Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

On random polynomials over finite fields

Richard Arratiaa1, A. D. Barboura2 and Simon Tavaréa3

a1 Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, U.S.A.

a2 Institut für Angewandte Mathematik, Universität Zurich, Rämistrasse 74, CH-8001 Zurich, Switzerland

a3 Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, U.S.A.

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.

(Received July 20 1992)