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Cactus barriers

Jarosław Buczyński

TL;DR

The paper addresses fundamental barriers to using linear rank methods for bounding border rank of tensors and polynomials by reframing the problem with cactus varieties. It develops a geometric framework in which ranks, border ranks, and cactus ranks are controlled via subvarieties of projective space and their cactus analogues, showing that linear rank methods can at best detect border cactus rank and not the true border rank in broad settings. A central contribution is proving that, for smooth and many classical varieties, the cactus barrier bounds any effort to surpass a certain threshold, with explicit connections to Grassmann cactus varieties and determinantal spines. The work clarifies the intrinsic limitations of determinant-based methods, highlights deformation-theoretic avenues for barrier-breaking, and lays out open questions on expanding barrier-breaking techniques beyond current Special cases.

Abstract

Determinantal methods for bounding the rank and border rank of tensors or polynomials are subject to a major barrier. For instance, it is known that using determinantal methods one cannot prove a lower bound for the border rank of a 3-way tensor of size m in each direction that exceeds 6m-4. We explain the precise geometric reason for this number (and analogous bounds in more general tensor spaces) using cactus varieties and, more generally, scheme theoretic methods in algebraic geometry.

Cactus barriers

TL;DR

The paper addresses fundamental barriers to using linear rank methods for bounding border rank of tensors and polynomials by reframing the problem with cactus varieties. It develops a geometric framework in which ranks, border ranks, and cactus ranks are controlled via subvarieties of projective space and their cactus analogues, showing that linear rank methods can at best detect border cactus rank and not the true border rank in broad settings. A central contribution is proving that, for smooth and many classical varieties, the cactus barrier bounds any effort to surpass a certain threshold, with explicit connections to Grassmann cactus varieties and determinantal spines. The work clarifies the intrinsic limitations of determinant-based methods, highlights deformation-theoretic avenues for barrier-breaking, and lays out open questions on expanding barrier-breaking techniques beyond current Special cases.

Abstract

Determinantal methods for bounding the rank and border rank of tensors or polynomials are subject to a major barrier. For instance, it is known that using determinantal methods one cannot prove a lower bound for the border rank of a 3-way tensor of size m in each direction that exceeds 6m-4. We explain the precise geometric reason for this number (and analogous bounds in more general tensor spaces) using cactus varieties and, more generally, scheme theoretic methods in algebraic geometry.
Paper Structure (11 sections, 11 theorems, 23 equations)

This paper contains 11 sections, 11 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Linear rank methods
  3. Border and cactus ranks
  4. Barriers in terms of cacti
  5. Prior work
  6. Strengthening of barriers
  7. Barrier breaking
  8. Finite subschemes
  9. Cacti
  10. Determinantal spines
  11. Appendix: from sets to algebraic varieties

Key Result

Theorem 1.3

With variety $X$ and matrix $M$ as above, if in addition $X\subset \left\{\mathop{\mathrm{rk}}\nolimits M \leqslant k\right\}$ then $\mathfrak{K}_{r}\left( X_0 \right) \subset \left\{\mathop{\mathrm{rk}}\nolimits M \leqslant k \cdot r\right\}$. In particular, if $g$ is an integer such that $\mathfra

Theorems & Definitions (16)

  • Definition 1.2
  • Theorem 1.3: Cactus barrier for linear rank method
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 3.1
  • ...and 6 more