Estimates for exponential sums with a large automorphism group

TL;DR

This paper improves classical bounds for certain exponential sums over finite fields by exploiting polynomials invariant under large automorphism groups, achieving tighter estimates in specific cases.

Contribution

It provides new bounds for exponential sums associated with polynomials invariant under large automorphism groups, extending classical Weil bounds.

Findings

Improved bounds by a factor of √q for specific polynomial classes

Applicable to polynomials invariant under translation or homothety

Results hold over sufficiently large field extensions

Abstract

We prove some improvements of the classical Weil bound for one variable additive and multiplicative character sums associated to a polynomial over a finite field $k=\Fq$ for two classes of polynomials which are invariant under a large abelian group of automorphisms of the affine line $\AAA^1_k$: those invariant under translation by elements of $k$ and those invariant under homotheties with ratios in a large subgroup of the multiplicative group of $k$. In both cases, we are able to improve the bound by a factor of $\sqrt{q}$ over an extension of $k$ of cardinality sufficiently large compared to the degree of $f$.