  In this work 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 $\beta > 1/2 - 1/(2(n+1))$. 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.
