  We present counting methods for some special classes of multivariate
polynomials over a finite field, namely the reducible ones, the s-powerful ones
(divisible by the s-th power of a nonconstant polynomial), and the relatively
irreducible ones (irreducible but reducible over an extension field). One
approach employs generating functions, another one uses a combinatorial method.
They yield exact formulas and approximations with relative errors that
essentially decrease exponentially in the input size.
