基于特征的主理想整环上计算格罗布纳基的算法
A Signature-based Algorithm for computing Computing Gröbner Bases over Principal Ideal Domains
摘要：
基于签名的算法已经成为域上多项式系统的格罗布纳基计算的标准方法，但如何将这些技术扩展到一般环中的系数还没有很好地理解。在本文中，我们提出了一个概念证明的签名为基础的算法计算格罗布纳基交换整环上。它改编自Möller算法（1988）的一般版本，该算法考虑在每一步通过多个多项式进行约简。该算法使用非递减签名执行约简，并且特别地，不会发生签名丢弃。当系数是从一个主要的理想域（例如，环的整数或环的一元多项式在一个字段），我们证明了正确性和终止的算法，我们展示了如何使用签名属性来实现经典的基于签名的标准，以消除一些冗余的减少。特别地，如果输入是规则序列，则算法在没有任何减少到0的情况下操作。我们已经在Magma中编写了该算法的玩具实现。早期的实验结果表明，该算法甚至可能是正确的，并终止于一个更一般的设置，多项式在一个唯一的因式分解域（例如，环的多元多项式在一个字段或PID）。
英文摘要：
Signature-based algorithms have become a standard approach for Gröbner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this paper, we present a proof-of-concept signature-based algorithm for computing Gröbner bases over commutative integral domains. It is adapted from a general version of Möller's algorithm (1988) which considers reductions by multiple polynomials at each step. This algorithm performs reductions with non-decreasing signatures, and in particular, signature drops do not occur. When the coefficients are from a principal ideal domain (e.g. the ring of integers or the ring of univariate polynomials over a field), we prove correctness and termination of the algorithm, and we show how to use signature properties to implement classic signature-based criteria to eliminate some redundant reductions. In particular, if the input is a regular sequence, the algorithm operates without any reduction to 0. We have written a toy implementation of the algorithm in Magma. Early experimental results suggest that the algorithm might even be correct and terminate in a more general setting, for polynomials over a unique factorization domain (e.g. the ring of multivariate polynomials over a field or a PID).
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