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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the poles of topological zeta functions


Authors: Ann Lemahieu, Dirk Segers and Willem Veys
Journal: Proc. Amer. Math. Soc. 134 (2006), 3429-3436
MSC (2000): Primary 14B05, 14J17, 32S05; Secondary 14E15, 32S25
Published electronically: June 9, 2006
MathSciNet review: 2240652
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Abstract: We study the topological zeta function $ Z_{top,f}(s)$ associated to a polynomial $ f$ 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 $ n=2$ and $ n=3$, the last two authors proved before that these are exactly the poles less than $ -(n-1)/2$. As the main result we prove that each rational number in the interval $ [-(n-1)/2,0)$ is contained in $ \mathcal{P}_n$.


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Additional Information

Ann Lemahieu
Affiliation: Departement Wiskunde, Celestijnenlaan 200B, B-3001 Leuven, Belgium
Email: ann.lemahieu@wis.kuleuven.be

Dirk Segers
Affiliation: Departement Wiskunde, Celestijnenlaan 200B, B-3001 Leuven, Belgium
Email: dirk.segers@wis.kuleuven.be

Willem Veys
Affiliation: Departement Wiskunde, Celestijnenlaan 200B, B-3001 Leuven, Belgium
Email: wim.veys@wis.kuleuven.be

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08512-1
PII: S 0002-9939(06)08512-1
Received by editor(s): January 25, 2005
Received by editor(s) in revised form: June 29, 2005
Published electronically: June 9, 2006
Communicated by: Michael Stillman
Article copyright: © Copyright 2006 American Mathematical Society