Several classes of permutation polynomials with trace functions over $$\mathbb {F}_{p^n}$$ F p n

Permutation polynomials over finite fields constitute an active research area and have important applications in many areas of science and engineering. In this paper, several classes of permutation polynomials with trace functions are presented over $$\mathbb {F}_{p^{n}} (p=2, 3)$$ by investigating the number of solutions to special equations.