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

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Science China Mathematics 2012年第1期55卷 1-72页

作者： FRIED Michael D. Department of Mathematics University of California at Irvine

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 *** 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 *** and Loeser applied this to add Chow motive coefficients to previously introduced zeta functions on a diophantine *** 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 *** encouraged Thompson to formulate the genus 0 problem on rational function monodromy *** and Thompson led its solution in *** look at two developments since the solution of Davenport's *** 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.

关键词： group representations normal varieties Galois stratification Davenport pair monodromy group primitive group covers fiber products Open Image Theorem Riemann's existence theorem genus zero problem

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Two Classes of Solvable Subgroups of SL（3, C） and Application for Differential Equations in C

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Chinese Quarterly Journal of Mathematics 2010年第1期25卷 30-36页

作者： ZHANG Shao-fei LI Hui-zhen Department of Mathematics Beijing University of Aeronautics and Astronautics Beijing 100083 China Department of Basic Courses Academy of Armored Force Engineering Beijing 100072 China

Structures of two classes of solvable subgroups in SL（3, C） are given in this paper, and the integrability of the 3-order Fuchsian equation which is integrable in the sense that its monodromy group is solvable is di... 详细信息

关键词： Fuehsian system solvable group general linear group special linear group monodromy group integrability

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