Counting Polynomials with Zeros of Given Multiplicities in Finite Fields

We consider the set of polynomials inrindeterminates over a finite field and with bounded degree. We give here a way to count the number of elements of some of its subsets, namely those sets defined by the multiplicities of their elements at some points of rq. The number of polynomials having at least one zero in a given finite field is computed as a particular applications.