On the Factorization of Differential Modules

Abstract

Abstract. Differential modules are modules over rings of linear (partial) differential operators which are finite-dimensional vector spaces. We present a generalization of the Beke-Schlesinger algorithm that factors differential modules. The method requires solving only one set of associated equations for each degree d of a potential factor.

Gorenstein theory for n-th differential modules

“…Zhou and Winklerb [18] introduce a series of algorithms to construct Gröbner bases for a class of difference-differential modules. Wu in [16] generalizes the Beke-Schlesinger algorithm that factors differential modules. The authors in [1] establish lower bounds on the class-a substitute for the length of a free complex, and on the rank of a differential module in terms of invariants of its homology.…”

A Jordan–Hölder Theorem for differential algebraic groups

“…Let (k, δ) be a differential field of characteristic zero with derivation δ and let k[δ] be the ring of linear differential operators with coefficients in k. It has been known for a long time (see [40] for a brief history and [2], [34], [46] for more recent algorithmic results) that one has a form of unique factorization for this ring:…”

Factoring Partial Differential Systems in Positive Characteristic

“…The problem of factoring D-finite partial differential systems in characteristic zero has been recently studied by Li, Schwarz and Tsarev in [21,22] (see also [28]). Ïn these articles, the authors show how to adapt Beke's algorithm (which factors ordinary differential systems, see [9] or [26, 4.2.1] and references therein) to the partial differential case.…”