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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].

MSC:
 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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