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

Let Fq[X] denote a polynomial ring over a finite field Fq with q elements. Let Pn be the set of monic polynomials over Fq of degree n. Assuming that each of the q possible monic polynomials in Pn 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 Pn. 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 Rényi’s problem, concerning the distribution of the difference of the (total) number of irreducibles and the number of distinct irreducibles, is also presented. AMS 1991 Mathematics subject classification: Primary 11T06; secondary 60C05.

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