Counting Reducible, Powerful, and Relatively Irreducible Multivariate Polynomials over Finite Fields
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, another one a combinatorial method. They yield approximations with relative errors that essentially decrease exponentially in the input size.
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