Geometric determination of the poles of highest and second highest order of Hodge and motivic zeta functions

Abstract

To any $f \in \C[x_{1}, \ldots, x_{n}] \setminus \C$ with $f(0) = 0$ one can associate the motivic zeta function. Another interesting singularity invariant of $f^{-1}\!\{0\}$ is the zeta function on the level of Hodge polynomials, which is actually just a specialization of the motivic one. In this paper we generalize for the Hodge zeta function the result of Veys which provided for $n = 2$ a complete geometric determination of the poles. More precisely we give in arbitrary dimension a complete geometric determination of the poles of order $n-1$ and $n$. We also show how to obtain the same results for the motivic zeta function.