Nonlinear methods for tensors: determinantal equations for secant varieties beyond cactus

TL;DR

This paper introduces Kronecker-Koszul flattenings, a new family of tensor flattenings that generate explicit equations for secant varieties, providing new border rank criteria and a novel proof for matrix multiplication tensor rank.

Contribution

It develops a new class of flattenings called Kronecker-Koszul, deriving explicit polynomial equations for secant varieties and establishing border rank criteria.

Findings

Derived new border rank criteria using minors of flattenings.

First explicit equations for secant varieties that do not vanish on cactus varieties.

Provided a computer-free proof that the matrix multiplication tensor has border rank 7.

Abstract

We present a family of flattening methods of tensors which we call Kronecker-Koszul flattenings, generalizing the famous Koszul flattenings and further equations of secant varieties studied among others by Landsberg, Manivel, Ottaviani and Strassen. We establish new border rank criteria given by vanishing of minors of Kronecker-Koszul flattenings. We obtain the first explicit polynomial equations -- tangency flattenings -- vanishing on secant varieties of Segre variety, but not vanishing on cactus varieties. Additionally, our polynomials have simple determinantal expressions. As another application, we provide a new, computer-free proof that the border rank of the $2\times2$ matrix multiplication tensor is $7$.