Poles of a Two-Variable P-Adic Complex Power

Strauss, Leon

doi:10.2307/1999166

Cited by 5 publications

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“…Analogous results have previously appeared in the literature for the p-adic zeta function [Lo88,St83], the topological zeta function [Ve97] and the so-called "naïve" motivic zeta function [Ro04]. 8.3.…”

Section: • Relationssupporting

confidence: 72%

Computing motivic zeta functions on log smooth models

Bultot

Nicaise

2019

Math. Z.

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We give an explicit formula for the motivic zeta function in terms of a log smooth model. It generalizes the classical formulas for snc-models, but it gives rise to much fewer candidate poles, in general. This formula plays an essential role in recent work on motivic zeta functions of degenerating Calabi–Yau varieties by the second-named author and his collaborators. As a further illustration, we explain how the formula for Newton non-degenerate polynomials can be viewed as a special case of our results.

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Section: • Relationssupporting

confidence: 72%

Computing motivic zeta functions on log smooth models

Bultot

Nicaise

2019

Math. Z.

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“…Numerous other works study the poles of the local zeta function [25,34,20,35,3,36,4,7,5,44,45,30,47,38,39,23,24]. Many authors have labored on the calculation of local zeta functions in various situations [28,37,1,5,42,43,21,29,30,9], and many works either use local zeta functions or else apply the methods developed for obtaining them [8,46,48,31,17,18,49,40,33,50].…”

Section: Introductionmentioning

confidence: 99%

A new generating function for calculating the Igusa local zeta function

Cowan

Katz

White

2017

Advances in Mathematics

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A new method is devised for calculating the Igusa local zeta function Z f of a polynomial f (x1, . . . , xn) over a p-adic field. This involves a new kind of generating function G f that is the projective limit of a family of generating functions, and contains more data than Z f . This G f resides in an algebra whose structure is naturally compatible with operations on the underlying polynomials, facilitating calculation of local zeta functions. This new technique is used to expand significantly the set of quadratic polynomials whose local zeta functions have been calculated explicitly. Local zeta functions for arbitrary quadratic polynomials over p-adic fields with p odd are presented, as well as for polynomials over unramified 2-adic fields of the form Q + L where Q is a quadratic form and L is a linear form where Q and L have disjoint variables. For a quadratic form over an arbitrary p-adic field with odd p, this new technique makes clear precisely which of the three candidate poles are actual poles.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Zeta functions for analytic mappings, log-principalization of ideals, and Newton polyhedra

Veys

Zúñiga–Galindo

2007

Trans. Amer. Math. Soc.

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In this paper we provide a geometric description of the possible poles of the Igusa local zeta function Z Φ (s, f) associated to an analytic mapping f = (f 1 ,. .. , f l) : U (⊆ K n) → K l , and a locally constant function Φ, with support in U , in terms of a log-principalizaton of the K [x] −ideal I f = (f 1 ,. .. , f l). 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 I f. We associate 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. 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).

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Poles of a Two-Variable P-Adic Complex Power

Cited by 5 publications

References 0 publications

Computing motivic zeta functions on log smooth models

Computing motivic zeta functions on log smooth models

A new generating function for calculating the Igusa local zeta function

Zeta functions for analytic mappings, log-principalization of ideals, and Newton polyhedra

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