Solutions of polynomial equations in several variables modulo a prime power

TL;DR

This paper presents a method to find solutions of multivariate polynomial equations modulo prime powers using a tree structure, enabling solution counting and applications like proving rationality in Igusa's theorem.

Contribution

It introduces a tree-based approach to determine solutions of polynomial equations modulo prime powers, facilitating solution enumeration and related proofs.

Findings

Solutions can be reconstructed from a tree structure called the trunk.

Method allows counting solutions without explicit enumeration.

Applied to prove a case of Igusa's theorem on rationality.

Abstract

We explain how to obtain the set of solutions of a multivariate polynomial equation modulo a power of a prime number. These solutions are determined by a tree, called the trunk, which makes it possible to reconstruct all solutions. We apply these methods to determine the number of solutions, without having to enumerate them. We also illustrate these techniques by proving a simple case of Igusa's theorem: the Poincar\'e series associated with a polynomial in two separated variables is rational.