Computing strong regular characteristic pairs with Gröbner bases
Computing strong regular characteristic pairs with Gröbner bases
复制标题
使用 Gröbner 基计算强正则特征对
DOI：
10.1016/j.jsc.2020.06.012
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发表时间：
2021
影响因子：
0.7
通讯作者：
Dongming Wang
中科院分区：
文献类型：
--
作者：
Rina Dong;Dongming Wang
多项式理想的 W 特征集是包含在理想的简化词典编纂 Gröbner 基中的最小三角形集。如果 G 是简化字典序 Gröbner 基，C 是理想 < G> 的 W 特征集，C 的饱和理想 sat (C) 等于 < G>，且 C 是正则，则一对 (G, C) 多项式集是强正则特征对。在本文中，我们证明，对于给定生成器的任何多项式理想 I，我们可以检测 I 是单位，或者通过计算 Gröbner 基构造强正则特征对 (G, C)，使得 I⊆ sat (C)=< G> 和 sat (C) 除 I，因此理想 I 可以分为饱和理想 sat (C) 和商理想 I: sat (C)。基于这种商分裂策略以及 Gröbner 基和理想计算，我们设计了一种简单的算法，将任意多项式集 F 分解为有限多个强正则特征对，从中获得 F 的零点的两种表示：一种是强正则 Gröbner 基，另一种是正则三角形集。我们提出了强正则特征对和特征分解的一些性质，并通过示例和实验结果说明了所提出的算法及其性能。
The W-characteristic set of a polynomial ideal is the minimal triangular set contained in the reduced lexicographical Gröbner basis of the ideal. A pair (G, C) of polynomial sets is a strong regular characteristic pair if G is a reduced lexicographical Gröbner basis, C is the W-characteristic set of the ideal< G>, the saturated ideal sat (C) of C is equal to< G>, and C is regular. In this paper, we show that for any polynomial ideal I with given generators one can either detect that I is unit, or construct a strong regular characteristic pair (G, C) by computing Gröbner bases such that I⊆ sat (C)=< G> and sat (C) divides I, so the ideal I can be split into the saturated ideal sat (C) and the quotient ideal I: sat (C). Based on this strategy of splitting by means of quotient and with Gröbner basis and ideal computations, we devise a simple algorithm to decompose an arbitrary polynomial set F into finitely many strong regular characteristic pairs, from which two representations for the zeros of F are obtained: one in terms of strong regular Gröbner bases and the other in terms of regular triangular sets. We present some properties about strong regular characteristic pairs and characteristic decomposition and illustrate the proposed algorithm and its performance by examples and experimental results.