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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 beevaluated at any real-valued polynomial. The resulting upper bound for themixed 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 workcapitalizes on the recent stratification of multiplicative character sums dueto Xu, and the resolution of the Vinogradov Mean Value Theorem in arbitrarydimensions.

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