Poles of Archimedean zeta functions for analytic mappings

Let f = (f1, . . . , fl) : U → Kl, with K = R or C, be a K-analytic mapping defined on an open set U ⊆ Kn, and let Φ be a smooth function on U with compact support. In this paper, we give a description of the possible poles of the local zeta function attached to (f , Φ) in terms of a log-principalization of the ideal If = (f1, . . . , fl). When f is a non-degenerate mapping, we give an explicit list for the possible poles of ZΦ(s, f) in terms of the equations of the supporting hyperplanes of a Newton polyhedron attached to f . These results extend the corresponding results of Varchenko to the case l ≥ 1 and K = R or C.