Concise tensors of minimal border rank

TL;DR

This paper characterizes concise tensors of minimal border rank in three-way tensor spaces for specific dimensions, using algebraic invariants and recent equations, advancing the understanding of tensor complexity.

Contribution

It introduces the 111-algebra invariant, solves defining equations for minimal border rank tensors at dimension 5, and classifies wild tensors in C^5⊗C^5⊗C^5.

Findings

Determined equations for minimal border rank tensors in C^5⊗C^5⊗C^5.

Introduced the 111-algebra invariant for concise tensors.

Classified wild minimal border rank tensors in C^5⊗C^5⊗C^5.

Abstract

We determine defining equations for the set of concise tensors of minimal border rank in $C^m\otimes C^m\otimes C^m$ when $m=5$ and the set of concise minimal border rank $1_*$-generic tensors when $m=5,6$. We solve this classical problem in algebraic complexity theory with the aid of two recent developments: the 111-equations defined by Buczy\'{n}ska-Buczy\'{n}ski and results of Jelisiejew-\v{S}ivic on the variety of commuting matrices. We introduce a new algebraic invariant of a concise tensor, its 111-algebra, and exploit it to give a strengthening of Friedland's normal form for $1$-degenerate tensors satisfying Strassen's equations. We use the 111-algebra to characterize wild minimal border rank tensors and classify them in $C^5\otimes C^5\otimes C^5$.