簇的近似根：使用高斯消元或SVD的全局方法
Approximate radical for clusters: a global approach using Gaussian elimination or SVD
AI总结：
本文提出一种基于Dickson引理的全局方法，通过计算迹矩阵的数值零空间来求零维理想簇的近似根，保证每个簇内恰有一个根，误差界为epsilon^2。
摘要：
我们提出一种基于Dickson引理的方法，用于计算C[x1, . . ., xm]中具有零簇的零维理想I的“近似根”：对于足够小的簇，近似根理想在每个簇中恰好有一个根。我们的方法是“全局”的，因为它不需要任何局部近似：它将问题简化为计算所谓的“迹矩阵”的数值零空间，该矩阵可由I的生成多项式计算。为了计算迹矩阵的数值零空间，我们建议使用带主元的高斯消元或奇异值分解。我们证明，如果I有k个不同的零簇，每个簇在最大范数下的半径至多为epsilon，则对迹矩阵进行k步高斯消元会得到一个子矩阵，其所有条目渐近地以epsilon^2为界。我们还证明，迹矩阵的第(k+1)个奇异值是O(epsilon^2)。得到的近似根在每个簇中有一个根，其坐标为簇的算术平均，误差项渐近地以epsilon^2为界。在单变量情况下，我们的方法为已知的近似无平方因子分解算法提供了一种更简单且精度更易理解的替代方案。
英文摘要：
We present a method based on Dickson's lemma to compute the "approximate radical" of a zero dimensional ideal I in C[x1, . . ., xm] which has zero clusters: the approximate radical ideal has exactly one root in each cluster for sufficiently small clusters. Our method is "global" in the sense that it does not require any local approximation of the zero clusters: it reduces the problem to the computation of the numerical nullspace of the so called "matrix of traces", a matrix computable from the generating polynomials of I. To compute the numerical nullspace of the matrix of traces we propose to use Gaussian elimination with pivoting or singular value decomposition. We prove that if I has k distinct zero clusters each of radius at most epsilon in the max-norm, then k steps of Gauss elimination on the matrix of traces yields a submatrix with all entries asymptotically bounded by epsilon^2. We also show that the (k+1)-th singular value of the matrix of traces is O(epsilon^2). The resulting approximate radical has one root in each cluster with coordinates which are the arithmetic mean of the cluster, up to an error term asymptotically bounded by epsilon^2. In the univariate case our method gives an alternative to known approximate square-free factorization algorithms which is simpler and its accuracy is better understood.
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