Counting factorisations of monomials over rings of integers modulo N

A sharp bound is obtained for the number of ways to express the monomial Xn as a product of linear factors over Z/pαZ. 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.