Numerical Algorithms for Dual Bases of Positive-Dimensional Ideals
Numerical Algorithms for Dual Bases of Positive-Dimensional Ideals
复制标题
正维理想双基的数值算法
DOI：
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发表时间：
2012
期刊：
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通讯作者：
Robert Krone
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作者：
Robert Krone
局部多项式环的理想可以通过计算关于局部单序的标准基来描述。然而，标准基算法在数值上并不稳定。相反，我们可以通过寻找湮灭它的对偶泛函的空间来数值描述理想，将问题简化为线性代数的问题。有几种已知的算法可以找到截断到任意指定度的对偶，这对于描述零维理想是有用的。我们提出了一个基于齐次化的正维情况的停止准则，保证找到所有初始单项理想的生成点。这在计算希尔伯特函数上有应用。
An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. However standard basis algorithms are not numerically stable. Instead we can describe the ideal numerically by finding the space of dual functionals that annihilate it, reducing the problem to one of linear algebra. There are several known algorithms for finding the truncated dual up to any specified degree, which is useful for describing zero-dimensional ideals. We present a stopping criterion for positive-dimensional cases based on homogenization that guarantees all generators of the initial monomial ideal are found. This has applications for calculating Hilbert functions.