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15 January 2014 Bounded-rank tensors are defined in bounded degree
Jan Draisma, Jochen Kuttler
Duke Math. J. 163(1): 35-63 (15 January 2014). DOI: 10.1215/00127094-2405170

Abstract

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.

Citation

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Jan Draisma. Jochen Kuttler. "Bounded-rank tensors are defined in bounded degree." Duke Math. J. 163 (1) 35 - 63, 15 January 2014. https://doi.org/10.1215/00127094-2405170

Information

Published: 15 January 2014
First available in Project Euclid: 8 January 2014

zbMATH: 1314.14109
MathSciNet: MR3161311
Digital Object Identifier: 10.1215/00127094-2405170

Subjects:
Primary: 14Q15
Secondary: 13E05, 15A69

Rights: Copyright © 2014 Duke University Press

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Vol.163 • No. 1 • 15 January 2014
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