Formal reduction theory of meromorphic differential equations: a group theoretic view

Abstract

One of the main goals of this paper is to develop an algorithm for reducing the first order (singular) system of differential equations: (f) f z = ^f to a Turrittin-Levelt canonical form. Here A(z) = z r A r + z r+] A r+ 4-, r < -1 and A r+m G flI(; Q m > 0. The reduction of (f) to a canonical form is implemented by the natural gauge adjoint action of GL(n; ) where ^is the algebraic closure of the field of formal Laurent series about 0 with at most a finite pole at 0. For example, it is shown that the irregular part of the canonical form (f) is determined by A r+m , 0 < ra < rc(| r| -1). The proofs utilize group theoretic techniques as well as the method of Galois descent. In particular almost all of the results generalize to the case where GL( n) and g I (n) are replaced by an arbitrary affine algebraic group G over C and its Lie algebra g.