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

作     者：FRIED Michael D. 

作者机构：Department of MathematicsUniversity of California at Irvine 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2012年第55卷第1期

页      面：1-72页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

主　　题：分类处理 变量分离方程 有限单群 引理 周期 zeta函数 剩余类域 分离变量 

摘      要：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,Lewis and 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.Denef and 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 by using the simple group classification.This encouraged Thompson to formulate the genus 0 problem on rational function monodromy groups.Guralnick and Thompson led its solution in stages.We look at two developments since the solution of Davenport s problem.Stemming from 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.