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: (†) df/dz=A(z)f to a Turrittin-Levelt canonical form. Here A(z) = zrAr + zr+1Ar+1 + …, r < - 1 and Ar+m ∈ g I ((n; C) m ≥ 0. The reduction of (†) to a canonical form is implemented by the natural gauge adjoint action of GL(n; f) where f 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 (†) is determined by Ar+m, 0 ≤ m < n(|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. © 1983 by Pacific Journal of Mathematics.