Concise Tensors of Minimal Border Rank

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 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 $ℂ_5⊗ℂ_5⊗ℂ_5$ .

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Recommended citation: Joachim Jelisiejew and Joseph M Landsberg & Arpan Pal. "Concise Tensors of Minimal Border Rank". Math. Ann. (2023).