代数数域上的Gröbner基
Gröbner Bases over Algebraic Number Fields
摘要：
虽然Buchberger算法在理论上允许我们计算任何域上的Gröbner基，但是在实践中，计算效率取决于基场的算法。考虑一个域$K = \mathbb{Q}(α)$，它是$\mathbb{Q}$的一个简单扩展，其中$α$是一个代数数，令$f \in \mathbb{Q}[t]$是$α$的最小多项式。本文提出了一种计算代数数域$K$上多项式环中Gröbner基的新方法.从Noro [Noro，2006]的想法出发，我们将$f$加入到要考虑的理想中，并添加$t$作为额外变量。但是，我们没有通过反转代数数来避免多余的S对约简，而是通过应用模块化方法来实现相同的目标[Arnold，2003; Böhm等人，2015; Idrees等人，2011]，即，通过从特性$p > 0$中的信息推断特性零中的信息。对于适当的素数$p$，极小多项式$f$在$\mathbb{F}_p$上是可约的。这使我们能够再次在第二个层次上对$f$的因素应用模块化方法。因此，该算法类似于分而治之的策略，特别是容易并行化。在当前状态下，该算法是概率的，因为对于其他模块化Gröbner基计算，有效的最终验证测试仅针对齐次理想或局部单项排序。所提出的时间表明，对于大多数的例子，我们的算法，它已被实现在SINGULAR，远远优于其他已知的方法。
英文摘要：
Although Buchberger's algorithm, in theory, allows us to compute Gröbner bases over any field, in practice, however, the computational efficiency depends on the arithmetic of the ground field. Consider a field $K = \mathbb{Q}(α)$, a simple extension of $\mathbb{Q}$, where $α$ is an algebraic number, and let $f \in \mathbb{Q}[t]$ be the minimal polynomial of $α$. In this paper we present a new efficient method to compute Gröbner bases in polynomial rings over the algebraic number field $K$. Starting from the ideas of Noro [Noro, 2006], we proceed by joining $f$ to the ideal to be considered, adding $t$ as an extra variable. But instead of avoiding superfluous S-pair reductions by inverting algebraic numbers, we achieve the same goal by applying modular methods as in [Arnold, 2003; Böhm et al., 2015; Idrees et al., 2011], that is, by inferring information in characteristic zero from information in characteristic $p > 0$. For suitable primes $p$, the minimal polynomial $f$ is reducible over $\mathbb{F}_p$. This allows us to apply modular methods once again, on a second level, with respect to the factors of $f$. The algorithm thus resembles a divide and conquer strategy and is in particular easily parallelizable. At current state, the algorithm is probabilistic in the sense that, as for other modular Gröbner basis computations, an effective final verification test is only known for homogeneous ideals or for local monomial orderings. The presented timings show that for most examples, our algorithm, which has been implemented in SINGULAR, outperforms other known methods by far.
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