. J(s, m, θ), (5.25) Proof. The result follows immediately from propositions 5.2, and 5.3, by using the fact that f(x, y) is arithmetically non degenerate with respect to Γ A (f) Some algebraic identities The following algebraic identities follows easily from (5.24). These identities will be used later on, for the explicit computation of certain p adic integrals. S 1 (s, i) := q (a+b)m d0ms (Di+1 d0)ms (1+sEi+1)[mτi] = (5.26) m=1 with τ i = D i+1 Di E i E i+1, A l := q (a+b) d 0 ms, if i = 0, 1 q = (a+b) d 0 ms Ei E i+1 1 l=0 ( q B i,l A l (γ i +δ i ) s) 1 q γ i δ i s { 0, if l 0, 1, if l = 0., if i = 1,.., r, B i,l := (a + b)l + [lτ i ] + s([lτ i ]E i+1 + ld i+1 ), γ i := (a + b)(e i E i+1 ) + (D i+1 D i ), δ i := D i+1 (E i E i+1 ) + (D i+1 D i )E i+1. (5.27) Sob a superisão da CPq/ICMC