Proof of the existence of certain triples of polynomials (Q2462255)

Let \(a\), \(b\) and \(c\) be complex coprime polynomials, not all constant, such that \(a+b+c=0\), denote their maximum degree by \(M\), and the numbers of their respective distinct roots by \(l\), \(h\) and \(k\). The \(abc\)--theorem for function fields claims that \(l+h+k>M\). \textit{L. N. Vaserstein} and \textit{E. R. Wheland} [Commun. Algebra 31, No. 2, 751--772 (2003; Zbl 1073.12001)] raised the question to describe all the possibilities for the degrees and the numbers of distinsct roots of polynomials \(a\), \(b\) amd \(c\) as above. The author proves that the only essential restriction is the above inequality. The proof relies on Riemann existence theorem, which is used to provide a complex rational function of degree \(M\) with suitable ramification conditions above \(0\), \(1\) and \(\infty\). Hence the problem is reduced to a purely combinatorial construction of certain permutations, and that is completely attained in the paper itself. The result is then generalized to polynomials with coefficients in any algebraically closed field of characteristic \(0\) or \(>M\) by specialization and good reduction methods.