Burgess bounds for short character sums evaluated at forms

Abstract

We establish a Burgess bound for short multiplicative character sums in arbitrary dimensions, in which the character is evaluated at a homogeneous form that belongs to a very general class of “admissible” forms. This $n$-dimensional Burgess bound is nontrivial for sums over boxes of sidelength at least $q^{\beta }$, with $\beta >1∕2-1∕\left(2\left(n+1\right)\right)$. This is the first Burgess bound that applies in all dimensions to generic forms of arbitrary degree. Our approach capitalizes on a recent stratification result for complete multiplicative character sums evaluated at rational functions, due to the second author.