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 tensor...

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Bibliographic Details
Published in:Mathematische annalen 2024-01, Vol.388 (3), p.2473-2517
Main Authors: Jelisiejew, Joachim, Landsberg, J. M., Pal, Arpan
Format: Article
Language:English
Subjects:
Canonical forms
Classification
Complexity theory
Mathematical analysis
Mathematics
Mathematics and Statistics
Matrix algebra
Tensors
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Summary: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 .
ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-023-02569-y