The functional decomposition of polynomials has been a topic of great interest and importance in pure and computer algebra and their applications. The structure of compositions of (suitably normalized)is well understood in many cases, but quite poorly when the degrees of both components are divisible by the characteristic p. This work investigates the decomposition of polynomials whose degree is a power of p. An (equal-degree) icollision is a set of i distinct pairs (g, h) of polynomials, all with the

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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 sth power of a nonconstant polynomial), and the relatively irreducible ones (irreducible but reducible over an extension field). One approach employs generating functions, and another one uses a combinatorial method. They yield exact formulas and approximations with relative errors that essentially decrease exponentially in the input size.

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