Counting reducible, powerful, and relatively irreducible multivariate polynomials over finite fields

Authors:
Joachim von zur Gathen;Alfredo Viola;Konstantin Ziegler
Affiliations:
B-IT, Universität Bonn, Bonn, Germany;Instituto de Computación, Universidad de la República, Montevideo, Uruguay;B-IT, Universität Bonn, Bonn, Germany
Venue:
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Year:
2010

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Abstract

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.