Schinzel's Problem: Imprimitive covers and the monodromy method

TL;DR

This paper advances the understanding of Schinzel's problem by developing a formula for branch cycles that characterizes when polynomial pairs are newly reducible, extending monodromy methods beyond primitive cases.

Contribution

It introduces a new formula for branch cycles that characterizes Schinzel pairs under certain conditions, enhancing monodromy techniques for imprimitive covers.

Findings

Derived a formula for branch cycles in Schinzel's problem.

Connected the formula to conditions by Avanzi, Gusic, and Zannier.

Extended monodromy methods to imprimitive covers.

Abstract

Schinzel's original problem was to describe when an expression f(x)-g(y), with f,g nonconstant and having complex coefficients, is reducible. We call such an (f,g) a Schinzel pair if this happens nontrivially: f(x)-g(y) is newly reducible. Fried accomplished this as a special case of a result in "http://www.math.uci.edu/~mfried/paplist-ff/dav-red.pdf">dav-red.pdf, when f is indecomposable. That work featured using primitive permutation representations. Even after 42 years going beyond using primitivity is a challenge to the monodromy method despite many intervening related papers (see http://www.math.uci.edu/~mfried/paplist-ff/UMStory.pdf">UMStory.pdf. Here we develop a formula for branch cycles that characterizes Schinzel pairs satisfying a condition of Avanzi, Gusic and Zannier and relate it to this ongoing story.