Mathematics in Computer Science

, Volume 10, Issue 4, pp 479–492 | Cite as

On the Connection Between Ritt Characteristic Sets and Buchberger–Gröbner Bases

Article

Abstract

For any polynomial ideal \(\mathcal {I}\), let the minimal triangular set contained in the reduced Buchberger–Gröbner basis of \(\mathcal {I}\) with respect to the purely lexicographical term order be called the W-characteristic set of \(\mathcal {I}\). In this paper, we establish a strong connection between Ritt’s characteristic sets and Buchberger’s Gröbner bases of polynomial ideals by showing that the W-characteristic set \(\mathbb {C}\) of \(\mathcal {I}\) is a Ritt characteristic set of \(\mathcal {I}\) whenever \(\mathbb {C}\) is an ascending set, and a Ritt characteristic set of \(\mathcal {I}\) can always be computed from \(\mathbb {C}\) with simple pseudo-division when \(\mathbb {C}\) is regular. We also prove that under certain variable ordering, either the W-characteristic set of \(\mathcal {I}\) is normal, or irregularity occurs for the jth, but not the \((j+1)\)th, elimination ideal of \(\mathcal {I}\) for some j. In the latter case, we provide explicit pseudo-divisibility relations, which lead to nontrivial factorizations of certain polynomials in the Buchberger–Gröbner basis and thus reveal the structure of such polynomials. The pseudo-divisibility relations may be used to devise an algorithm to decompose arbitrary polynomial sets into normal triangular sets based on Buchberger–Gröbner bases computation.

Keywords

Characteristic set Gröbner basis Irregularity structure Polynomial ideal Triangular decomposition 

Mathematics Subject Classification

13P10 13P15 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.LMIB, SKLSDE, School of Mathematics and Systems ScienceBeihang UniversityBeijingChina
  2. 2.HCIC, SMS International, Guangxi University for NationalitiesNanningChina
  3. 3.LIP6, Centre National de la Recherche ScientifiqueParis Cedex 16France

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