On the poles of topological zeta functions

Ann Lemahieu (1) , Dirk Segers (2) , Willem Veys (3)

1 UNS - Université Nice Sophia Antipolis (1965 - 2019)
2 Department of Mathematics
3 Department of Mathematics

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 Pn := {s0 | ∃f ∈ C[x1,. .. , xn] : Z top,f (s) has a pole in s0}. We show that {−(n−1)/2−1/i | i ∈ Z>1} is a subset of Pn; 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 Pn.

Domaines: Mathématiques [math]

Ann Lemahieu, Dirk Segers, Willem Veys. On the poles of topological zeta functions. Proceedings of the American Mathematical Society, 2006, 134 (12), pp.3429-3436. ⟨hal-01279911⟩