The asymptotic behaviour of the number of solutions of polynomial congruences

TL;DR

This paper investigates the precise relationship between Igusa's p-adic zeta function poles and the asymptotic count of solutions to polynomial congruences, addressing a gap in existing literature.

Contribution

It provides a rigorous clarification of how Igusa's p-adic zeta function poles influence the asymptotic behavior of polynomial congruence solutions.

Findings

Established a clear connection between zeta function poles and solution asymptotics

Resolved a gap in the theoretical understanding of polynomial congruences

Enhanced the mathematical framework relating p-adic analysis to polynomial solutions

Abstract

One mentions in a lot of papers that the poles of Igusa's p-adic zeta function determine the asymptotic behavior of the number of solutions of polynomial congruences. However, no publication clarifies this connection precisely. We try to get rid of this gap.