The poles of Igusa zeta integrals and the unextendability of semi-invariant distributions

TL;DR

This paper explores how the poles of Igusa zeta integrals relate to the unextendability of semi-invariant distributions, providing bounds and criteria under algebraic conditions using generalized semi-invariant distributions.

Contribution

It introduces a new criterion linking pole orders of Igusa zeta integrals to the unextendability of semi-invariant distributions, utilizing the concept of generalized semi-invariant distributions.

Findings

Upper bounds for pole orders of Igusa zeta integrals

A criterion for unextendability of semi-invariant distributions

Application of generalized semi-invariant distributions

Abstract

We investigate the relationship between the poles of Igusa zeta integrals and the unextendability of semi-invariant distributions. Under some algebraic conditions, we obtain an upper bound for the order of the poles of Igusa zeta integral, and by using the order of the poles we give a criterion on the unextendability of semi-invariant distributions. A key ingredient of our method is the idea of generalized semi-invariant distributions.