Gaps in the pairs (border rank,symmetric rank) for symmetric tensors

TL;DR

This paper characterizes the possible symmetric tensor ranks up to a certain bound for symmetric tensors with a fixed border rank, revealing gaps in the relationship between border rank and symmetric rank.

Contribution

It provides a detailed description of the symmetric tensor ranks for tensors with a given border rank, highlighting gaps and new insights into their rank structure.

Findings

Identifies gaps in the pairs (border rank, symmetric rank) for symmetric tensors.

Provides bounds and conditions for symmetric tensor ranks given border rank.

Enhances understanding of the rank structure of symmetric tensors in algebraic geometry.

Abstract

Fix integers $m \ge 2$, $s\ge 5$ and $d\ge 2s+2$. Here we describe the possible symmetric tensor ranks $\le 2d+s-7$ of all symmetric tensors (or homogeneous degree $d$ polynomials) in $m+1$ variables with border rank $s$.