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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Newton polyhedra, unstable faces and the poles of Igusa’s local zeta function
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by Kathleen Hoornaert PDF
Trans. Amer. Math. Soc. 356 (2004), 1751-1779 Request permission


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
  • Received by editor(s): March 12, 2002
  • Published electronically: December 15, 2003
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 1751-1779
  • MSC (2000): Primary 11S40, 11D79; Secondary 14M25, 52B20, 14G10
  • DOI:
  • MathSciNet review: 2031040