On inverses of permutation polynomials of the form x( x^s -a) ^(q^m-1)/s over 𝔽_q^n

The inverse of a class of permutation polynomials (PPs) of the form x(x^s -a)^(q^m - 1)/s over 𝔽_q^n is given. More simplified expressions of some subclasses of this family are presented, such as x(x^q-1 - a)^q+1 and x(x^q+1 - a)^q-1 over 𝔽_q^3 , x(x^2 -a)^3 and x(x^3 -a)^2 over 𝔽_7^n . These expressions and some known results solve the problem of determining the inverses of all PPs of degree 7 over finite fields. In addition, an explicit criteria for x(x^s -a)^(q - 1)/s to be an involution of 𝔽_q is established.