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Generalized Loewy decomposition of D-modules

: Grigoriev, D.; Schwarz, F.


Association for Computing Machinery -ACM-:
ISSAC 2005, ACM-SIGSAM International Symposium on Symbolic and Algebraic Computation. Proceedings : Beijing, China, July 24-27, 2005
New York: ACM Press, 2005
ISBN: 1-59593-095-7
International Symposium on Symbolic and Algebraic Computation (ISSAC) <2005, Beijing>
Fraunhofer SCAI ()
partial differential equation; D-module; Computer-Algebra

Starting from the well-known factorization of linear ordinary differential equations, we define the generalized Loewy decomposition for a ${\\cal D}$-module. To this end, for any module $I$, overmodules $J\\supseteq I$ are constructed. They subsume the conventional factorization as special cases. Furthermore, the new concept of the module of relative syzygies $Syz(I,J)$ is introduced. The invariance of this module and its solution space w.r.t. the set of generators is shown. We design an algorithm which constructs the Loewy-decomposition for finite-dimensional and some kinds of general ${\\cal D}$-modules. These results are applied for solving various second- and third-order linear partial differential equations.