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H. Davenport's Problem asks: What can we expect of two polynomials, over theintegers, with the same ranges on almost all residue class fields? This stoodout among many separated variable problems posed by Davenport, D.J. Lewis andA. Schinzel.  By bounding the degrees, but expanding the maps and variables in Davenport'sProblem, Galois stratification enhanced the separated variable theme, solvingan Ax and Kochen problem from their Artin Conjecture work. J. Denef and F.Loeser applied this to add Chow motive coefficients to previously introducedzeta functions on a diophantine statement.  By restricting the variables, but leaving the degrees unbounded, we found thestriking distinction between Davenport's problem over the rationals, solved byapplying the Branch Cycle Lemma, and its generalization over any number field,solved using the simple group classification. This encouraged J. Thompson toformulate the genus 0 problem on rational function monodromy groups. R.Guralnick and Thompson led its solution in stages.  We look at at two developments since the solution of Davenport's problem.  * Stemming from C. MacCluer's 1967 thesis, identifying a general class ofproblems, including Davenport's, as monodromy precise.  * R(iemann) E(xistence) T(heorem)'s role as a converse to problemsgeneralizing Davenport's, and Schinzel's (on reducibility).  We use these to consider: Going beyond the simple group classification tohandle imprimitive groups; and what is the role of covers and correspondencesin going from algebraic equations to zeta functions with Chow motivecoefficients.

Variables separated equations: Strikingly different roles for the Branch Cycle Lemma and the finite simple group classification