  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.
