Bounded-rank tensors are defined in bounded degree
Bounded-rank tensors are defined in bounded degree
复制标题
有界秩张量以有界度定义
DOI：
10.1215/00127094-2405170
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发表时间：
2011
影响因子：
2.5
通讯作者：
J. Kuttler
中科院分区：
文献类型：
--
作者：
J. Draisma;J. Kuttler
秩至多为k的矩阵定义为k+1次多项式在其元素（即其（（k+1）×（k+1））-子行列式）中消失，而与矩阵的大小无关。我们证明了一个定性的类似的任意维张量，其中矩阵对应于二维张量。更具体地说，我们证明了，对于每个k存在一个上限d=d（k），使得边界秩最多为k的张量由次数最多为d的多项式的消失来定义，而不管张量的维数和它在每个维度上的大小。我们的证明涉及传递到向量空间的张量幂的无限维极限，其元素我们称之为无限维张量，并以关键的方式利用这一极限的对称性。
Matrices of rank at most k are defined by the vanishing of polynomials of degree k+1 in their entries (namely, their ((k+1)×(k+1))-subdeterminants), regardless of the size of the matrix. We prove a qualitative analogue of this statement for tensors of arbitrary dimension, where matrices correspond to two-dimensional tensors. More specifically, we prove that for each k there exists an upper bound d=d(k) such that tensors of border rank at most k are defined by the vanishing of polynomials of degree at most d, regardless of the dimension of the tensor and regardless of its size in each dimension. Our proof involves passing to an infinite-dimensional limit of tensor powers of a vector space, whose elements we dub infinite-dimensional tensors, and exploiting the symmetries of this limit in crucial ways.