Factorizations of root-based polynomial compositions

AbstractLet Fq denote the finite field of order q=pr, p a prime and r a positive integer, and let f(x) and g(x) denote monic polynomials in Fq[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 Fq 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