Concise tensors of minimal border rank
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abstract
We determine defining equations for the set of concise tensors of minimal border rank in $C^motimes C^motimes C^m$ 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 $C^5otimes C^5otimes C^5$.
