Monodromy, the Riemann-Hilbert Problem, and the Differential Galois Group

Abstract

Let U be an open connected subset of the complex sphere P¹=C∪{∞} and let Y′=AY be a differential equation on U, with A an n×n matrix with coefficients that are meromorphic functions on U. We assume that the equation is regular at every point p∈U. Thus, for any point p∈U, the equation has n independent solutions y₁,…, y _n consisting of vectors with coordinates in C({z−p}). It is known ([132], chap. 9; [225], p. 5) that these solutions converge in a disk of radius ρ, where ρ is the distance from p to the complement of U. These solutions span an n-dimensional vector space denoted by V _p . If we let F _p be a matrix whose columns are the n independent solutions y₁,…, y _n then F _p is a fundamental matrix with entries in C({z−p}). One can normalize F _p such that F _p (p) is the identity matrix. The question we are interested in is:

Does there exist on all of U, a solution space for the equation having dimension n?