Variables separated polynomials, the genus 0 problem and moduli spaces
Michael D. Fried · 1999
. The monodromy method---featuring braid group action---first appeared as a moduli space approach for finding solutions of arithmetic problems that produce reducible variables separated curves. Examples in this paper illustrate its most interesting aspect: investigating the moduli space of exceptions to a specific diophantine outcome. Explicit versions of Hilbert's irreducibility theorem and Davenport's problem fostered this technique and motivated the genus 0 problem started by J. Thompson and taken up by many group theorists. We review progress on the genus 0 problem in 0 characteristic, and its quite different contributions in positive characteristic. Example: Let f and h be polynomials with coefficients in a number field K. The classification of finite simple groups shows there is a bound on exceptional degrees for f to the following result. If f is indecomposable and h is not a composition with f , then f(x) \\Gamma h(y) is irreducible. This answered challenge problems...