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Symmetrization maps and minimal border rank Comon's conjecture

Tomasz Mańdziuk, Emanuele Ventura

Abstract

One of the fundamental open problems in the field of tensors is the border Comon's conjecture: given a symmetric tensor $F\in(\mathbb{C}^n)^{\otimes d}$ for $d\geq 3$, its border and symmetric border ranks are equal. In this paper, we prove the conjecture for large classes of concise tensors in $(\mathbb{C}^n)^{\otimes d}$ of border rank $n$, i.e., tensors of minimal border rank. These families include all tame tensors and all tensors whenever $n\leq d+1$. Our technical tools are border apolarity and border varieties of sums of powers.

Symmetrization maps and minimal border rank Comon's conjecture

Abstract

One of the fundamental open problems in the field of tensors is the border Comon's conjecture: given a symmetric tensor for , its border and symmetric border ranks are equal. In this paper, we prove the conjecture for large classes of concise tensors in of border rank , i.e., tensors of minimal border rank. These families include all tame tensors and all tensors whenever . Our technical tools are border apolarity and border varieties of sums of powers.

Paper Structure

This paper contains 7 sections, 31 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Preliminaries
  4. Notation
  5. The diagonal morphism
  6. A morphism between border varieties of sums of powers
  7. A morphism in the other direction

Key Result

Theorem 1

Let $n\leq r\leq \binom{n+2}{2}$. There exists a morphism (called desymmetrizing) $\Upsilon: \underline{\mathrm{VSP}}(p_F,r)\rightarrow \underline{\mathrm{VSP}}(F,r)$.

Theorems & Definitions (70)

  • Conjecture 1.1: Border Comon's conjecture for minimal border rank
  • Theorem : Theorem \ref{['theo:1']}
  • Theorem : Theorem \ref{['thm:equality_of_br_in_terms_of_vspb']}
  • Theorem : Corollary \ref{['cor:nleq d+1']}
  • Theorem : Corollary \ref{['cor:special_case_nonwil']}
  • Theorem : Theorem \ref{['thm:sharp']}
  • Definition 2.1: Apolar ideals
  • Proposition 2.2: Ga23
  • Definition 2.3: Border rank
  • Definition 2.4: Smoothable rank
  • ...and 60 more