Number of irreducible polynomials and pairs of relatively prime polynomials in several variables over finite fields

Authors:
Xiang-Dong Hou;Gary L. Mullen
Affiliations:
Department of Mathematics, University of South Florida, 4202 East Fowler Ave., Tampa, FL 33620, USA;Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
Venue:
Finite Fields and Their Applications
Year:
2009

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Abstract

We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the total degree and the vector degree, are considered. We show that the number of irreducibles can be computed recursively by degree and that the number of relatively prime pairs can be expressed in terms of the number of irreducibles. We also obtain asymptotic formulas for the number of irreducibles and the number of relatively prime pairs. The asymptotic formulas for the number of irreducibles generalize and improve several previous results by Carlitz, Cohen and Bodin.