On the poles of topological zeta functions

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) vertical bar there exists f is an element of C[x(1),..., x(n)] : Z(top,) (f) (s) has a pole in s(0)}. We show that {-( n- 1)/ 2-1/ i vertical bar i is an element of 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 than -( n - 1)/2. As the main result we prove that each rational number in the interval [-( n- 1)/ 2, 0) is contained in P-n.