Poles of Archimedean zeta functions for analytic mappings

TL;DR

This paper characterizes the poles of local zeta functions associated with analytic mappings, providing explicit lists for non-degenerate cases and extending previous results to more general settings, simplifying pole determination.

Contribution

It extends Varchenko's results to multiple mappings and arbitrary real or complex fields, offering explicit pole lists and simplifying the pole detection process.

Findings

Explicit pole lists for non-degenerate mappings

Extension of Varchenko's results to multiple mappings

Shorter candidate pole lists from Newton polyhedron

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.