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Smallest poles of Igusa's and topological zeta functions and solutions of polynomial congruences

Abstract : Igusa's p-adic zeta function is associated to a polynomial f in several variables over the integers and to a prime p. It is a meromorphic function which encodes for every i the number of solutions M_i of f=0 modulo p^i. The intensive study of Igusa's p-adic zeta function by using an embedded resolution of f led to the introduction of the topological zeta function. This geometric invariant of the zero locus of a polynomial f in several variables over the complex numbers was introduced in the early nineties by Denef and Loeser. It is a rational function which they obtained as a limit of Igusa's p-adic zeta functions and which is defined by using an embedded resolution.
I have studied the smallest poles of the topological zeta function and the smallest real parts of the poles of Igusa's p-adic zeta function. For n=2 and n=3, I obtained results by using an embedded resolution of singularities. I discovered that the smallest real part of a pole of Igusa's p-adic zeta function is related with the divisibility of the M_i by powers of p. I obtained a general theorem on the divisibility of the M_i by powers of p, which I used to obtain the optimal lower bound for the real part of a pole of Igusa's p-adic zeta function in arbitrary dimension n. I obtained this lower bound also for the topological zeta function by taking the limit.
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https://tel.archives-ouvertes.fr/tel-00006134
Contributor : Dirk Segers <>
Submitted on : Sunday, May 23, 2004 - 9:52:54 PM
Last modification on : Saturday, July 20, 2019 - 1:13:01 AM
Long-term archiving on: : Friday, April 2, 2010 - 8:22:24 PM

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  • HAL Id : tel-00006134, version 1

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Dirk Segers. Smallest poles of Igusa's and topological zeta functions and solutions of polynomial congruences. Mathematics [math]. Université Catholique de Louvain, 2004. English. ⟨tel-00006134⟩

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