Burgess bounds for short character sums evaluated at forms II: the mixed case

This work proves a Burgess bound for short mixed character sums in n dimensions. The non-principal multiplicative character of prime conductor q may be evaluated at any "admissible" form, and the additive character may be evaluated at any real-valued polynomial. The resulting upper bound for the mixed character sum is nontrivial when the length of the sum is at least q(beta) with beta > 1/2 - 1/(2(n + 1)) in each coordinate. This work capitalizes on the recent stratification of multi-plicative character sums due to Xu, and the resolution of the Vinogradov Mean Value Theorem in arbitrary dimensions.