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Regular linear systems on \({\mathbb{C}}P^ 1\) and their monodromy groups. (English) Zbl 0814.34006
Camacho, C. (ed.) et al., Complex analytic methods in dynamical systems. Proceedings of the congress held at Instituto de Matemática Pura e Aplicada, IMPA, Rio de Janeiro, Brazil, January 1992. Paris: Société Mathématique de France, Astérisque. 222, 259-283 (1994).
A meromorphic linear system of differential equations on \(\mathbb{C}\text{P}^ 1\) can be presented in the form \(dX/dt= A(t)X\), where \(A(t)\) is a meromorphic on \(\mathbb{C}\text{P}^ 1\) \(n\times n\) matrix function. Denote its poles \(a_ 1, \dots, a_{p+1}\), \(p\geq 1\). The monodromy operator \(M_ j\) corresponding to a usual contour is the linear operator mapping the matrix \(B\in \text{GL}(n, \mathbb{C})\) onto the value of the analytic continuation of the solution of system which equals \(B\) for the \(t= a\) along the contour encircling \(a_ j\), positively oriented. It is natural to consider \(\text{GL}(n, \mathbb{C})^ p\) as the space of monodromy groups of a regular system on \(\mathbb{C}\text{P}^ 1\) with \(p+1\) prescribed poles. The author defines an analytic stratification of \(\text{GL}(n, \mathbb{C})^ p\) by the Jordan normal forms of the operators \(M_ 1,\dots, M_{p+1}\) and the possible reducibility of the group \(\{M_ 1,\dots, M_ p\}\). Fixing the Jordan normal form of \(M_ 1,\dots, M_ p\) is equivalent to restricting the matrix-function \(M_{p+1}= (M_ 1\cdots M_ p)^{-1}\) to a smooth analytic subvariety of \(\text{GL}(n, \mathbb{C})^ p\). The basic aim of this paper is to begin the study of the stratification of \(\text{GL}(n, \mathbb{C})^ p\) and the smoothness of the strata and superstrata.
For the entire collection see [Zbl 0797.00019].

34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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