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VARIABLES SEPARATED EQUATIONS AND FINITE SIMPLE GROUPS

H. Davenport’s Problem asks: What can we expect of two polynomials, over Z, with the same ranges on almost all residue class fields? This stood out among many separated variable problems posed by Davenport, D.J. Lewis and A. Schinzel. By bounding the degrees, but expanding the maps and variables in Davenport’s Problem, we introduced Galois stratification. We thus enhanced the separated variable theme, solving an Ax and Kochen problem from their Artin Conjecture work. J. Denef and F. Loeser applied this to add Chow motive coefficients to previously introduced zeta functions on a diophantine statement. By restricting the variables, but leaving the degrees unbounded, we found the striking distinction between Davenport’s problem over Q and its generalization over any number field. We used this to encourage J. Thompson to formulate the genus 0 problem on rational function monodromy groups. R. Guralnick and Thompson solved this in stages. Then, separately, Guralnick and P. Müller (and other group theorists), found more practical forms of it. We look at at two developments since the solution of Davenport’s problem. • Stemming from C. MacCluer’s 1967 thesis, identifying a general class of problems, including Davenport’s, as monodromy precise. • The role of R(iemann’s) E(xistence) T(heorem) as a converse to problems generalizing Davenport’s. Using these developments we consider the significance of the kernel in going from 1st order equations to zeta functions with Chow motive coefficients.