Variables Separated Equations and Finite Simple Groups:
Mike Fried, UC Irvine
My talk will start with a variables
separated (algebraic) equation:
(*) f(x)-g(y)=0,
f and g polynomials.
I will discuss the roots of a paper "Relating two genus 0
problems of John Thompson," from Thompson's 70th birthday volume. Davenport's problem asked what
polynomial pairs ( f, g ) over Q
have their ranges on Z/p (the integers mod p, p a prime) the same for almost all p.
Davenport's precise conjecture was only 'almost
true.' From this came the Genus 0 Problem: Showing that
rational functions have severely limited monodromy groups. This was the work
of many group theorists, foremost of Bob Guralnick, starting with his
collaboration with Thompson.
By walking through Davenport's problem with
hindsight, we get lessons on two general tools about algebraic
equations. We celebrate that by attending to two questions:
- What allows us to produce equations from branch cycles?
- What (finite) groups arise in 'nature'?
Each phrase addresses an aspect of problems based on
equations. That is, we seem to need algebraic equations. Yet why, and
how much do we lose in using more easily manipulated surregates for
them? The full story is in UMStory.html.