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On the poles of topological zeta functions

Authors:	Ann Lemahieu Dirk Segers Willem Veys

Institution:	Departement Wiskunde, Celestijnenlaan 200B, B-3001 Leuven, Belgium ; Departement Wiskunde, Celestijnenlaan 200B, B-3001 Leuven, Belgium ; Departement Wiskunde, Celestijnenlaan 200B, B-3001 Leuven, Belgium

Abstract:	We study the topological zeta function $Z_{top,f}(s)$ associated to a polynomial with complex coefficients. This is a rational function in one variable, and we want to determine the numbers that can occur as a pole of some topological zeta function; by definition these poles are negative rational numbers. We deal with this question in any dimension. Denote $\mathcal{P}_n := \{ s_0 \mid \exists f \in \mathbb{C}x_1,\ldots, x_n] \, : \, Z_{top,f}(s)$ has a pole in $s_0 \}$ . We show that $\{-(n-1)/2-1/i \mid i \in \mathbb{Z}_{>1}\}$ is a subset of $\mathcal{P}_n$ ; for and , the last two authors proved before that these are exactly the poles less than . As the main result we prove that each rational number in the interval is contained in $\mathcal{P}_n$ .

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