On a conjecture about a class of permutation trinomials

TL;DR

This paper proves a conjecture about specific permutation trinomials over finite fields, characterizing all parameter pairs that make these polynomials permutations of the field.

Contribution

It provides a complete characterization of when the given class of trinomials permute the finite field, resolving a conjecture posed by previous researchers.

Findings

Identifies all parameter pairs (,) for permutation property

Proves the conjecture for even q

Characterizes permutation trinomials explicitly

Abstract

We prove a conjecture by Tu, Zeng, Li, and Helleseth concerning trinomials $f_{\alpha,\beta}(x)= x + \alpha x^{q(q-1)+1} + \beta x^{2(q-1)+1} \in \mathbb{F}_{q^2}[x]$, $\alpha\beta \neq 0$, $q$ even, characterizing all the pairs $(\alpha,\beta)\in \mathbb{F}_{q^2}^2$ for which $f_{\alpha,\beta}(x)$ is a permutation of $\mathbb{F}_{q^2}$.