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Additional parameters in inverse monodromy problems. (English. Russian original) Zbl 1159.34059
Sb. Math. 197, No. 12, 1753-1773 (2006); translation from Mat. Sb. 197, No. 12, 43-64 (2006).
The title of the article refers to the fact that, in general, the number of parameters determining a rank \(p\) Fuchsian ODE with \(n\) singular points is less than the number of parameters determining a (reducible) rank \(p\) monodromy representation of the fundamental group of \(\overline{\mathbb C}\backslash\{a_1,\dots,a_n\}\). In fact, an order \(p\) scalar Fuchsian ODE solving such an inverse monodromy problem, besides singularities at \(a_1, \dots, a_n\) with the prescribed monodromy, has a number of apparent singularities (at these points, the coefficients of the ODE are singular while all its solutions are holomorphic). In the matrix case, one of the singular points among \(a_1, \dots, a_n\) can be non-Fuchsian regular.
Here, using the method of holomorphic vector bundles developed by Bolibrukh, the authors obtain an estimate for the number of apparent singularities for the scalar Fuchsian ODE and an estimate for the Poincaré rank of a unique non-Fuchsian regular singularity for the matrix ODE. These general estimates supplement the known results for irreducible representations. Also, applying similar approach, the authors describe a meromorphic reduction to a polynomial Birkhoff’s normal form of a linear reducible (i.e., block upper-triangular) matrix ODE defined in a neighborhood of its irregular singularity and estimate the increment of the Poincaré rank of the resulting irregular singular point.

34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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