SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 2, Page 883-912, January 2021.
We propose a multilinear algebra framework to solve systems of polynomial equations with simple roots. We translate connections between univariate polynomial root-finding, eigenvalue decompositions, and harmonic retrieval to their higher-order counterparts: a canonical polyadic decomposition (CPD) that exploits shift invariance structures in the null space of the Macaulay matrix reveals the roots of the polynomial system. The new framework allows us to use numerical CPD algorithms for solving systems of polynomial equations. For the same degree of the Macaulay matrix as in numerical polynomial algebra/polynomial numerical linear algebra, the CPD is interpreted as the joint eigenvalue decomposition of the multiplication tables. In our approach the degree can also be lower. Affine roots and roots at infinity can be handled in the same way. With minor modifications, the technique can be used to estimate approximate roots of overconstrained systems.

中文翻译：

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多项式方程组、高阶张量分解和多维谐波检索：统一框架。第一部分：规范的多元分解

SIAM 矩阵分析与应用杂志，第 42 卷，第 2 期，第 883-912 页，2021 年 1 月。
我们提出了一个多线性代数框架来求解具有简单根的多项式方程组。我们将单变量多项式求根、特征值分解和调和检索之间的联系转化为它们的高阶对应项：利用Macaulay 矩阵零空间中的移位不变性结构的规范多元分解（CPD）揭示了多项式系统的根. 新框架允许我们使用数值 CPD 算法来求解多项式方程组。对于与数值多项式代数/多项式数值线性代数中相同阶的麦考莱矩阵，CPD 被解释为乘法表的联合特征值分解。在我们的方法中，度数也可以更低。仿射根和无穷远根可以用同样的方式处理。