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Article

Multiple Factorizations of Bivariate Linear Partial Differential Operators

Author:

Ekaterina ShemyakovaAuthors Info & Claims

CASC '09: Proceedings of the 11th International Workshop on Computer Algebra in Scientific Computing

Pages 299 - 309

https://doi.org/10.1007/978-3-642-04103-7_26

Published: 01 October 2009 Publication History

Abstract

We study the case when a bivariate Linear Partial Differential Operator (LPDO) of orders three or four has several different factorizations.

We prove that a third-order bivariate LPDO has a first-order left and right factors such that their symbols are co-prime if and only if the operator has a factorization into three factors, the left one of which is exactly the initial left factor, and the right one is exactly the initial right factor. We show that the condition that the symbols of the initial left and right factors are co-prime is essential, and that the analogous statement "as it is" is not true for LPDOs of order four.

Then we consider completely reducible LPDOs, which are defined as an intersection of principal ideals. Such operators may also be required to have several different factorizations. Considering all possible cases, we ruled out some of them from the consideration due to the first result of the paper. The explicit formulae for the sufficient conditions for the complete reducibility of an LPDO were found also.

References

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Cited By

Giesbrecht MHeinle ALevandovskyy V(2016)Factoring linear partial differential operators in n variablesJournal of Symbolic Computation10.1016/j.jsc.2015.11.01175:C(127-148)Online publication date: 1-Jul-2016
https://dl.acm.org/doi/10.1016/j.jsc.2015.11.011
Giesbrecht MHeinle ALevandovskyy VNagasaka KWinkler FSzanto A(2014)Factoring linear differential operators in n variablesProceedings of the 39th International Symposium on Symbolic and Algebraic Computation10.1145/2608628.2608667(194-201)Online publication date: 23-Jul-2014
https://dl.acm.org/doi/10.1145/2608628.2608667

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Information

Published In

cover image Guide Proceedings

CASC '09: Proceedings of the 11th International Workshop on Computer Algebra in Scientific Computing

October 2009

391 pages

ISBN:9783642041020

Editors:
Vladimir P. Gerdt
Joint Institute for Nuclear Research, Laboratory of Information Technologies, Dubna, Russia
,
Ernst W. Mayr
Technische Universität München`, Institut für Informatik, Garching, Germany
,
Evgenii V. Vorozhtsov
Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, Russia

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 October 2009

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Cited By

Giesbrecht MHeinle ALevandovskyy V(2016)Factoring linear partial differential operators in n variablesJournal of Symbolic Computation10.1016/j.jsc.2015.11.01175:C(127-148)Online publication date: 1-Jul-2016
https://dl.acm.org/doi/10.1016/j.jsc.2015.11.011
Giesbrecht MHeinle ALevandovskyy VNagasaka KWinkler FSzanto A(2014)Factoring linear differential operators in n variablesProceedings of the 39th International Symposium on Symbolic and Algebraic Computation10.1145/2608628.2608667(194-201)Online publication date: 23-Jul-2014
https://dl.acm.org/doi/10.1145/2608628.2608667

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