Duke Mathematical Journal

Bounded-rank tensors are defined in bounded degree

Jan Draisma and Jochen Kuttler

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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.

Article information

Duke Math. J., Volume 163, Number 1 (2014), 35-63.

First available in Project Euclid: 8 January 2014

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Zentralblatt MATH identifier

Primary: 14Q15: Higher-dimensional varieties
Secondary: 15A69: Multilinear algebra, tensor products 13E05: Noetherian rings and modules


Draisma, Jan; Kuttler, Jochen. Bounded-rank tensors are defined in bounded degree. Duke Math. J. 163 (2014), no. 1, 35--63. doi:10.1215/00127094-2405170. https://projecteuclid.org/euclid.dmj/1389190324

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