# Burgess bounds for short character sums evaluated at forms

## TL;DR

This paper extends Burgess bounds to short multiplicative character sums evaluated at general forms in arbitrary dimensions, providing a new nontrivial bound applicable to generic forms of any degree.

## Contribution

It establishes the first multidimensional Burgess bound for generic forms of arbitrary degree, broadening the scope of character sum estimates.

## Key findings

- Burgess bound proven for sums over boxes with side length at least q^{}
- Bound applies to all dimensions and generic forms of any degree
- Utilizes recent stratification results for rational functions

## Abstract

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.