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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 \({\mathbb {C}}^m{\mathord { \otimes } }{\mathbb {C}}^m{\mathord { \otimes } }{\mathbb {C}}^m\) 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 \({\mathbb {C}}^5{\mathord { \otimes } }{\mathbb {C}}^5{\mathord { \otimes } }{\mathbb {C}}^5\).

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Notes

  1. After this paper was submitted, A. Conca pointed out an explicit example of a 111-abundant, not 111-sharp tensor when \(m=9\). We do not know if such exist when \(m=6,7,8\). The example is a generalization of Example 4.6.

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Authors and Affiliations

  1. Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097, Warsaw, Poland

    Joachim Jelisiejew

  2. Department of Mathematics, Texas A &M University, College Station, TX, 77843-3368, USA

    J. M. Landsberg & Arpan Pal

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  1. Joachim Jelisiejew
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  2. J. M. Landsberg
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Correspondence to Joachim Jelisiejew.

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Landsberg supported by NSF grants AF-1814254 and AF-2203618. Jelisiejew supported by National Science Centre grant 2018/31/B/ST1/02857.

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Jelisiejew, J., Landsberg, J.M. & Pal, A. Concise tensors of minimal border rank. Math. Ann. (2023). https://doi.org/10.1007/s00208-023-02569-y

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