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Counting reducible, powerful, and relatively irreducible multivariate polynomials over finite fields. (English) Zbl 1348.11094
Authors’ abstract: We present counting methods for some special cases 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, and another one uses a combinatorial method. They yield exact formulas and approximations with relative errors that essentially decrease exponentially in the input size.

##### MSC:
 11T06 Polynomials over finite fields 12E20 Finite fields (field-theoretic aspects) 12Y05 Computational aspects of field theory and polynomials (MSC2010) 05A15 Exact enumeration problems, generating functions
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