Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Variables separated equations: Strikingly different roles for the Branch
  Cycle Lemma and the Finite Simple Group Classification   H. Davenport's Problem asks: What can we expect of two polynomials, over the
integers, 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 the rationals, 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.

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The categories of this paper are: q-fin
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The categories of this paper are: math-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: cond-mat
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The categories of this paper are: alg-geom
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The categories of this paper are: gr-qc
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The categories of this paper are: acc-phys
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The categories of this paper are: astro-ph
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The categories of this paper are: stat
0
The categories of this paper are: hep-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-bio
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The categories of this paper are: plasm-ph
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The categories of this paper are: funct-an