Motivic Milnor fiber of a quasi-ordinary hypersurface

Abstract

Let f : (Cd+1, 0) -> (C, 0) be a germ of complex analytic function such that its zero level defines an irreducible germ of quasi-ordinary hypersurface (S, 0) subset of (Cd+1, 0). We describe the motivic Igusa zeta function, the motivic Milnor fibre and the Hodge-Steenbrink spectrum of f at 0 in terms of topological invariants of (S, 0) subset of (Cd+1, 0).