文章基本信息

标题：A Column Space Based Approach to Solve Systems of Multivariate Polynomial Equations ⁎ ⁎
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作者：Christof Vermeersch ; Bart De Moor
期刊名称：IFAC PapersOnLine
印刷版ISSN：2405-8963
出版年度：2021
卷号：54
期号：9
页码：137-144
DOI：10.1016/j.ifacol.2021.06.144
语种：English
出版社：Elsevier
摘要：AbstractWe propose a novel approach to solve systems of multivariate polynomial equations, using the column space of the Macaulay matrix that is constructed from the coefficients of these polynomials. A multidimensional realization problem in the null space of the Macaulay matrix results in an eigenvalue problem, the eigenvalues and eigenvectors of which yield the common roots of the system. Since this null space based algorithm uses well-established numerical linear algebra tools, like the singular value and eigenvalue decomposition, it finds the solutions within machine precision. In this paper, on the other hand, we determine a complementary approach to solve systems of multivariate polynomial equations, which considers the column space of the Macaulay matrix instead of its null space. This approach works directly on the data in the Macaulay matrix, which is sparse and structured. We provide a numerical example to illustrate our new approach and to compare it with the existing null space based root-finding algorithm.
关键词：KeywordsMacaulay matrixmultivariate polynomialsnumerical algorithmsrealization theorymatrix algebra