Solutions of polynomial equations in several variables modulo a prime power

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 Poincaré series associated with a polynomial in two separated variables is rational.