Vol. 109, No. 1, 1983

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

Donald George Babbitt and V. S. Varadarajan

Vol. 109 (1983), No. 1, 1–80

One of the main goals of this paper is to develop an algorithm for reducing the first order (singular) system of differential equations:

df = A(z)f                          (†)

to a Turrittin-Levelt canonical form. Here A(z) = zrAr + zr+1Ar+1 + , r < 1 and Ar+m gl(n;C) m 0. The reduction of () 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 () 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 gl(n) are replaced by an arbitrary affine algebraic group G over C and its Lie algebra g.

Mathematical Subject Classification 2000
Primary: 34A20
Secondary: 12H05, 14D05, 14D25
Received: 7 January 1982
Revised: 28 April 1982
Published: 1 November 1983
Donald George Babbitt
V. S. Varadarajan
Department of Mathematics
Univ of California, Los Angeles
Los Angeles CA 90095-1555
United States