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), 17511779
MSC (2000):
Primary 11S40, 11D79; Secondary 14M25, 52B20, 14G10
Published electronically:
December 15, 2003
MathSciNet review:
2031040
Fulltext PDF Free Access
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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 nondegenerate 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:
http://dx.doi.org/10.1090/S0002994703035074
PII:
S 00029947(03)035074
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
