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Reduced forms of linear differential systems and the intrinsic Galois-Lie algebra of Katz. (English) Zbl 1448.12002

The paper is devoted to the direct problem of differential Galois theory. It generalizes the results of a previous paper with the participation of one of the authors [A. Aparicio-Monforte et al., J. Pure Appl. Algebra 217, No. 8, 1504–1516 (2013; Zbl 1272.12016)].
Calculation of the Galois group \(G\) of a differential equation (1) \(\delta\text{Y=AY,}\) where \(A\in M_{F}(n)\), over some differential field \(F\) of characteristic zero with an algebraically closed field of constants, is a nontrivial problem. In some cases, for example, when the \(G\) is a connected algebraic group and the cohomology set \(H^{1}(F,G)\) is trivial, by a linear transformation \(Y\longmapsto PY\:(P\in \mathrm{GL}_{F}(n))\) system (1) can be reduced to the form \(Y=BY,\) where \(B\in \mathrm{Lie}_{F}(G))\). Using the matrix \(B\) of the reduced form of equation (1), one can calculate the Lie algebra of the group \(G\), which contains important information about the structure of the group \(G\).
In the paper, a number of results are obtained regarding the reduction of systems of the form (1) to the reduced form. In particular, it is shown that the system (1) is “in reduced form if and only if any differential module in a construction admits a constant basis”.

MSC:

12H05 Differential algebra
12G05 Galois cohomology
34M03 Linear ordinary differential equations and systems in the complex domain
34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms

Citations:

Zbl 1272.12016
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References:

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