可对角化的高次形式与对称张量
Diagonalizable Higher Degree Forms and Symmetric Tensors
摘要：
我们提供了简单的标准和算法，用于将齐次多项式表示为独立线性型的幂和，或者等效地，用于将对称张量分解为线性无关向量的1阶对称张量的和。该准则依赖于两个方面的高次形式，即哈里森的代数理论和一些代数几何性质。所提出的算法是基本的，纯粹基于求解线性和二次方程。此外，作为我们的标准和算法的副产品，人们可以很容易地决定是否齐次多项式或对称张量是正交或酉可分解。
英文摘要：
We provide simple criteria and algorithms for expressing homogeneous polynomials as sums of powers of independent linear forms, or equivalently, for decomposing symmetric tensors into sums of rank-1 symmetric tensors of linearly independent vectors. The criteria rely on two facets of higher degree forms, namely Harrison's algebraic theory and some algebro-geometric properties. The proposed algorithms are elementary and based purely on solving linear and quadratic equations. Moreover, as a byproduct of our criteria and algorithms one can easily decide whether or not a homogeneous polynomial or symmetric tensor is orthogonally or unitarily decomposable.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
