Symmetrization maps and minimal border rank Comon's conjecture

TL;DR

This paper proves Comon's conjecture for large classes of symmetric tensors with minimal border rank, using border apolarity and border varieties of sums of powers.

Contribution

It establishes the conjecture for concise tensors of border rank n, including all tame tensors and tensors with n ≤ d+1.

Findings

Proves border Comon's conjecture for large classes of tensors.

Identifies classes of tensors where border and symmetric border ranks coincide.

Utilizes border apolarity and border varieties of sums of powers as key tools.

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.