On the Degree of Igusa's Local Zeta Function

Denef, Jan

doi:10.2307/2374583

Cited by 136 publications

(173 citation statements)

References 11 publications

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“…Let (X, h) be such a resolution, using now all notations of paragraph 0. DENEF [Dl,thm 2.4 So all real poles of Z(s) are part of the set {-^/7v, \ i e S}. These poles are connected with certain eigenvalues of monodromy by the following conjecture.…”

Section: Rr^+i) Then Igusa's Local Zeta Function Of / Is (The Meromomentioning

confidence: 98%

Poles of Igusa's local zeta function and monodromy

Veys¹

1993

Bul. Soc. Math. France

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Section: Rr^+i) Then Igusa's Local Zeta Function Of / Is (The Meromomentioning

confidence: 98%

Poles of Igusa's local zeta function and monodromy

Veys¹

1993

Bul. Soc. Math. France

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“…[3]). The essential numerical data of π f are the pairs (N i , ν i ) for i ∈ J with J = I \ I ′ and where I ′ is the set of indices i in I such that (N i , ν i ) = (1, 1) and such that E i does not intersect another E j with (N j , ν j ) = (1, 1).…”

Section: Lemma ([8] Lemmamentioning

confidence: 99%

Igusa and Denef-Sperber conjectures on nondegenerate p-adic exponential sums

Cluckers

2008

Duke Math. J.

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Abstract. -We prove the intersection of Igusa's Conjecture of [Igusa, J., Lectures on forms of higher degree, Lect. math. phys., Springer-Verlag, 59 (1978) IntroductionLet f be a polynomial over Z in n variables. Consider the "global" exponential sumwhere N varies over the positive integers. In order to bound |S f (N)| in terms of N, it is enough to boundin terms of m > 0 and prime numbers p. When f is nondegenerate in several senses related to its Newton polyhedron, specific bounds which depend uniformly on m and p have been conjectured by Igusa and by Denef -Sperber. We prove these bounds, thus solving a conjecture by Denef and Sperber from a 1990 manuscript [7] (published in 2001 [8]), and the nondegenerate case of Igusa's conjecture for exponential sums from the introduction of his book [10].One of the main points of this article is that, while for finite field exponential sums like S f (p) one knows that the weights and Betti numbers have some uniform behaviour for big p, for p-adic exponential sums S f (p m ) one does not yet completely

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“…La fonction zêta d'Igusa motivique est un analogue dans le contexte de l'intégration motivique de la fonction zêta d'Igusa locale associéeà un polynôme (voir [7]à ce sujet).…”

Section: Introductionunclassified

Espaces d'arcs et invariants d'Alexander

Guibert¹

2002

Comment. Math. Helv.

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Abstract. We compute the motivic Igusa zeta function of Denef-Loeser associated with a two variables irreducible serie and use this result to give a new proof of the formula expressing the Hodge-Steenbrink spectrum in terms of the Puiseux data. We study a generalisation of the motivic Igusa function to a family of functions and show that this Igusa function is related with the Alexander invariants of the family. Using this result, we obtain a formula for the Alexander plolynomial of a plane curve. Mathematics

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On the Degree of Igusa's Local Zeta Function

Cited by 136 publications

References 11 publications

Poles of Igusa's local zeta function and monodromy

Poles of Igusa's local zeta function and monodromy

Igusa and Denef-Sperber conjectures on nondegenerate p-adic exponential sums

Espaces d'arcs et invariants d'Alexander

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