Equidistribution and independence of Gauss sums

TL;DR

This paper proves a broad equidistribution and independence result for Gauss sums associated with multiple monomials over finite fields, extending previous findings and characterizing relations among these sums.

Contribution

It introduces a general equidistribution theorem for Gauss sums of multivariate monomials, expanding the understanding of their independence and relations.

Findings

Gauss sums are shown to be equidistributed and independent under broad conditions.

Any algebraic relation among these Gauss sums is generated by known classical identities.

The results unify and extend previous special cases for Gauss and Jacobi sums.

Abstract

We prove a general independent equidistribution result for Gauss sums associated to $n$ monomials in $r$ variable multiplicative characters over a finite field, which generalizes several previous equidistribution results for Gauss and Jacobi sums. As an application, we show that any relation satisfied by these Gauss sums must be a combination of the conjugation relation $G(\chi)G(\overline\chi)=\pm q$, Galois conjugation invariance and the Hasse-Davenport product formula.