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Toric Eigenvalue Methods for Solving Sparse Polynomial Systems
arXiv - CS - Symbolic Computation Pub Date : 2020-06-18 , DOI: arxiv-2006.10654
Mat\'ias R. Bender, Simon Telen

We consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact toric variety $X$. Our starting point is a homogeneous ideal $I$ in the Cox ring of $X$, which gives a global description of this subscheme. It was recently shown that eigenvalue methods for solving this problem lead to robust numerical algorithms for solving (nearly) degenerate sparse polynomial systems. In this work, we give a first description of this strategy for non-reduced, zero-dimensional subschemes of $X$. That is, we allow isolated points with arbitrary multiplicities. Additionally, we investigate the regularity of $I$ to provide the first universal complexity bounds for the approach, as well as sharper bounds for weighted homogeneous, multihomogeneous and unmixed sparse systems, among others. We disprove a recent conjecture regarding the regularity and prove an alternative version. Our contributions are illustrated by several examples.

中文翻译:

求解稀疏多项式系统的复曲面特征值方法

我们考虑计算紧凑复曲面变量 $X$ 的零维子方案中点的齐次坐标的问题。我们的起点是 $X$ 的 Cox 环中的齐次理想 $I$,它给出了这个子方案的全局描述。最近表明,用于解决这个问题的特征值方法导致用于解决(几乎)退化稀疏多项式系统的稳健数值算法。在这项工作中,我们首先描述了这种针对 $X$ 的非简化零维子方案的策略。也就是说,我们允许具有任意多重性的孤立点。此外,我们研究了 $I$ 的规律性,以提供该方法的第一个通用复杂性界限,以及为加权同构、多同构和非混合稀疏系统等提供更清晰的界限。我们反驳了最近关于规律性的猜想,并证明了一个替代版本。几个例子说明了我们的贡献。
更新日期:2020-06-19
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