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    Ivchenko, G. I. and Medvedev, Yu. I. 2008. Random polynomials over a finite field. Discrete Mathematics and Applications, Vol. 18, Issue. 1, p. 1.

    Ивченко, Григорий Иванович Ivchenko, Grigorii Ivanovich Медведев, Юрий Иванович and Medvedev, Yurii Ivanovich 2008. Случайные многочлены над конечным полем. Дискретная математика, Vol. 20, Issue. 1, p. 3.

    Ivchenko, G. I. and Medvedev, Yu. I. 2003. Goncharov's Method and Its Development in the Analysis of Different Models of Random Permutations. Theory of Probability & Its Applications, Vol. 47, Issue. 3, p. 518.

    Hwang, Hsien-Kuei 1999. Asymptotics of poisson approximation to random discrete distributions: an analytic approach. Advances in Applied Probability, Vol. 31, Issue. 02, p. 448.

    Schmutz, Eric 1995. The order of a typical matrix with entries in a finite field. Israel Journal of Mathematics, Vol. 91, Issue. 1-3, p. 349.

    Barbour, A. D. and Tavaré, Simon 1994. A Rate for the Erdős-Turán Law. Combinatorics, Probability and Computing, Vol. 3, Issue. 02, p. 167.

    Hansen, Jennie C. 1994. Order statistics for decomposable combinatorial structures. Random Structures & Algorithms, Vol. 5, Issue. 4, p. 517.

  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 114, Issue 2
  • September 1993, pp. 347-368

On random polynomials over finite fields

  • Richard Arratia (a1), A. D. Barbour (a2) and Simon Tavaré (a1)
  • DOI:
  • Published online: 24 October 2008

We consider random monic polynomials of degree n over a finite field of q elements, chosen with all qn possibilities equally likely, factored into monic irreducible factors. More generally, relaxing the restriction that q be a prime power, we consider that multiset construction in which the total number of possibilities of weight n is qn. We establish various approximations for the joint distribution of factors, by giving upper bounds on the total variation distance to simpler discrete distributions. For example, the counts for particular factors are approximately independent and geometrically distributed, and the counts for all factors of sizes 1, 2, …, b, where b = O(n/log n), are approximated by independent negative binomial random variables. As another example, the joint distribution of the large factors is close to the joint distribution of the large cycles in a random permutation. We show how these discrete approximations imply a Brownian motion functional central limit theorem and a Poisson-Dirichiet limit theorem, together with appropriate error estimates. We also give Poisson approximations, with error bounds, for the distribution of the total number of factors.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
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