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On local Weyl equivalence of higher order Fuchsian equations. (English) Zbl 1327.34145

Let \(F\) be the differential field of meromophic germs at the origin. The authors consider the problem of classifying linear differential operators over \(F\). This problem is closely related to the inverse problem of the Galois theory of differential fields which, as it is known (see [C. Mitschi and M. F. Singer, J. Pure Appl. Algebra 110, No. 2, 185–194 (1996; Zbl 0854.12001)]), has a nontrivial solution for the field \(F\). Different kinds of equivalence of operators (Levi, Weil, Fuchs) are discussed. However, in the article, the authors restrict the consideration to Fuchsian operators, what essentially simplifies the task of classification. From the equivalence of these operators over \(\mathbb{C}((x))\) it follows their equivalence over \(F\). All zeros of Fuchsian operators are generated over \(F\) by zeros of operators \({y}'={{z}^{-1}},{y}'{{y}^{-1}}=\lambda {{z}^{-1}}(\lambda \in \mathbb{C})\). This allows them to obtain a number of results ”which are similar but not completely parallel to the known results on local (holomorphic and meromorphic) gauge equivalence of systems of first order equations”. Note that two Fuchsian operators are W-equivalent if and only if they have the same logarithmic type and match the modules generated by the roots of their characteristic equations (see [N. V. Grigorenko, Mat. Sb., Nov. Ser. 109(151), 355–364 (1979; Zbl 0416.12013)]).

MSC:

34M25 Formal solutions and transform techniques for ordinary differential equations in the complex domain
34M03 Linear ordinary differential equations and systems in the complex domain
34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
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