Skip to main content
×
Home
    • Aa
    • Aa

On random polynomials over finite fields

  • Richard Arratia (a1), A. D. Barbour (a2) and Simon Tavaré (a1)
Abstract
Abstract

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.

Copyright
References
Hide All
[1]Arratia R. A. and Tavaré S.. Independent process approximations for random combinatorial structures. Advances in Mathematics (1993). (In the press.)
[2]Arratia R. A., Barbour A. D. and Tavaré S.. Poisson process approximations for the Ewens Sampling Formula. Ann. Appl. Prob. 2 (1992), 519535.
[3]Barbour A. D.. Comment on a paper of Arratia, Goldstein and Gordon. Statistical Science (1990), 425427.
[4]Barbour A. D., Holst L. and Janson S.. Poisson approximation. Oxford University Press, 1992.
[5]Car M.. Factorization dans Fq(X). C.R. Acad. Sci. Paris Ser. I. 294 (1982), 147150.
[6]Car M.. Ensembles de polynômes irréductibles et théorèmes de densité. Acta Arith. 44 (1984), 323342.
[7]Csörgo˝ M. and Révész P.. Strong approximation in probability and statistics. Academic Press, 1981.
[8]de Laurentis J. M. and Pittel B. G.. Random permutations and Brownian motion. Pacific J. Math. 119 (1985), 287301.
[9]Diaconis P. and Pitman J. W.. Unpublished lecture notes. Statistics Department, University of California, Berkeley (1986).
[10]Diaconis P., McGrath M. and Pitman J. W.. Cycles and descents of random permutations. Preprint (1992).
[11]Donnelly P. and Joyce P.. Continuity and weak convergence of ranked and size-biassed permutations on an infinite simplex. Stoch. Proc. Applns 31 (1989), 89103.
[12]Feller W.. The fundamental limit theorems in probability. Bull. Amer. Math. Soc. 51 (1945), 800832.
[13]Feller W.. An introduction to probability theory and its applications. Vol I. Wiley, New York, 1950.
[14]Flajolet P. and Soria M.. Gaussian limiting distributions for the number of components in combinatorial structures. J. Comb. Th. A 53 (1990), 165182.
[15]Gessel I. M. and Reutenauer C.. Counting permutations with given cycle structure and descent set. Preprint (1991).
[16]Hansen J.. Order statistics for decomposable combinatorial structures. Preprint (1991).
[17]Kolchin V. F.. Random mappings. Optimization Software, Inc., New York, 1986.
[18]Komlós J., Major P. and Tusnády G.. An approximation of partial sums of independent RV-s, and the sample DF. I, Z. Wahrscheinlichkeitstheorie verw. Geb. 32 (1975), 111131.
[19]Kurtz T. G.. Strong approximation theorems for density dependent Markov chains. Stoch. Proc. Applns 6 (1978), 223240.
[20]Lidl R. and Niederreiter H.. Introduction to finite fields and their applications. Cambridge University Press, 1986.
[21]Metropolis N. and Rota G.-C.. Witt vectors and the algebra of necklaces. Adv. Math. 50 (1983), 95125.
[22]Metropolis N. and Rota G.-C.. The cyclotomic identity. In Contemporary Mathematics 34, (American Mathematical Society, 1984), pp. 1927.
[23]Rachev S. T.. Probability metrics and the stability of stochastic models. Wiley, 1991.
[24]Rényi A.. Théorie des elements saillants d'une suite d'observations. Coll. Comb. Meth. Prob. Th. Mathematisk Institut, Aarhus Universitet (1962), 104115.
[25]Shepp L. A. and Lloyd S. P.. Ordered cycle lengths in a random permutation. Trans. Amer. Math. Soc. 121 (1966), 340357.
[26]Vershik A. M. and Shmidt A. A.. Limit measures arising in the theory of groups I. Theor. Prob. Applns 22 (1977), 7985.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 5 *
Loading metrics...

Abstract views

Total abstract views: 43 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd October 2017. This data will be updated every 24 hours.