Explicit tensors of border rank at least 2d−2 in K^d ⊗ K^d ⊗ K^d in arbitrary characteristic

ABSTRACT

For tensors in ${\mathbb{C}}^d\otimes {\mathbb{C}}^d\otimes {\mathbb{C}}^d$, Landsberg provides non-trivial equations for tensors of border rank 2d−3 for d even and 2d−5 for d odd in Landsberg [Non-triviality of equations and explicit tensors in ${\mathbb{C}}^m\otimes {\mathbb{C}}^m\otimes {\mathbb{C}}^m$ of border rank at least 2m−2. J Pure Appl Algebra. 2015;219(8):3677–3684]. In Derksen and Makam [On non-commutative rank and tensor rank, Linear Multilinear Algebra, published online 2017], we observe that Landsberg's method can be interpreted in the language of tensor blow-ups of matrix spaces, and using concavity of blow-ups, we improve the case for odd d from 2d−5 to 2d−4. The purpose of this paper is to show that the aforementioned results extend to tensors in $K^d\otimes K^d\otimes K^d$ for any field K.

KEYWORDS:

• Border rank
• lower bounds
• Koszul flattenings
• linear subspaces of matrices
• tensor blow-ups

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors were supported by Division of Mathematical Sciences NSF [grant number DMS-1601229].