Permutation polynomials of the form $x+\gamma \mathrm{Tr}(H(x))$

TL;DR

This paper investigates permutation polynomials of a specific form involving trace functions over finite fields, providing bounds on parameter sets and characterizing cases of maximal size.

Contribution

It introduces properties of the set of parameters for which these polynomials permute, establishes an upper bound, and characterizes when this bound is achieved.

Findings

Upper bound for the size of the set of parameters, reaching up to $q^n - q^{n-1}$.

When the bound is reached, the trace function must be linear over _q.

For certain classes over _{q^2}, the parameter sets are small, often trivial.

Abstract

Given a polynomial \( H(x) \) over \(\mathbb{F}_{q^n}\), we study permutation polynomials of the form \( x + \gamma \mathrm{Tr}(H(x)) \) over \(\mathbb{F}_{q^n}\). Let \[P_H=\{\gamma\in \mathbb{F}_{q^n} : x+\gamma \mathrm{Tr}(H(x))~\text{is a permutation polynomial}\}.\] We present some properties of the set \(P_H\), particularly its relationship with linear translators. Moreover, we obtain an effective upper bound for the cardinality of the set \(P_H\) and show that the upper bound can reach up to $q^n - q^{n - 1}$. Furthermore, we prove that when the cardinality of the set \(P_H\) reaches this upper bound, the function \(\mathrm{Tr}(H(x))\) must be an \(\mathbb{F}_q\)-linear function. Finally, we study two classes of functions $H(x)$ over \(\mathbb{F}_{q^2}\) and determine the corresponding sets $P_H$. The sizes of these sets $P_H$ are all relatively small, even only including the trivial case.