Aufsatz
Complementary decompositions of monomial ideals and involutive bases
Abstract
Complementary decompositions of monomial ideals - also known as Stanley decompositions - play an important role in many places in commutative algebra. In this article, we discuss and compare several algorithms for their computation. This includes a classical recursive one, an algorithm already proposed by Janet and a construction proposed by Hironaka in his work on idealistic exponents. We relate Janet’s algorithm to the Janet tree of the Janet basis and extend this idea to Janet-like bases to obtain an optimised algorithm. We show that Hironaka’s construction terminates, if and only if the monomial ideal is quasi-stable. Furthermore, we show that in this case the algorithm of Janet determines the same decomposition more efficiently. Finally, we briefly discuss how these results can be used for the computation of primary and irreducible decompositions.
Citation
In: Applicable Algebra in Engineering, Communication and Computing Volume 33 / issue 6 (2022-06-25) , S. 791-821 ; eissn:1432-0622Sponsorship
Gefördert im Rahmen des Projekts DEALCitation
@article{doi:10.17170/kobra-202301047293,
author={Hashemi, Amir and Orth, Matthias and Seiler, Werner M.},
title={Complementary decompositions of monomial ideals and involutive bases},
journal={Applicable Algebra in Engineering, Communication and Computing},
year={2022}
}
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2023-01-04T10:00:54Z 2023-01-04T10:00:54Z 2022-06-25 doi:10.17170/kobra-202301047293 http://hdl.handle.net/123456789/14326 Gefördert im Rahmen des Projekts DEAL eng Namensnennung 4.0 International http://creativecommons.org/licenses/by/4.0/ Monomial ideals Combinatorial decompositions Involutive bases Quasi-stable ideals Primary decompositions 510 Complementary decompositions of monomial ideals and involutive bases Aufsatz Complementary decompositions of monomial ideals - also known as Stanley decompositions - play an important role in many places in commutative algebra. In this article, we discuss and compare several algorithms for their computation. This includes a classical recursive one, an algorithm already proposed by Janet and a construction proposed by Hironaka in his work on idealistic exponents. We relate Janet’s algorithm to the Janet tree of the Janet basis and extend this idea to Janet-like bases to obtain an optimised algorithm. We show that Hironaka’s construction terminates, if and only if the monomial ideal is quasi-stable. Furthermore, we show that in this case the algorithm of Janet determines the same decomposition more efficiently. Finally, we briefly discuss how these results can be used for the computation of primary and irreducible decompositions. open access Hashemi, Amir Orth, Matthias Seiler, Werner M. doi:10.1007/s00200-022-00569-0 05E40 13P10 Monomiales Ideal Zerlegung <Mathematik> Kommutative Algebra publishedVersion eissn:1432-0622 issue 6 Applicable Algebra in Engineering, Communication and Computing 791-821 Volume 33 false
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