Schinzel's Problem: Imprimitive covers and the monodromy method

Details Schinzel's Problem: Imprimitive covers and the monodromy method Schinzel's Problem: Imprimitive covers and the monodromy method

2011-04-10

arxiv.org/abs/1104.1740v3

Mathematics

Algebraic Geometry

15 pages, 1 figure, to Appear in Acta Arithmetica early 2012 for the 75th birthday volume for Andrzej Schinzel

Scientific paper

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.

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