Further results on permutation polynomials from trace functions

Abstract

For a prime p and positive integers m, n, let ${{\mathbb {F}}}_q$ be a finite field with $q=p^m$ elements and ${{\mathbb {F}}}_{q^n}$ be an extension of ${{\mathbb {F}}}_q.$ Let h(x) be a polynomial over ${{\mathbb {F}}}_{q^n}$ satisfying the following conditions: (i) ${\mathrm{Tr}}_m^{nm}(x)\circ h(x)=\tau (x)\circ {\mathrm{Tr}}_m^{nm}(x)$; (ii) For any $s \in {{\mathbb {F}}}_{q}$, h(x) is injective on ${\mathrm{Tr}}_m^{nm}(x)^{-1}(s),$ where $\tau (x)$ is a polynomial over ${{\mathbb {F}}}_{q}.$ For $b,c \in {{\mathbb {F}}}_q,$ $\delta \in {{\mathbb {F}}}_{q^n}$, and positive integers i, j, d with $q\equiv \pm 1 \pmod {d}$, we propose a class of permutation polynomials of the form

$$\begin{aligned} b({\mathrm{Tr}}_m^{nm}(x)+\delta )^{1+\frac{i(q^n-1)}{d}}+c({\mathrm{Tr}}_m^{nm}(x)+\delta )^{1+\frac{j(q^n-1)}{d}}+h(x) \end{aligned}$$

over ${{\mathbb {F}}}_{q^n}$ by employing the Akbary–Ghioca–Wang (AGW) criterion in this paper. Accordingly, we also present the permutation polynomials of the form

$$\begin{aligned} b({\mathrm{Tr}}_m^{nm}(x)+\delta )^{1+\frac{i(q^n-1)}{d}}+h(x) \end{aligned}$$

by letting $c=0$ and choosing some special i, which covered some known results of this form.