A Poisson * Negative Binomial Convolution Law for Random Polynomials over Finite Fields

Abstract

Let Fq[X] denote a polynomial ring over a finite field Fq with q elements. Let n be the set of monic polynomials over Fq of degree n. Assuming that each of the qn possible monic polynomials in n is equally likely, we give a complete characterization of the limiting behavior of P(Ωn=m) as n∞ by a uniform asymptotic formula valid for m≥1 and n−m∞, where Ωn represents the number (multiplicities counted) of irreducible factors in the factorization of a random polynomial in n. The distribution of Ωn is essentially the convolution of a Poisson distribution with mean log n and a negative binomial distribution with parameters q and q−1. Such a convolution law exhibits three modes of asymptotic behaviors: when m is small, it behaves like a Poisson distribution; when m becomes large, its behavior is dominated by a negative binomial distribution, the transitional behavior being essentially a parabolic cylinder function (or some linear combinations of the standard normal law and its iterated integrals). As applications of this uniform asymptotic formula, we derive most known results concerning P(Ωn=m) and present many new ones like the unimodality of the distribution. The methods used are widely applicable to other problems on multiset constructions. An extension to Renyi's problem, concerning the distribution of the difference of the (total) number of irreducibles and the number of distinct irreducibles, is also presented. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 17–47, 1998