The compositional inverses of the permutation polynomials of the form $x+\gamma {\mathrm{Tr}}_q^{q^n}(H(x))$ over ${\mathbb{F}}_{q^n}$

This paper focuses on computing the compositional inverses of permutation polynomials of the form $x+\gamma {\mathrm{Tr}}_q^{q^n}(H(x))$ over finite fields via the local method. We explicitly construct compositional inverses for four families of permutation polynomials of this type over ${\mathbb{F}}_{q^2}$, another four families over ${\mathbb{F}}_{q^3}$, and one general family over ${\mathbb{F}}_{q^n}$. The closed-form inverse expressions derived in this work supplement the theory of trace permutation polynomials.