Multiple Exponential and Character Sums with Monomials

TL;DR

This paper develops new bounds for multivariate exponential sums with monomials over short intervals and extends these techniques to sums involving multiplicative characters, surpassing previous limitations.

Contribution

It introduces novel bounds for exponential sums with monomials and applies these methods to multiplicative characters, breaking the $p^{1/4}$ barrier for prime moduli.

Findings

New bounds for multivariate exponential sums with monomials

Estimates for sums with multiplicative characters beyond previous limits

Breakthrough in the range of variables for prime moduli

Abstract

We obtain new bounds of multivariate exponential sums with monomials, when the variables run over rather short intervals. Furthermore, we use the same method to derive estimates on similar sums with multiplicative characters to which previously known methods do not apply. In particular, in the multiplicative characters modulo a prime $p$ we break the barrier of $p^{1/4}$ for ranges of individual variables.