Abstract
We study a singular system of ordinary differential equations, Lu ≡ {zp+1DIN−A(z)}u = 0, where z ∈ C, p ≥ 0, D = d/dz and A(z) is an N square matrix of holomorphic functions in a neighborhood of z = 0. We call a matrix of operators L ≡ zp+1DIN−A(z) := (p, A(z)) a system, for short.
In this paper we introduce a notion of T-expansion of a matrix function A(z), which gives a summation expression of A(z) different from the usual Taylor expansion. The idea comes from the result by L. R. Volevič [Vol], where he studied a general matrix of partial differential operators A(∂x) and he presented a way of finding out a leading part from the matrix operators which we call Volevič's lemma (cf. Section 3).
By using the T-expansion of A(z), we obtain an algorithm of the reduction procedure of the system L into a decomposition by irreducible subsystems (cf. Theorem Aδ and (4.23) in Subsection 4.3). From this decomposition we can define the Newton polygon N(L) by taking the characteristic polynomial of each irreducible subsystem in Definitions 2.2 and 2.3. The importance of the Newton polygon N(L) will be shown by proving an index formula of the operator L on a formal Gevrey space ${\cal G}$s (1 ≤ s ≤ ∞) in Theorem C(∞), which is obtained from the vertical coordinate of an associated vertex of N(L). This is an extension of J.-P. Ramis's results [Ram1,2] for single operators. The index formula is proved by applying the index formula for general matrix of ordinary differential operators obtained in a joint paper with M. Yoshino [M-Y2]. Many other problems concerned with the study of the singular system L = (p, A(z)) are studied. For example, in Subsection 4.4 we give a structure of fundamental matrix solution of Lu = 0 in exact form. In other words, the reduction procedure into Hukuhara-Turrittin's canonical form is exactly shown. The reduction procedure seems to peel one piece of peel of an onion one piece.
