On the poles of topological zeta functions

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 $\mathcal {P}_n$.