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The next gap in the subrank of 3-tensors

Fulvio Gesmundo, Jeroen Zuiddam

Abstract

Recent works of Costa-Dalai, Christandl-Gesmundo-Zuiddam, Blatter-Draisma-Rupniewski, and Briët-Christandl-Leigh-Shpilka-Zuiddam have investigated notions of discreteness and gaps in the possible values that asymptotic tensor ranks can take. In particular, it was shown that the asymptotic subrank and asymptotic slice rank of any nonzero 3-tensor is equal to 1, equal to 1.88, or at least 2 (over any field), and that the set of possible values of these parameters is discrete (in several regimes). We determine exactly the next gap, showing that the asymptotic subrank and asymptotic slice rank of any nonzero 3-tensor is equal to 1, equal to 1.88, equal to 2, or at least 2.68.

The next gap in the subrank of 3-tensors

Abstract

Recent works of Costa-Dalai, Christandl-Gesmundo-Zuiddam, Blatter-Draisma-Rupniewski, and Briët-Christandl-Leigh-Shpilka-Zuiddam have investigated notions of discreteness and gaps in the possible values that asymptotic tensor ranks can take. In particular, it was shown that the asymptotic subrank and asymptotic slice rank of any nonzero 3-tensor is equal to 1, equal to 1.88, or at least 2 (over any field), and that the set of possible values of these parameters is discrete (in several regimes). We determine exactly the next gap, showing that the asymptotic subrank and asymptotic slice rank of any nonzero 3-tensor is equal to 1, equal to 1.88, equal to 2, or at least 2.68.
Paper Structure (15 sections, 14 theorems, 14 equations)

This paper contains 15 sections, 14 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Concrete gaps.
  3. General structure.
  4. New results and methods.
  5. Gaps in the asymptotic subrank
  6. Basic definitions
  7. Previously known gaps
  8. New result
  9. Preliminary Results
  10. Rank of slice-spans under restriction
  11. Degenerating to the trivial algebra
  12. Classification of matrix subspaces of small rank
  13. Proof of \ref{['thm:new']}
  14. Open problems
  15. Acknowledgements.

Key Result

Theorem 1

Let $n_1, n_2, n_3 \in \mathbb{N}$ be arbitrary. For every nonzero $T \in \mathbb{K}^{n_1} \otimes \mathbb{K}^{n_2} \otimes \mathbb{K}^{n_3}$ exactly one of the following is true:

Theorems & Definitions (23)

  • Theorem 1: cgz
  • Theorem 2: cgz
  • Theorem 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • Lemma 7
  • ...and 13 more