## Duke Mathematical Journal

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

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

Jan Draisma and Jochen Kuttler

#### Abstract

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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/00127094-2405170

**Mathematical Reviews number (MathSciNet)**

MR3161311

**Zentralblatt MATH identifier**

1314.14109

**Subjects**

Primary: 14Q15: Higher-dimensional varieties

Secondary: 15A69: Multilinear algebra, tensor products 13E05: Noetherian rings and modules

#### 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