Variables Separated Equations and Finite Simple Groups: Davenport's
problem is to figure out the nature of two polynomials over a number
field having the same ranges on almost all residue class fields of the
number field. Solving this problem initiated the monodromy method.
That included two new tools: the B(ranch)C(ycle)L(emma) and the
Hurwitz monodromy group. By walking through Davenport's problem with
hindsight, variables separated equations let us simplify lessons on
using these tools. We attend to these general questions:
1. What allows us to produce branch cycles, and what was their effect
on the Genus 0 Problem (of Guralnick/Thompson)?
2. What is in the kernel of the Chow motive map, and how much is it
captured by using (algebraic) covers?
3. What groups arise in 'nature' (a 'la a paper by R. Solomon)?
Each phrase addresses formulating problems based on equations. We seem
to need explicit algebraic equations. Yet why, and how much do we lose/
gain in using more easily manipulated surrogates for them? To make
this clear we consider the difference in the result for Davenport's
Problem and that for its formulation over finite fields, using a
technique of R. Abhyankar.