On inverses of permutation polynomials of the form x (x(s) - a)((qm-1)/s) over F-qn

The inverse of a class of permutation polynomials (PPs) of the form x(x(s) - a) (q(m-1)/s) over F-qn 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-q3, x(x(2) - a)(3) and x(x(3) - a)(2) over F-7n. 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.