Exact Divisibility of Exponential Sums Associated to Binomials over finite fields

TL;DR

This paper precisely determines the divisibility of exponential sums linked to binomials over finite fields, providing insights into permutation properties and value set sizes of these polynomials.

Contribution

It introduces exact divisibility results for exponential sums of binomials with degree constraints and offers a new criterion for permutation polynomials over finite fields.

Findings

Identifies families of binomials that do not permute _p

Provides a lower bound for the size of binomial value sets

Develops a new divisibility-based permutation criterion

Abstract

In this paper we compute the exact divisibility of exponential sums associated to binomials $F(X)=aX^{d_1} +b X^{d_2}$. In particular, for the case where $\max\{d_1,d_2\}\leq\sqrt{p-1}$, the exact divisibility is computed. As a byproduct of our results, we obtain families of binomials that do not permute $\mathbb{F}_p$, and a lower bound for the sizes of value sets of binomials over $\mathbb{F}_p$. Additionally, we obtain a new criterion to determine if a polynomial defines or not a permutation of $\mathbb{F}_p$ that depends on the divisibility of the exponential sum associated to the polynomial.