## Duke Mathematical Journal

### Bounded-rank tensors are defined in bounded degree

#### Abstract

Matrices of rank at most $k$ are defined by the vanishing of polynomials of degree $k+1$ in their entries (namely, their $((k+1)\times(k+1))$-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 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

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

Dates
First available in Project Euclid: 8 January 2014

https://projecteuclid.org/euclid.dmj/1389190324

Digital Object Identifier
doi:10.1215/00127094-2405170

Mathematical Reviews number (MathSciNet)
MR3161311

Zentralblatt MATH identifier
1314.14109

#### Citation

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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