A generalization of Mason's theorem for four polynomials
Let \(f\), \(g\), \(h\), \(k \in {\mathbb C}[t]\) be non-constant relatively prime polynomials satisfying \(f+g+h=k\), and write \(n_0(F)\) for the number of distinct roots of a polynomial \(F\). The author proves that the maximum of the degrees of \(f\), \(g\), \(h\), and \(k\) is bounded above by \(2n_0(fghk)-3\). This is a generalization of Mason's Theorem for three polynomials, in which the maximal degree of \(f\), \(g\), and \(h\), where \(f+g=h\) and \((f,g,h)=1\), is bounded above by \(n_0(fgh)-1\). The proof makes use of the logarithmic derivative, some elementary linear algebra, and divisibility properties. As an application, the author deduces that if \(n_1, \dots, n_4\) are integers with \(n_i \geq 8\) for each \(i\), then there is no solution of the generalized Fermat equation \(a(t)^{n_1} + b(t)^{n_2} + c(t)^{n_3} = d(t)^{n_4}\) with \(a\), \(b\), \(c\), and \(d\) non-constant relatively prime polynomials over \({\mathbb C}\).
scientific article; zbMATH DE number 2102275