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Concise tensors of minimal border rank

Joachim Jelisiejew, J. M. Landsberg, Arpan Pal

TL;DR

The paper advances the classification of concise tensors with minimal border rank in $\mathbb{C}^m\otimes\mathbb{C}^m\otimes\mathbb{C}^m$ by introducing the 111-algebra and the 111-equations, and by connecting these to module-theoretic ADHM/Quot-geometry frameworks. It delivers explicit defining equations for $m=5$ and extends the 111-abundance analysis to $m\le 6$ under $1_*$-genericity, showing that 111-equations can suffice where Strassen’s and End-closed equations fail. The authors provide a detailed classification in the corank-one 111-abundant, $m=5$ case (five isomorphism types up to symmetry) and construct normal forms strengthening Friedland’s results, along with two independent proofs of minimal border rank for the degenerate cases. They further develop obstructions to minimal border rank via the 111-algebra, relate these obstructions to deformation theory and the Quot scheme, and connect cactus rank, smoothable rank, and 111-algebra smoothability, highlighting potential implications for algebraic complexity and Strassen-type laser methods. The work thus offers a robust algebraic framework for understanding minimal border rank tensors and lays groundwork for leveraging deformation theory to overcome longstanding lower bound barriers in tensor rank problems.

Abstract

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

Concise tensors of minimal border rank

TL;DR

The paper advances the classification of concise tensors with minimal border rank in by introducing the 111-algebra and the 111-equations, and by connecting these to module-theoretic ADHM/Quot-geometry frameworks. It delivers explicit defining equations for and extends the 111-abundance analysis to under -genericity, showing that 111-equations can suffice where Strassen’s and End-closed equations fail. The authors provide a detailed classification in the corank-one 111-abundant, case (five isomorphism types up to symmetry) and construct normal forms strengthening Friedland’s results, along with two independent proofs of minimal border rank for the degenerate cases. They further develop obstructions to minimal border rank via the 111-algebra, relate these obstructions to deformation theory and the Quot scheme, and connect cactus rank, smoothable rank, and 111-algebra smoothability, highlighting potential implications for algebraic complexity and Strassen-type laser methods. The work thus offers a robust algebraic framework for understanding minimal border rank tensors and lays groundwork for leveraging deformation theory to overcome longstanding lower bound barriers in tensor rank problems.

Abstract

We determine defining equations for the set of concise tensors of minimal border rank in when and the set of concise minimal border rank -generic tensors when . 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 -degenerate tensors satisfying Strassen's equations. We use the 111-algebra to characterize wild minimal border rank tensors and classify them in .
Paper Structure (50 sections, 30 theorems, 73 equations)

This paper contains 50 sections, 30 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Results on tensors of minimal border rank
  3. The 111-algebra and its uses
  4. Previous work on tensors of minimal border rank in $\mathbb{C}^m{\mathord{ \otimes } } \mathbb{C}^m{\mathord{ \otimes } } \mathbb{C}^m$
  5. Open questions and future directions
  6. 111-abundant, not 111-sharp tensors
  7. 111-abundant $1$-degenerate tensors
  8. 111-abundance v. classical equations
  9. 111-abundance in the symmetric case
  10. The locus of concise, 111-sharp tensors
  11. 111-algebra in the symmetric case
  12. The Segre-Veronese variety
  13. Strassen's laser method
  14. Overview
  15. Definitions/Notation
  16. ...and 35 more sections

Key Result

Proposition 1.2

Let $T\in \mathbb{C}^m{\mathord{ \otimes } } \mathbb{C}^m{\mathord{ \otimes } } \mathbb{C}^m$ be concise. If $T$ satisfies the 111-equations then it also satisfies Strassen's equations and the End-closed equations. If $T$ is $1_A$ generic, then it satisfies the 111-equations if and only if it satisf

Theorems & Definitions (70)

  • Example 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Definition 1.10
  • ...and 60 more