Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components
Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components
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DOI：
10.1137/s0036142900372549
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发表时间：
2000-11
期刊：
影响因子：
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通讯作者：
A. Sommese;J. Verschelde;C. Wampler
中科院分区：
文献类型：
--
作者：
A. Sommese;J. Verschelde;C. Wampler
在工程和应用数学中，出现了多项式系统，其解集包含不同维度和多重性的分量。在这篇文章中，我们提出的算法，同伦延续的基础上，计算的解集的主要分解中所包含的几何信息。特别是，忽略多重性，我们的算法布局的解决方案的集合分解成不可约的组件，通过查找，在每个维度上，每个组件上的通用点。作为副产品，计算还确定了每个组件的程度和其多重性的上限。边界是尖锐的（即，等于1）的简化组件。该算法基本上使用通用的投影和插值，并且如果需要，可以精确地描述每个不可约分量作为有限数量的多项式的公共零点。
In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As by-products, the computation also determines the degree of each component and an upper bound on its multiplicity. The bound is sharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroes of a finite number of polynomials.