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Decomposition of polynomial sets into characteristic pairs


Authors: Dongming Wang, Rina Dong and Chenqi Mou
Journal: Math. Comp. 89 (2020), 1993-2015
MSC (2010): Primary 68W30; Secondary 13P10, 13P15
DOI: https://doi.org/10.1090/mcom/3504
Published electronically: January 7, 2020
Uncorrected version: Original version posted January 7, 2020
Corrected version: This version of the article replaces an earlier version, due to a publisher error
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Abstract: A characteristic pair is a pair $ (\mathcal {G}, \mathcal {C})$ of polynomial sets in which $ \mathcal {G}$ is a reduced lexicographic Gröbner basis and $ \mathcal {C}$ is the minimal triangular set contained in $ \mathcal {G}$. It is said to be normal (or strong normal) if $ \mathcal {C}$ is normal (or $ \mathcal {C}$ is normal and its saturated ideal equals the ideal generated by $ \mathcal {G}$). In this paper, we show that any finite polynomial set $ \mathcal {P}$ can be decomposed algorithmically into finitely many (strong) normal characteristic pairs with associated zero relations, which provide representations for the zero set of $ \mathcal {P}$ in terms of those of Gröbner bases and those of triangular sets. The algorithm we propose for the decomposition makes use of the inherent connection between Ritt characteristic sets and lexicographic Gröbner bases and is based essentially on the structural properties and the computation of lexicographic Gröbner bases. Several nice properties about the decomposition and the resulting (strong) normal characteristic pairs, in particular relationships between the Gröbner basis and the triangular set in each pair, are established. Examples are given to illustrate the algorithm and some of the properties.


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Additional Information

Dongming Wang
Affiliation: BDBC – LMIB – School of Mathematical Sciences, Beihang University, Beijing 100191, China; Centre National de la Recherche Scientifique, 75794 Paris cedex 16, France
Email: dongming.wang@lip6.fr

Rina Dong
Affiliation: LMIB – School of Mathematical Sciences, Beihang University, Beijing 100191, China
Email: rina.dong@buaa.edu.cn

Chenqi Mou
Affiliation: BDBC – LMIB – School of Mathematical Sciences, Beihang University, Beijing 100191, China
Email: chenqi.mou@buaa.edu.cn

DOI: https://doi.org/10.1090/mcom/3504
Keywords: Characteristic pair, normal triangular set, lexicographic Gr\"obner basis, zero decomposition
Received by editor(s): April 20, 2017
Received by editor(s) in revised form: October 21, 2019
Published electronically: January 7, 2020
Additional Notes: The third author is the corresponding author
This work was supported partially by the National Natural Science Foundation of China (NSFC 11771034 and 11971050).
Article copyright: © Copyright 2020 American Mathematical Society