A proof of the set-theoretic version of the salmon conjecture

TL;DR

This paper proves that the set of 4x4x4 tensors with border rank at most 4 can be characterized exactly by specific polynomial equations of degrees 5, 6, and 9, confirming a set-theoretic version of the Salmon conjecture.

Contribution

It establishes the defining polynomial equations for the border rank 4 tensors in the 4x4x4 case, confirming the set-theoretic version of the Salmon conjecture.

Findings

The variety of border rank at most 4 tensors is defined by degree 5, 6, and 9 polynomials.

The polynomial equations are the Strassen commutative conditions, Landsberg-Manivel polynomials, and symmetrization conditions.

This result confirms the set-theoretic version of the Salmon conjecture for 4x4x4 tensors.

Abstract

We show that the irreducible variety of 4 x 4 x 4 complex valued tensors of border rank at most 4 is the zero set of polynomial equations of degree 5 (the Strassen commutative conditions), of degree 6 (the Landsberg-Manivel polynomials), and of degree 9 (the symmetrization conditions).