The additive index of polynomials over finite fields

TL;DR

This paper introduces the additive index of polynomials over finite fields and explores its applications in understanding value sets, character sums, and permutation polynomials.

Contribution

It presents the first study of the additive index concept and demonstrates its usefulness in finite field polynomial analysis.

Findings

Additive index provides new bounds on polynomial value sets.

It offers novel character sum estimates.

Characterizes permutation polynomials using additive indices.

Abstract

In this paper we introduce the additive analogue of the index of a polynomial over finite fields. We study several problems in the theory of polynomials over finite fields in terms of their additive indices, such as value set sizes, bounds on multiplicative character sums, and characterizations of permutation polynomials.