关于主理想整环上Buchberger算法的两个签名变体
On Two Signature Variants Of Buchberger's Algorithm Over Principal Ideal Domains
摘要：
基于特征的算法大大提高了域上多项式系统的格罗布纳基算法的性能。此外，它们产生额外的数据，这些数据可以用于计算理想的合冲模或计算输入生成元的系数。在本文中，我们研究了两个变种的Buchberger算法计算格罗布纳基上的主理想域，除了签名。第一个算法是从Kandri-Rody和Kapur的算法中改进而来的，而第二个算法则是利用了L。Pan（1989）和D. Lichtblau（2012年）。算法之间的构造差异必然导致与签名兼容的操作以及可用于丢弃元素的标准的差异。我们证明了这两种算法是正确的，并讨论了它们的相对性能在岩浆的原型实现。
英文摘要：
Signature-based algorithms have brought large improvements in the performances of Gröbner bases algorithms for polynomial systems over fields. Furthermore, they yield additional data which can be used, for example, to compute the module of syzygies of an ideal or to compute coefficients in terms of the input generators. In this paper, we examine two variants of Buchberger's algorithm to compute Gröbner bases over principal ideal domains, with the addition of signatures. The first one is adapted from Kandri-Rody and Kapur's algorithm, whereas the second one uses the ideas developed in the algorithms by L. Pan (1989) and D. Lichtblau (2012). The differences in constructions between the algorithms entail differences in the operations which are compatible with the signatures, and in the criteria which can be used to discard elements. We prove that both algorithms are correct and discuss their relative performances in a prototype implementation in Magma.
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