Schinzel's problem: imprimitive covers and the monodromy method (Q2919664)

From the abstract: Schinzel's original problem was to describe expressions \(f(x)-g(y),\) with \(f,g\in\mathbb{C}[x]\) nonconstant, that are reducible. Call \((f,g)\) a \textit{Schinzel pair} if this happens non-trivially. When \(f\) is decomposable [\textit{M. D. Fried}, Illinois J. Math., 17, No. 1, 128--146 (1973; Zbl 0266.14013)] solved Schinzel's Problem as a corollary.NEWLINENEWLINE... We take here the next step to consider the problem left by \textit{R. M. Avanzi} and \textit{U. M. Zannier} [Compos. Math., 139, No. 3, 263--295,(2003; Zbl 1050.14020)] and the second author [\textit{I. Gusić}, ``Reducibility of \(f(x)-cf(y)\)'', preprint 2010]. Consider those \(f\) for which there is a \(g=\alpha\circ f,\) with \(\alpha\in \mathrm{PGL}_2(\mathbb{C})\) satisfying an essential condition for possible Schinzel pairs: The Galois closure of the covers \(f,g:\mathbb{P}_x^1\to\mathbb{P}_x^1\) are the same. Then, from those find \((f,g)\) that are Schinzel pairs.