On the poles of topological zeta functions

Details On the poles of topological zeta functions On the poles of topological zeta functions

2005-07-06

arxiv.org/abs/math/0507121v1

Mathematics

Algebraic Geometry

Scientific paper

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 P_n := {s_0 | \exists f in C[x_1,..., x_n] : Z_{top,f}(s) has a pole in s_0}. We show that {-(n-1)/2-1/i | i in Z_{>1}} is a subset of P_n; for n=2 and n=3, the last two authors proved before that these are exactly the poles less then -(n-1)/2. As main result we prove that each rational number in the interval [-(n-1)/2,0) is contained in P_n.

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