Poles of Archimedean zeta functions for analytic mappings

Abstract

In this paper, we give a description of the possible poles of the local zeta function attached to a complex or real analytic mapping in terms of a log-principalization of an ideal associated to the mapping. When the mapping is a non-degenerate one, we give an explicit list for the possible poles of the corresponding local zeta function in terms of the normal vectors to the supporting hyperplanes of a Newton polyhedron attached to the mapping, and some additional vectors (or rays) that appear in the construction of a simplicial conical subdivision of the first orthant. These results extend the corresponding results of Varchenko to the case l\geq1, and K=R or C. In the case l=1 and K=R, Denef and Sargos proved that the candidates poles induced by the extra rays required in the construction of a simplicial conical subdivision can be discarded from the list of candidate poles. We extend the Denef-Sargos result arbitrary l\geq1. This yields in general a much shorter list of candidate poles, that can moreover be read off immediately from the Newton polyhedron.