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On the holomorphy conjecture for Igusa's local zeta function

Authors: Jan Denef and Willem Veys
Journal: Proc. Amer. Math. Soc. 123 (1995), 2981-2988
MSC: Primary 11S40; Secondary 32S40
MathSciNet review: 1283546
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Abstract: To a polynomial f over a p-adic field K and a character $ \chi $ of the group of units of the valuation ring of K one associates Igusa's local zeta function $ Z(s,f,\chi )$, which is a meromorphic function on $ \mathbb{C}$. Several theorems and conjectures relate the poles of $ Z(s,f,\chi )$ to the monodromy of f; the so-called holomorphy conjecture states roughly that if the order of $ \chi $ does not divide the order of any eigenvalue of monodromy of f, then $ Z(s,f,\chi )$ is holomorphic on $ \mathbb{C}$. We prove mainly that if the holomorphy conjecture is true for $ f({x_1}, \ldots ,{x_{n - 1}})$, then it is true for $ f({x_1}, \ldots ,{x_{n - 1}}) + x_n^k$ with $ k \geq 3$, and we give some applications.

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Keywords: Igusa's local zeta function, monodromy
Article copyright: © Copyright 1995 American Mathematical Society

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