PIDs上强格罗布纳基的基于符号的Möller算法
Signature-based Möller's algorithm for strong Gröbner bases over PIDs
摘要：
基于签名的算法是目前计算域上多项式系统的格罗布纳基的最新和最有效的方法。最近，这些技术的可能扩展到一般的环引起了一些作者的注意。在本文中，我们提出了一个基于签名的版本的Möller的经典变种的Buchberger的算法计算强格罗布纳基在主理想域（或PID）。它确保了签名在算法过程中不会减少，这使得可以应用经典的签名标准进行进一步优化。特别是，在F5准则下，穆勒算法的签名版本计算了一个格罗布纳基，而没有为一个由正则序列给出的多项式系统归零。我们还展示了如何Buchberger的链标准可以实现，以便与签名兼容。我们证明了算法的正确性和终止性。此外，我们在Magma中编写了一个玩具实现，允许我们定量地比较消除S对的各种标准的效率。
英文摘要：
Signature-based algorithms are the latest and most efficient approach as of today to compute Gröbner bases for polynomial systems over fields. Recently, possible extensions of these techniques to general rings have attracted the attention of several authors. In this paper, we present a signature-based version of Möller's classical variant of Buchberger's algorithm for computing strong Gröbner bases over Principal Ideal Domains (or PIDs). It ensures that the signatures do not decrease during the algorithm, which makes it possible to apply classical signature criteria for further optimization. In particular, with the F5 criterion, the signature version of Möller's algorithm computes a Gröbner basis without reductions to zero for a polynomial system given by a regular sequence. We also show how Buchberger's chain criterion can be implemented so as to be compatible with the signatures. We prove correctness and termination of the algorithm. Furthermore, we have written a toy implementation in Magma, allowing us to quantitatively compare the efficiency of the various criteria for eliminating S-pairs.
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