Cyclotomic enumeration of polynomials

Using the cyclotomic identity we compute sums over d-tuples of monic polynomials in F_q[x] weighted by the multiplicity of their irreducible factors. As consequences we determine explicit expressions for the number of d-tuples of polynomials such that their greatest common divisor is rth power free. We also compute the number of monic polynomials where the multiplicity of each irreducible factor belongs to the monoid generated by two relatively prime integers.