On commutative semigroups of polynomials with respect to composition

In the first part, commutative semigroups of Dickson polynomials with respect to composition over integral domains are studied. Subsequently we investigate commutative semigroups of polynomials with respect to composition over fields or integral domains. all such semigroups which contain only polynomials of positive even degree and at least one polynomial of each positive even degree over a field of characteristic ≠2 and ≠3 are determined. Further, all commutative semigroups which contain only polynomials of odd degree and at least one polynomial of each odd degree over fields of characteristic ≠3 and ≠5 are characterised. Where appropriate the corresponding results for polynomials over integral domains are given.