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

TL;DR

This paper characterizes when a linear differential system is in reduced form using algebraic modules and explores properties of Katz's intrinsic Galois Lie algebra, providing explicit criteria and insights.

Contribution

It generalizes previous results by establishing an equivalence between reduced form and the existence of a constant basis in algebraic modules, with explicit criteria and properties of Katz's Galois algebra.

Findings

Characterization of reduced form via algebraic modules

Explicit criteria for reduced form in differential systems

Properties of Katz's intrinsic Galois Lie algebra

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.