Oscillating integrals and Newton polyhedra

Abstract

We establish a principal value integral formula, for the residue of the largest non-trivial candidate pole of the real or complex local zeta function associated to an analytic germ f, which is non-degenerate with respect to its Newton polyhedron. In particular, up to an easy non-zero factor, this residue only depends on the (tau_0)-principal part of f, where tau_0 is the smallest face of the Newton polyhedron intersecting the diagonal. This formula allows us to prove some vanishing results for the residue. More precisely, we prove that the residue vanishes when tau_0 is unstable, and we give a partial proof of the reverse implication in the complex case. We also deduce an explicit formula for the residue, in the case where tau_0 is a simplex of codimension 1, and the only points of the support of f on tau_0 are its vertices.