engleski

Schinzel's problem: Imprimitive covers and the monodromy method

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 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. 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.

Davenport's Problem; Schinzel's Problem; factorization of variables separated polynomials; Riemann's Existence Theorem; wreath products; imprimitive groups

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