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

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• 作者：Dongming Wang
• 关键词：Characteristic set ; Gröbner basis ; Irregularity structure ; Polynomial ideal ; Triangular decomposition
• 刊名：Mathematics in Computer Science
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：10
• 期：4
• 页码：479-492
• 全文大小：501 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Computer Science, general
  
• 出版者：Springer Basel
• ISSN：1661-8289
• 卷排序：10

文摘

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.