Effective formal reduction of linear differential systems
In the introduction, the author considers the formal reduction of the first-order linear differential system \[ x^{q+1}{dY\over dx}= A(x)Y,\tag{1} \] where \(A\) is an \(n\)-dimensional formal power series matrix over a field \(K\subset\mathbb{C}\) and \(q> 0\) is an integer, called the Poincaré rank of the system. The theory establishes the existence of a formal fundamental matrix solution (F.F.M.S.) which represents \(n\) linearly independent formal solutions to system (1). For the theoretical results see \textit{W. Balser}, \textit{W. B. Jurkat} and \textit{D. A. Lutz} [Funkc. Ekvacioj, Ser. Int. 22, 197-221 (1979; Zbl 0434.34002), ibid. 257-283 (1979; Zbl 0473.34028), and J. Math. Anal. Appl. 71, 48-94 (1979; Zbl 0415.34008)] and \textit{W. Wasow} [Funkc. Ekvacioj, Ser. Int. 10, 107-122 (1967; Zbl 0155.42102)]. In this note, the author presents a new method and new technologies for the direct formal reduction and the computation of a F.F.M.S. In section 2, the author considers classical results of the theory of linear differential systems. The splitting lemma is an important tool in the classical algorithms for the formal reduction. Section 3 contains the important ideas and results of the paper. The author generalizes the notion of simple systems [see \textit{M. A. Barkatou} and \textit{E. Pflügel}, in: O. Gloor (ed.), Proceedings of the 1998 international symposium on symbolic and algebraic computation. ISSAC '98, Rostock/Germany, August 13-15 1998. New York, NY: ACM Press, 268-275 (1998; Zbl 0928.65082)] by introducing the notion of \(k\)-simple systems, and obtains a generalization of the classical splitting lemma. In section 4 are presented a new characterization of the notion of Moser-irreducible and super-irreducible forms of linear differential systems [see \textit{A. Hilali} and \textit{A. Wazner}, Ann. Inst. Fourier 36, No. 3, 67-81 (1986; Zbl 0595.34006)]. In the first part of section 5 the \(\rho\)-reduction algorithm is introduced. Practical aspects are presented of an implementation of this algorithm in a computer algebra system and some examples are given. section 6 contains notes on formal solutions and a conclusion. For other details see the author's comprehensive references.
Moser-irreducible and super-irreducible forms
scientific article; zbMATH DE number 1394479