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Newton polyhedra, unstable faces and the poles of Igusa’s local zeta function

Author: Kathleen Hoornaert
Journal: Trans. Amer. Math. Soc. 356 (2004), 1751-1779
MSC (2000): Primary 11S40, 11D79; Secondary 14M25, 52B20, 14G10
Published electronically: December 15, 2003
MathSciNet review: 2031040
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Abstract: In this paper we examine when the order of a pole of Igusa’s local zeta function associated to a polynomial $f$ is smaller than “expected”. We carry out this study in the case that $f$ is sufficiently non-degenerate with respect to its Newton polyhedron $\Gamma (f)$, and the main result of this paper is a proof of one of the conjectures of Denef and Sargos. Our technique consists in reducing our question about the polynomial $f$ to the same question about polynomials $f_\mu$, where $\mu$ are faces of $\Gamma (f)$ depending on the examined pole and $f_\mu$ is obtained from $f$ by throwing away all monomials of $f$ whose exponents do not belong to $\mu$. Secondly, we obtain a formula for Igusa’s local zeta function associated to a polynomial $f_\mu$, with $\mu$ unstable, which shows that, in this case, the upperbound for the order of the examined pole is obviously smaller than “expected”.

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Additional Information

Kathleen Hoornaert
Affiliation: Department of Mathematics, Catholic University Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium

Keywords: Igusa zeta function, Newton polyhedron, congruences, $p$-adic integrals
Received by editor(s): March 12, 2002
Published electronically: December 15, 2003
Article copyright: © Copyright 2003 American Mathematical Society