Concise tensors of minimal border rank

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 $1_{\ast }$-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 Buczyńska-Buczyński and results of Jelisiejew-Šivic 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 $C^5\otimes C^5\otimes C^5$.

arXiv commentv3, corrected a typo in statement of Theorem 1.8