Punctual Hilbert scheme and certified approximate singularities
Punctual Hilbert scheme and certified approximate singularities
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DOI：
10.1145/3373207.3404024
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发表时间：
2020-02
期刊：
影响因子：
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通讯作者：
Angelos Mantzaflaris;B. Mourrain;Á. Szántó
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文献类型：
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作者：
Angelos Mantzaflaris;B. Mourrain;Á. Szántó
本文给出了一种新的方法来证明一个邻近多项式系统有一个奇异孤立根，并计算了它的重数结构。更准确地说，给定多项式系统f =（f1，...，fN）∈ C[x1，.，xn]N，我们提出了一个扩展的收缩系统的牛顿迭代，在正则性条件下，局部收敛到f的一个小变形，使得这个变形系统有一个精确的奇异根.迭代同时收敛到奇异根的坐标和描述根处多重结构的所谓逆系统的系数。利用α-理论检验证明了算法的二次收敛性，并给出了变形大小和逼近误差的界.该方法依赖于分析的准时希尔伯特计划，我们提供了一个新的描述。我们特别表明，它的一些地层可以合理地参数化，并利用这些参数化的认证。在数值实验中，我们展示了如何近似逆系统可以计算作为牛顿迭代的起点和快速数值收敛到奇异根与其多重性结构，我们的标准认证。
In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root and we compute its multiplicity structure. More precisely, given a polynomial system f = (f1, ..., fN) ∈ C[x1, ..., xn]N, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of f such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so-called inverse system that describes the multiplicity structure at the root. We use α-theory test to certify the quadratic convergence, and to give bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria.