Explicit tensors of border rank at least $2d-2$ in $K^d \otimes K^d \otimes K^d$ in arbitrary characteristic

TL;DR

This paper extends Landsberg's border rank equations for tensors in three-dimensional spaces to arbitrary fields, improving bounds for odd dimensions using tensor blow-up techniques and concavity properties.

Contribution

It generalizes border rank equations from complex tensors to tensors over any field, enhancing bounds for odd dimensions via tensor blow-up methods.

Findings

Extended border rank equations to arbitrary fields.

Improved bounds for odd dimensions from 2d-5 to 2d-4.

Applied tensor blow-up techniques to generalize results.

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 were found by Landsberg. In previous work, 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 $K$.