On the Number of Factorizations of Polynomials over Finite Fields

Abstract

Motivated by coding applications, two enumeration problems are considered: the number of distinct divisors of a degree-m polynomial over $\mathbb{F} = GF(q)$, and the number of ways a polynomial can be written as a product of two polynomials of degree at most n over $\mathbb{F}$. For the two problems, bounds are obtained on the maximum number of factorizations, and a characterization is presented for polynomials attaining that maximum. Finally, expressions are presented for the average and the variance of the number of factorizations, for any given m (resp., n).