Rational Solutions of the Schlesinger System and Isoprincipal Deformations of Rational Matrix Functions I

Abstract

Abstract. In the present paper we discuss the general facts, concerning the Schlesinger system: the τ -function, the local factorization of solutions of Fuchsian equations and holomorphic deformations. We introduce the terminology "isoprincipal" for the deformations of Fuchsian equations with general (not necessarily non-resonant) matrix coefficients, corresponding to solutions of the Schlesinger system. Every isoprincipal deformation is isomonodromic. The converse is also true in the non-resonant case, but not in general.In the forthcoming sequel we shall give an explicit description of a class of rational solutions of the Schlesinger system, based on the techniques, developed here, and the realization theory for rational matrix functions. NOTATIONS.• C stands for the complex plane;• C stands for the extended complex plane (= the Riemann sphere): C = C ∪ ∞; • C n stands for the n -dimensional complex space; • in the coordinate notation, a point t ∈ C n is written as t = (t 1 , . . . , t n ) ; • C n * is the set of those points t = (t 1 , . . . , t n ) ∈ C n , whose coordinates t 1 , . . . , t n are pairwise different: C n * = C n \ 1≤i,j≤n, i =j {t : t i = t j } ; • M k stands for the set of all k × k matrices with complex entries; • [ . , . ] denotes the commutator: for A, B ∈ M k , [A, B] = AB − BA;• I stands for the unity matrix of the appropriate dimension; • R(M k ) stands for the set of all rational M k -valued functions R with detR(z) ≡ 0; • P(R) stands for the set of all poles of the function R, N (R) stands for the set of all poles of the function R −1 ; P(R) is said to be the pole set of the function R, N (R) is said to be the zero set of the function R.