Counting factorisations of monomials over rings of integers modulo $N$

Abstract

A sharp bound is obtained for the number of ways to express the monomial $X^n$ as a product of linear factors over $\mathbb{Z}/p^{\alpha }\mathbb{Z}$. The proof relies on an induction-on-scale procedure which is used to estimate the number of solutions to a certain system of polynomial congruences. The method also applies to more general systems of polynomial congruences that satisfy a non-degeneracy hypothesis.

21 pages

References in corpus (1)

• Exponential sums and polynomial congruences in two variables: the quasi-homogeneous case