On Waring rank jumps via critical rank-one approximations

TL;DR

This paper explores how eigenvectors, as critical rank-one approximations, influence the Waring rank of symmetric tensors, revealing specific conditions under which they increase or decrease the rank.

Contribution

It characterizes the relationship between eigenvectors and Waring rank in symmetric tensors, especially for binary forms, using apolar action and Bombieri-Weyl product techniques.

Findings

Eigenvectors can increase or decrease Waring rank depending on form properties.

For binary forms, the set admitting an eigenvector in minimal decomposition has codimension one.

Subgeneric binary forms of rank less than (d+1)/2 always see rank increase from eigenvectors.

Abstract

We investigate whether eigenvectors, also known as critical rank-one approximations, of a symmetric tensor can be used to increase or decrease its Waring rank. First, we study the variety of degree-d rank-r forms which admit an eigenvector as part of a minimal Waring decomposition. In the case of binary forms, we show that this is of codimension-one in the r-th secant variety of the rational normal curve. On the other hand, we prove that for any binary form of rank less than (d+1)/2 (subgeneric), any eigenvector increases the rank. Additionally, when the degree is odd, the same holds for generic forms of generic rank. Our approach employs the strict relation between the apolar action and the Bombieri-Weyl product.