Cactus barriers

TL;DR

This paper explains the geometric reasons behind the limitations of determinantal methods in bounding tensor border rank, using cactus varieties and scheme-theoretic algebraic geometry.

Contribution

It provides a geometric explanation for the known barriers in determinantal methods for tensor border rank bounds, connecting algebraic geometry with tensor complexity.

Findings

Identifies the geometric barrier using cactus varieties.

Explains the scheme-theoretic reasons for the bounds.

Connects algebraic geometry concepts with tensor rank limitations.

Abstract

Determinantal methods for bounding the rank and border rank of tensors or polynomials are subject to a major barrier. For instance, it is known that using determinantal methods one cannot prove a lower bound for the border rank of a 3-way tensor of size m in each direction that exceeds 6m-4. We explain the precise geometric reason for this number (and analogous bounds in more general tensor spaces) using cactus varieties and, more generally, scheme theoretic methods in algebraic geometry.