# Permutation Binomials over Finite Fields

## TL;DR

This paper determines the exact count of elements in finite fields for which specific binomials permute the field, using algebraic curves and rational points, focusing on cases where r equals 2 and 3.

## Contribution

It provides the first exact enumeration of permutation binomials of a particular form over finite fields for r=2 and r=3, linking polynomial properties to algebraic curves.

## Key findings

- Exact counts of permutation binomials for r=2 and r=3
- Methodology connecting polynomials to algebraic curves
- Enhanced understanding of permutation polynomial structures

## Abstract

Let $\mathbb F_q$ denote the finite field with $q$ elements. In this paper we use the relationship between suitable polynomials and number of rational points on algebraic curves to give the exact number of elements $a\in \mathbb F_q$ for which the binomial $x^n(x^{(q-1)/r} + a)$ is a permutation polynomial in the cases $r = 2$ and $r = 3$.