2010 Fiscal Year Final Research Report

Analysis of an algebraic structure of a degenerate Garnier system from a viewpoint of algebraic solutions

Research Project

Project/Area Number	20540207
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Single-year Grants
Section	一般
Research Field	Global analysis
Research Institution	Mie University
Principal Investigator	KAWAMUKO Hiroyuki Mie University, 教育学部, 准教授 (00303719)
Project Period (FY)	2008 – 2010
Keywords	ガルニエ系 / 有理解 / 代数解 / 差分パンルヴェ方程式
Research Abstract	(1) We consider degenerate Garnier systems G(3,1,1) (which is defined by H.Kimura) and G(5/2,1,1) (which is defined by H.Kawamuko), and find all rational solutions of each equation. (2) We consider degenerate Garnier system G(3,2) (which is defined by H.Kimura), and find all algebraic solutions of G (3,2). (3) By using the results (1) and (2), we show that there is no birational transformation between these systems. (4) We consider a third order Fuchsian differential equation which has three regular singular points on Riemann sphere, and show that the Schlesinger transformation is equivalent to a difference VI Painlev\'e equation. Moreover, we can define a difference Garnier system in a similar way.