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

Abstract

Abstract. 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, Galois stratification 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, solved by applying the Branch Cycle Lemma, and its generalization over any number field, solved using the simple group classification. This encouraged J. Thompson to formulate 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 of problems, including Davenport's, as monodromy precise.• R(iemann) E(xistence) T(heorem)'s role as a converse to problems generalizing Davenport's, and Schinzel's (on reducibility). We use these to consider: Going beyond the simple group classification to handle imprimitive groups; and what is the role of covers and correspondences in going from algebraic equations to zeta functions with Chow motive coefficients.

30 years of collaboration

“…[79] and the monograph [151]) and the important contributions due to Fried (cf. the survey article [93]). …”

Schinzel's problem: Imprimitive covers and the monodromy method

“…Comments on [Ba02] For f, g ∈ C[x], Schinzel's problem was to describe those cases when (1.1) f (x) − g(y) factors nontrivially as a polynomial in two variables. The topic is in [Sc71]; [Fr12] has many relevant references. With K a number field, let O K be its ring of integers, p p p a prime ideal of O K , and O K /p p p its residue class field.…”

“…With the indecomposable assumption, solutions to Davenport's and Schinzel's problems were essentially the same( solved in [Fr73, Thm. 1]; see [Fr12,thm. 4…”

“…= n is odd, use Ex. 1.5 to see the replacement for (1.11) (as in [Fr12,Lem. 7.4]) has a factor of degree 1, the rest of degree 2.…”

Twisted Galois stratification

“…When compared to the earlier work of Ax [2], made more explicit by Kiefe [14], which showed that every formula in the language of rings is equivalent to a formula with a single (bounded) existential quantifier, the fundamental achievement of the Galois stratification was the effective (in fact primitive recursive) nature of its quantifier elimination procedure. Moreover, the precise description of formulae in terms of Galois covers was particularly well-suited for applications of geometric and number-theoretic nature, for example in Fried's work on Davenport's problem [8]. In our opinion, the most impressive application was in the work of Denef and Loeser on arithmetic motivic integration in [5].…”