Factorizations of root-based polynomial compositions (Q5948974)

Let \(F_q\) be a finite field, \(G\) a nonempty subset of the algebraic closure \(\Gamma_q\) of \(F_q\) that is invariant under the Frobenius automorphism \(\alpha \rightarrow \sigma(\alpha) = \alpha^q\), and \(\diamond\) a binary operation on \(G\) satisfying \(\sigma(\alpha\diamond\beta) = \sigma(\alpha)\diamond\sigma(\beta)\) for all \(\alpha, \beta \in G\). Additionally let \(M_G[q,x]\) denote the set of monic polynomials over \(F_q\) whose roots lie in \(G\). Then the composed product of two polynomials \(f,g \in M_G[q,x]\), denoted \(f \diamond g\), is the polynomial \(f \diamond g = \prod_\alpha\prod_\beta(x-(\alpha\diamond\beta))\) where the \(\diamond\)-products are taken over all roots \(\alpha, \beta\) of \(f, g\). NEWLINENEWLINENEWLINEThe author generalizes a result of \textit{J. V. Brawley} and \textit{L. Carlitz} [Discrete Math. 65, 115-139 (1987; Zbl 0615.05007)], who showed that \(f \diamond g\) is irreducible if and only if \(f, g\) are irreducible and \(\gcd(m,n) = 1\) where \(m = \deg(f)\), \(n = \deg(g)\) are the degrees of the polynomials \(f\) and \(g\). In particular for \(f, g\) irreducible it is shown that the number of distinct irreducible factors of \(f \diamond g\) in \(M_G[q,x]\) is upper bounded by \(d = \gcd(m,n)\) and the degree of each factor divides \(h = \text{lcm}(m,n)\). NEWLINENEWLINENEWLINEMoreover it is proved that the range \(H_{m,n}\) for the degrees of irreducible factors of \(f \diamond g\) is a subset of \(D = \{h/l:l|\bar{d}\}\) where \(\bar{d}\) is the largest factor of \(d\) such that \(\gcd(\bar{d},m/d) = \gcd(\bar{d},n/d) = 1\), and conditions under which we have \(H_{m,n} = D\) are provided. NEWLINENEWLINENEWLINEAdditionally conditions are given under which the upper bound on the number of distinct irreducible factors of \(f \diamond g\) is met for the general case as well as for the cases that \(G = \Gamma_q\) and \(\diamond\) is the usual addition respectively the multiplication on \(\Gamma_q\).