Concise tensors of minimal border rank

05/11/2022

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by   Joachim Jelisiejew, et al.

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We determine defining equations for the set of concise tensors of minimal border rank in C^m⊗ C^m⊗ 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 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⊗ C^5⊗ C^5.