Location of Repository

Matrices of rank at most k are defined by the vanishing of polynomials of degree k + 1 in their entries (namely, their (k + 1)-times-(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 sizes 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 way.Comment: Corrected the formulation of Corollary 1.4---thanks to Sonja Petrovic

Topics:
Mathematics - Algebraic Geometry, 14Q15, 15A69, 13E05

Year: 2011

DOI identifier: 10.1215/00127094-2405170

OAI identifier:
oai:arXiv.org:1103.5336

Provided by:
arXiv.org e-Print Archive

Download PDF:To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.