The compositional inverses of permutation polynomials from trace functions over finite fields

In this paper, we present the compositional inverses of several classes permutation polynomials of the form $\sum_{i=1}^kb_i\left({\rm Tr}_m^{mn}(x)^{t_i}+\delta\right)^{s_i}+f_1(x)$, where $1\leq i \leq k,$ $s_i$ are positive integers, $b_i \in \mathbb{F}_{p^m},$ $p$ is a prime and $f_1(x)$ is a polynomial over $\mathbb{F}_{p^{mn}}$ satisfying the following conditions: (i) ${\rm Tr}_m^{mn}(x) \circ f_1(x)=\varphi(x) \circ {\rm Tr}_m^{mn}(x),$ where $\varphi(x)$ is a polynomial over $\mathbb{F}_{p^m};$ (ii) For any $a \in \mathbb{F}_{p^m},$ $f_1(x)$ is injective on ${\rm Tr}_m^{mn}(a)^{-1}.$