A complete characterization of a family of permutation trinomials over $\mathbb F_{p^2}$

Abstract

Let $p>3$ be a prime and let $$f_{λ_1,λ_2}(x)=x^{p^2-p+1}+λ_1x^{p^2}+λ_2x^{2p-1}\in\mathbb F_{p^2}[x].$$ We determine all pairs $(λ_1,λ_2)\in(\mathbb F_{p^2})^2$ for which $f_{λ_1,λ_2}$ is a permutation polynomial of $\mathbb F_{p^2}$. The final classification consists of three explicit families. The first one is the binomial case $λ_1=0$. The other two are obtained from the condition $λ_2=cλ_1^3$, with $c\in \mathbb F_{p}^{*}$, and are defined by two simple equations involving the norm $λ_1^{p+1}$. The proof is based on the AGW criterion and on the study of a quartic curve naturally associated with the rational function induced on the unit circle $μ_{p+1}$.