Multivariate Hensel lifting without assuming as many variables as equations

Hensel lifting is a technique for taking solutions modulo p to systems of polynomial equations with integer coefficients, and "lifting" them to solutions mod p2,p4,p8, etc. One of the original applications of Hensel lifting was in algorithms for the factorization of integer polynomials, where the system of equations has the same number of variables as equations. While Hensel lifting in that scenario is well-covered in the literature, textbooks, courses, and online notes, in this expository note we are interested in the general case, where the number of variables can be more or less than the number of equations. Our exposition is entirely elementary, assuming no background beyond integer polynomials and modular arithmetic.