# Incomplete Gauss sums modulo primes

## TL;DR

This paper presents a new bound for incomplete Gauss sums modulo primes using Vinogradov's method, reducing the problem to counting solutions to two systems of congruences, one related to Vinogradov's mean value theorem.

## Contribution

It introduces a novel approach to bounding incomplete Gauss sums by analyzing two systems of congruences, one of which is a new consideration in the field.

## Key findings

- Improved bounds for incomplete Gauss sums in certain ranges
- Reduction of the problem to counting solutions of two systems of congruences
- Introduction of a new system of congruences not previously studied

## Abstract

We obtain a new bound for incomplete Gauss sums modulo primes. Our argument falls under the framework of Vinogradov's method which we use to reduce the problem under consideration to bounding the number of solutions to two distinct systems of congruences. The first is related to Vinogradov's mean value theorem, although the second does not appear to have been considered before. Our bound improves on current results in the range $N\ge q^{2k^{-1/2}+O(k^{-3/2})}$.