Concise tensors of minimal border rank

Abstract

We determine defining equations for the set of concise tensors of minimal border rank in $${\mathbb {C}}^m{\mathord { \otimes } }{\mathbb {C}}^m{\mathord { \otimes } }{\mathbb {C}}^m$$ when $$m=5$$ and the set of concise minimal border rank $$1_*$$ -generic tensors when $$m=5,6$$ . We solve the classical problem in algebraic complexity theory of classifying minimal border rank tensors in the special case $$m=5$$ . Our proofs utilize 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 $${\mathbb {C}}^5{\mathord { \otimes } }{\mathbb {C}}^5{\mathord { \otimes } }{\mathbb {C}}^5$$ .