正维理想的对偶基的数值算法
Numerical Algorithms for Dual Bases of Positive-Dimensional Ideals
摘要：
局部多项式环的理想可以通过计算关于局部单项序的标准基来描述。然而，标准基算法在数值上并不稳定。相反，我们可以通过寻找使理想零化的对偶泛函空间来数值地描述理想，从而将问题简化为线性代数的问题。有几个已知的算法可以找到任意次数的截断对偶，这对描述零维理想很有用。我们提出了一个基于均匀化理论的正维情形的停止准则，该准则保证了初始单项理想的所有生成元都被找到，这对于计算Hilbert函数有一定的应用。
英文摘要：
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.
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