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

TL;DR

This paper establishes a precise bound on the number of factorizations of monomials over rings of integers modulo N, using an induction method to analyze solutions to polynomial congruences.

Contribution

It introduces a novel induction-on-scale approach to bound factorizations of monomials over modular rings, extending to general polynomial systems under non-degeneracy conditions.

Findings

Derived a sharp bound for monomial factorizations over rac{p^{\u03b1}}{rac{p^{\u03b1}}} rac{p^{\u03b1}}{rac{p^{b1}}}rac{p^{b1}}

Extended the method to general polynomial systems with non-degeneracy hypothesis

Provided a new inductive technique for estimating solutions to polynomial congruences.

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.