On the poles of maximal order of the topological zeta function

The global and local topological zeta functions are singularity invariants associated to a polynomial f and its germ at 0, respectively. By denition, these zeta functions are rational functions in one variable, and their poles are negative rational numbers. In this paper we study their poles of maximal possible order. When f is non-degenerate with respect to its Newton polyhedron, we prove that its local topological zeta function has at most one such pole, in which case it is also the largest pole; we give a similar result concerning the global zeta function. Moreover, for any f we show that poles of maximal possible order are always of the form −1=N with N a positive integer. 0.1. To f 2 C[x1; : : : ; xn] is associated a singularity invariant, called the topo-logical zeta function of f, which is expressed as follows in terms of an embedded resolution of f−1f0g An. For simplicity of notation, suppose that f(0) = 0. Let h: X! An be an embedded resolution of f−1f0g. We denote by Ei, i 2 S