Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonlinear methods for tensors: determinantal equations for secant varieties beyond cactus

Matěj Doležálek, Mateusz Michałek

TL;DR

The paper develops Kronecker-Koszul flattenings to generate explicit determinantal equations for Secant varieties of Segre embeddings, addressing the long-standing cactus-secant barrier by using nonlinear embeddings. Central to the approach is the tangency flattening, a specific Kronecker-Koszul construction whose minors vanish on the $n$-th secant variety but not on the $n$-th cactus variety for all $n\ge14$, yielding new border rank lower bounds and a purely algebraic proof that the border rank of the $2\times2$ matrix multiplication tensor is $7$. The authors also connect to Koszul flattenings via algebras, showing when quadratic Kronecker-Koszul flattenings outperform classical Koszul ones, and illustrate applications to matrix multiplication and to detecting nonsmoothable algebras through minor flattenings. They discuss broader implications for Hilbert schemes, the Bodensee program, and potential extensions, including more contractions and multi-input constructions, suggesting a rich program for improving lower bounds and deriving new secant-variety equations.

Abstract

We present a family of flattening methods of tensors which we call Kronecker-Koszul flattenings, generalizing the famous Koszul flattenings and further equations of secant varieties studied among others by Landsberg, Manivel, Ottaviani and Strassen. We establish new border rank criteria given by vanishing of minors of Kronecker-Koszul flattenings. We obtain the first explicit polynomial equations -- tangency flattenings -- vanishing on secant varieties of Segre variety, but not vanishing on cactus varieties. Additionally, our polynomials have simple determinantal expressions. As another application, we provide a new, computer-free proof that the border rank of the $2\times2$ matrix multiplication tensor is $7$.

Nonlinear methods for tensors: determinantal equations for secant varieties beyond cactus

TL;DR

The paper develops Kronecker-Koszul flattenings to generate explicit determinantal equations for Secant varieties of Segre embeddings, addressing the long-standing cactus-secant barrier by using nonlinear embeddings. Central to the approach is the tangency flattening, a specific Kronecker-Koszul construction whose minors vanish on the -th secant variety but not on the -th cactus variety for all , yielding new border rank lower bounds and a purely algebraic proof that the border rank of the matrix multiplication tensor is . The authors also connect to Koszul flattenings via algebras, showing when quadratic Kronecker-Koszul flattenings outperform classical Koszul ones, and illustrate applications to matrix multiplication and to detecting nonsmoothable algebras through minor flattenings. They discuss broader implications for Hilbert schemes, the Bodensee program, and potential extensions, including more contractions and multi-input constructions, suggesting a rich program for improving lower bounds and deriving new secant-variety equations.

Abstract

We present a family of flattening methods of tensors which we call Kronecker-Koszul flattenings, generalizing the famous Koszul flattenings and further equations of secant varieties studied among others by Landsberg, Manivel, Ottaviani and Strassen. We establish new border rank criteria given by vanishing of minors of Kronecker-Koszul flattenings. We obtain the first explicit polynomial equations -- tangency flattenings -- vanishing on secant varieties of Segre variety, but not vanishing on cactus varieties. Additionally, our polynomials have simple determinantal expressions. As another application, we provide a new, computer-free proof that the border rank of the matrix multiplication tensor is .
Paper Structure (16 sections, 15 theorems, 70 equations, 1 figure)

This paper contains 16 sections, 15 theorems, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries & notation
  3. Kronecker-Koszul flattenings
  4. Kronecker-Young flattenings
  5. Koszul flattenings
  6. Structure tensors of algebras
  7. Comparison of Koszul flattenings with a quadratic flattening
  8. Applications
  9. Matrix multiplication
  10. Distinguishing cactus and secant varieties
  11. Minor flattenings and algebras
  12. Future directions
  13. Hilbert schemes
  14. Relations to the Bodensee program
  15. Special tensors and equations of secant varieties
  16. ...and 1 more sections

Key Result

Theorem 1.1

For all $n\geq14$, minors of size $n(n-1)(n-2)+1$ of the tangency flattenings vanish on the $n$-th secant variety of the Segre variety in ${\mathbb{P}}(\mathbb{C}^n\otimes\mathbb{C}^n\otimes\mathbb{C}^n)$ but do not vanish on its $n$-th cactus variety.

Figures (1)

  • Figure 1: Graph $G$ obtained as the union of the clique $\color{red}\lambda_{1,1}=\{1,2,3\}$ and the clique $\color{blue}\lambda_{2,1}=\{3,4\}$. All other sets in the partitions are singletons, contributing no edges.

Theorems & Definitions (41)

  • Theorem 1.1
  • Remark 3.1
  • Example 3.2
  • Theorem 3.3
  • proof
  • Definition 3.4
  • Corollary 3.5
  • proof
  • Example 3.6
  • Example 3.7
  • ...and 31 more