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

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.

The art of algorithmic guessing in gfun

“…The famous method of creative telescoping propagated by Zeilberger [Zei91], and constantly improved in the last decades, allows for finding and proving linear recurrences/differential equations for sequences given as explicit sums or functions given as integrals; see for example [Chy14] for a great exposition of many achievements in this field and [CK17] for open problems. We also mention very recent works by Bostan, Rivoal, Salvy [BRS21] and by Barkatou, Cluzeau, Di Vizio, Weil [BCDVW20] which allow for practical proofs of transcendence and algebraicity of given D-functions. Finally, as already mentioned, P-recursive sequences and D-finite functions happen to form a great class for efficient guessing algorithms.…”

Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory

“…In general, the matrix of such a differential systems is not in so(3, C K ); then, the results of the previous section would not apply directly. However, using the constructive theory of reduced forms from [7,9], we may compute effectively a gauge transformation matrix P such that, letting Y = P Z, the new unknown Z satisfies a system Z ′ = −BZ with B ∈ so(3, K). Then the results of the previous sections apply: we may solve using solutions of second-order equations and construct families of equations of similar shapes via Darboux transformation.…”

Differential Galois Theory and Integration

“…where the block diagonal part is in reduced form (see [BCDVW20,BCWDV16] for this). We will first reduce the south-east part (which is of the same form as (1)) into a form 𝐴 2 0 𝑆 𝐴 3…”