A proof of a conjecture on permutation trinomials

TL;DR

This paper proves a conjecture about permutation trinomials over finite fields using algebraic curves and number theory, confirming the polynomial's permutation property in specific finite field cases.

Contribution

It provides a rigorous proof of a permutation polynomial conjecture using algebraic geometry and number theory methods, advancing understanding of permutation polynomials.

Findings

Confirmed the permutation property of the polynomial over finite fields

Applied algebraic curves and number theory techniques in proof

Extended the class of known permutation trinomials

Abstract

In this paper we use algebraic curves and other algebraic number theory methods to show the validity of a permutation polynomial conjecture regarding $f(X)=X^{q(p-1)+1} +\alpha X^{pq}+X^{q+p-1}$, on finite fields $\mathbb{F}_{q^2}, q=p^k$, from [A. Rai, R. Gupta, {\it Further results on a class of permutation trinomials}, Cryptogr. Commun. 15 (2023), 811--820].