On Characteristic Decomposition and Quasi-characteristic Decomposition

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11661)


In this paper, the concepts of quasi-characteristic pair and quasi-characteristic decomposition are introduced. The former is a pair \((\mathcal {G}, \mathcal {C})\) of a reduced lexicographic Gröbner basis \(\mathcal {G}\) and the W-characteristic set \(\mathcal {C}\) which is regular and extracted from \(\mathcal {G}\); the latter is the decomposition of a polynomial set into finitely many quasi-characteristic pairs with associated zero relations. We show that the quasi-characteristic decomposition of any polynomial set can be obtained algorithmically regardless of the variable ordering condition. A new algorithm is presented for computing characteristic decomposition when the variable ordering condition is always satisfied, otherwise it degenerates to compute the quasi one. Some properties of quasi-characteristic pairs and decomposition are proved, and examples are given to illustrate the algorithm.


Quasi-characteristic decomposition Quasi-characteristic pair W-characteristic set Regular set Gröbner basis 



The authors would like to thank the anonymous reviewers for their detailed and helpful comments on an earlier version of this paper.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.LMIB–School of Mathematics and Systems Science, Beihang UniversityBeijingChina
  2. 2.Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang UniversityBeijingChina

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