Estimates for short character sums evaluated at homogeneous polynomials

Abstract Let be a prime. We prove bounds on short Dirichlet character sums evaluated at a class of homogeneous polynomials in arbitrary dimensions. In every dimension, this bound is nontrivial for sums over boxes with side lengths as short as for any . Our methods capitalize on the relationship between characters mod and characters over finite field extensions as well as bounds on the multiplicative energy of sets in products of finite fields.