Bilinear Systems with Two Supports: Koszul Resultant Matrices, Eigenvalues, and Eigenvectors
Bilinear Systems with Two Supports: Koszul Resultant Matrices, Eigenvalues, and Eigenvectors
复制标题
具有两个支持的双线性系统：Koszul 合成矩阵、特征值和特征向量
DOI：
10.1145/3208976.3209011
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Elias P. Tsigaridas
中科院分区：
文献类型：
--
作者：
M. Bender;J. Faugère;Angelos Mantzaflaris;Elias P. Tsigaridas
计算代数几何中的一个基本问题是结果的计算。一个核心问题是何时以及如何将其计算为矩阵的行列式，该矩阵的元素是输入多项式直至符号的系数。对于非混合多齐次系统，即由具有相同支持的多齐次多项式组成的系统，这个问题是很好理解的。然而，对于混合系统，即由具有不同支持的多项式组成的系统，人们知之甚少。我们考虑涉及两个不同支撑的双线性系统的多齐次合力的计算。我们提出了一种构造性方法，将结果表示为 Koszul 合成矩阵的精确行列式，该矩阵是由 Koszul 复合体中的映射构造的矩阵。 ％我们利用所得矩阵提出一种算法来解决此类系统。在此过程中，我们将经典特征值和特征向量准则扩展到更通用的设置。我们对特征值准则的扩展适用于一般类型的矩阵，包括 Sylvester 型和 Koszul 型矩阵。
A fundamental problem in computational algebraic geometry is the computation of the resultant. A central question is when and how to compute it as the determinant of a matrix whose elements are the coefficients of the input polynomials up-to sign. This problem is well understood for unmixed multihomogeneous systems, that is for systems consisting of multihomogeneous polynomials with the same support. However, little is known for mixed systems, that is for systems consisting of polynomials with different supports. We consider the computation of the multihomogeneous resultant of bilinear systems involving two different supports. We present a constructive approach that expresses the resultant as the exact determinant of a Koszul resultant matrix, that is a matrix constructed from maps in the Koszul complex. % We exploit the resultant matrix to propose an algorithm to solve such systems. In the process we extend the classical eigenvalues and eigenvectors criterion to a more general setting. Our extension of the eigenvalues criterion applies to a general class of matrices, including the Sylvester-type and the Koszul-type ones.
DOI：
10.1007/978-3-662-43414-7_18
发表时间：
2013-08
期刊：
--
影响因子：
--
作者：
A. Joux
通讯作者：
A. Joux