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Number of irreducible polynomials and pairs of relatively prime polynomials in several variables over finite fields. (English) Zbl 1190.11063
Authors’ summary: 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.

MSC:
 11T06 Polynomials over finite fields
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References:
 [1] Adams, W.; Loustaunau, P., An introduction to Gröbner bases, Grad. stud. math., vol. 3, (1994), Amer. Math. Soc. Providence, RI · Zbl 0803.13015 [2] Bender, E.A.; Goldman, J.R., On the applications of Möbius inversion in combinatorial analysis, Amer. math. monthly, 82, 789-803, (1975) · Zbl 0316.05001 [3] Benjamin, A.T.; Bennett, C.D., The probability of relatively prime polynomials, Math. mag., 80, 196-202, (2007) · Zbl 1204.11059 [4] Bodin, A., Number of irreducible polynomials in several variables over finite fields, Amer. math. monthly, 115, 7, 653-660, (2008) · Zbl 1219.12003 [5] Bourbaki, N., Elements of mathematics, algebra I, (1989), Springer-Verlag Berlin [6] Carlitz, L., The arithmetic of polynomials in a Galois field, Amer. J. math., 54, 39-50, (1932) · JFM 58.0150.01 [7] Carlitz, L., The distribution of irreducible polynomials in several indeterminates, Illinois J. math., 7, 371-375, (1963) · Zbl 0118.26002 [8] Carlitz, L., The distribution of irreducible polynomials in several indeterminates. II, Canad. J. math., 17, 261-266, (1965) · Zbl 0135.01704 [9] Cohen, S.D., The distribution of irreducible polynomials in several indeterminates over a finite field, Proc. edinb. math. soc., 16, 1-17, (1968) · Zbl 0172.05305 [10] Cohen, S.D., Further arithmetical functions in finite fields, Proc. edinb. math. soc., 16, 349-363, (1969) · Zbl 0188.11101 [11] Cohen, S.D., Some arithmetical functions in finite fields, Glasg. math. J., 11, 21-36, (1970) · Zbl 0209.35903 [12] Corteel, S.; Savage, C.; Wilf, H.; Zeilberger, D., A pentagonal number sieve, J. combin. theory ser. A, 82, 186-192, (1998) · Zbl 0910.05008 [13] Hou, X.; Mullen, G.L., Number of irreducible polynomials and pairs of relatively prime polynomials in several variables over finite fields · Zbl 1190.11063 [14] Lidl, R.; Niederreiter, H., Finite fields, Encyclopedia math. appl., vol. 20, (1997), Cambridge Univ. Press Cambridge [15] Reifegerate, A., On an involution concerning pairs of polynomials in $$F_2$$, J. combin. theory ser. A, 90, 216-220, (2000) [16] von zur Gathen, J., Counting reducible and singular bivariate polynomials, Finite fields appl., 14, 944-978, (2008) · Zbl 1192.12003 [17] von zur Gathen, J.; Gerhard, J., Modern computer algebra, (1999), Cambridge Univ. Press Cambridge, UK · Zbl 0936.11069
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