Determination of a Type of Permutation Trinomials over Finite Fields, II

TL;DR

This paper characterizes all permutation trinomials over finite fields of the form $ax+bx^q+x^{2q-1}$ in $F_{q^2}$, extending previous work on subclasses with coefficients in $F_q$.

Contribution

It provides a complete classification of permutation trinomials of a specific form over $F_{q^2}$, generalizing earlier results.

Findings

All permutation trinomials of the given form are identified.

The subclass with coefficients in $F_q$ was previously characterized.

The paper extends the classification to all such trinomials over $F_{q^2}$.

Abstract

Let $q$ be a prime power. We determine all permutation trinomials of $\Bbb F_{q^2}$ of the form $ax+bx^q+x^{2q-1}\in\Bbb F_{q^2}[x]$. The subclass of such permutation trinomials of $\Bbb F_{q^2}$ with $a,b\in\Bbb F_q$ was determined in a recent paper by the author.