Solutions of polynomial equations in several variables modulo a prime power
Number Theory
2026-02-25 v1 Algebraic Geometry
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.
Cite
@article{arxiv.2602.20649,
title = {Solutions of polynomial equations in several variables modulo a prime power},
author = {Arnaud Bodin and Christian Drouin},
journal= {arXiv preprint arXiv:2602.20649},
year = {2026}
}
Comments
31 pages, appendix with an explicit example