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

Author:
Kathleen Hoornaert

Translated by:

Journal:
Trans. Amer. Math. Soc. **356** (2004), 1751-1779

MSC (2000):
Primary 11S40, 11D79; Secondary 14M25, 52B20, 14G10

DOI:
https://doi.org/10.1090/S0002-9947-03-03507-4

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 is smaller than ``expected''. We carry out this study in the case that is sufficiently non-degenerate with respect to its Newton polyhedron , 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 to the same question about polynomials , where are faces of depending on the examined pole and is obtained from by throwing away all monomials of whose exponents do not belong to . Secondly, we obtain a formula for Igusa's local zeta function associated to a polynomial , with 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

DOI:
https://doi.org/10.1090/S0002-9947-03-03507-4

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