A Poisson negative binomial convolution law for randompolynomials over nite elds

Let F q X] denote a polynomial ring over a nite eld F q with q elements. Let P n be the set of monic polynomials over F q of degree n. Assuming that each of the q n possible monic polynomials in P n is equally likely, we give a complete characterization of the limiting behavior of P((n = m) as n ! 1 by a uniform asymptotic formula valid for m 1 and n?m ! 1, where n represents the number (multiplicities counted) of irreducible factors in the factorization of a random polynomial in P n. The distribution of n 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((n = 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 R enyi's problem, concerning the distribution of the diierence of the (total) number of irreducibles and the number of distinct irreducibles, is also presented.