Factorizations of root-based polynomial compositions

摘要

Let F_q denote the finite field of order q = p~r, p a prime and r a positive integer, and let f(x) and g(x) denote monic polynomials in F_q[x] of degrees m and n, respectively. Brawley and Carlitz (Discrete Math. 65 (1987) 115-139) introduce a general notion of root-based polynomial composition which they call the composed product and denote by f ◇ g. They prove that f ◇ g is irreducible over F_q if and only if f and g are irreducible with gcd(m,n) = 1. In this paper, we extend Brawley and Carlitz's work by examining polynomials which are composed products of irreducibles of non-coprime degrees. We give an upper bound on the number of distinct factors of f ◇ g, and we determine the possible degrees that the factors of f ◇ g can assume. We also determine when the bound on the number of factors of f ◇ g is met.