Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 054, 13 pages      arXiv:1912.10567      https://doi.org/10.3842/SIGMA.2020.054

Reduced Forms of Linear Differential Systems and the Intrinsic Galois-Lie Algebra of Katz

Moulay Barkatou a, Thomas Cluzeau a, Lucia Di Vizio b and Jacques-Arthur Weil a
a) XLIM, UMR7252, Université de Limoges et CNRS, 123 avenue Albert Thomas, 87060 Limoges Cedex, France
b) Université Paris-Saclay, UVSQ, CNRS, Laboratoire de mathématiques de Versailles, 78000, Versailles, France

Received January 20, 2020, in final form June 04, 2020; Published online June 17, 2020

Abstract
Generalizing the main result of [Aparicio-Monforte A., Compoint E., Weil J.-A., J. Pure Appl. Algebra 217 (2013), 1504-1516], we prove that a linear differential system is in reduced form in the sense of Kolchin and Kovacic if and only if any differential module in an algebraic construction admits a constant basis. Then we derive an explicit version of this statement. We finally deduce some properties of the Lie algebra of Katz's intrinsic Galois group.

Key words: linear differential systems; differential Galois theory; Lie algebras; reduced forms.

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