Poles of the Igusa local zeta function of some hybrid polynomials

León-Cardenal, Edwin; Ibadula, Denis; Segers, Dirk

doi:10.1016/j.ffa.2013.07.009

Cited by 5 publications

(9 citation statements)

References 10 publications

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“…Remark 1.5. By Theorem 1.3, we obtain the same three poles as the case considered in [10], but we also get a new extra pole, which depends on the characteristic of K and the parameter l.…”

Section: Introduction and The Main Resultssupporting

confidence: 55%

“…where M (q −s , χ) is a polynomial with complex coefficients. This explains why in our situation that l > 1, we can obtain a new extra candidate pole which does not appear in the case l = 1 due to [10].…”

Section: B2mentioning

confidence: 77%

“…Hybrid polynomials were first introduced by Hauser (see [4] or [5]) to study the resolution of singularities in positive characteristic. When r = l = 1 and t k+1 is a p-th power of an arbitrary unit from O K , León-Cardenal, Ibadula and Segers [10] used the method of [13] to prove that Z f (s, χ) is a rational function of q −s . More explicitly, they provided a list with just three candidate poles of Z f (s, χ).…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…where α ∈ O × K , β is a p-th power of an arbitrary unit from O K and n is a positive integer with p | (n + l). If l = 1, then the local zeta function Z f (s, χ) was studied in [10]. In this paper, we are able to treat with the general case l > 1.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

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Igusa local zeta functions of a class of hybrid polynomials

Yin,

Hong

2016

Preprint

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In this paper, we study the Igusa's local zeta functions of a class of hybrid polynomials with coefficients in a non-archimedean local field of positive characteristic. Such class of hybrid polynomial was first introduced by Hauser in 2003 to study the resolution of singularities in positive characteristic. We prove the rationality of these local zeta functions and describe explicitly their poles. The proof is based on Igusa's stationary phase formula.

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“…Remark 1.5. By Theorem 1.3, we obtain the same three poles as the case considered in [10], but we also get a new extra pole, which depends on the characteristic of K and the parameter l.…”

Section: Introduction and The Main Resultssupporting

confidence: 55%

Section: B2mentioning

confidence: 77%

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Section: Introduction and The Main Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Igusa local zeta functions of a class of hybrid polynomials

Yin,

Hong

2016

Preprint

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“…However the stationary phase formula, introduced by Igusa, has proved to be useful in several cases, see e.g. [12,20] an the references therein.…”

Section: Introductionmentioning

confidence: 99%

Igusa’s Local Zeta Functions and Exponential Sums for Arithmetically Non Degenerate Polynomials

Albarracin-Mantilla¹,

León-Cardenal²

2018

J. Théor. Nombres Bordeaux

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We study the twisted local zeta function associated to a polynomial in two variables with coefficients in a non-Archimedean local field of arbitrary characteristic. Under the hypothesis that the polynomial is arithmetically non degenerate, we obtain an explicit list of candidates for the poles in terms of geometric data obtained from a family of arithmetic Newton polygons attached to the polynomial. The notion of arithmetical non degeneracy due to Saia and Zúñiga-Galindo is weaker than the usual notion of non degeneracy due to Kouchnirenko. As an application we obtain asymptotic expansions for certain exponential sums attached to these polynomials.

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A survey of Igusa’s local zeta function

Meuser¹

2016

ajm

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The origin and development of the Igusa local zeta function, by Igusa and others, is presented. In particular, we discuss the various conjectures Igusa made and the notable results that have so far been obtained. We also explain how topological and motivic zeta functions arose from the Igusa local zeta function and present the current status of the analogous conjectures. Igusa's conjecture on exponential sums that are related to his zeta function is described, and along with the progress made towards its resolution.

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Poles of the Igusa local zeta function of some hybrid polynomials

Cited by 5 publications

References 10 publications

Igusa local zeta functions of a class of hybrid polynomials

Igusa local zeta functions of a class of hybrid polynomials

Igusa’s Local Zeta Functions and Exponential Sums for Arithmetically Non Degenerate Polynomials

A survey of Igusa’s local zeta function

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