Symmetric subrank and its border analogue

TL;DR

This paper investigates the asymptotic behavior of symmetric subrank and border subrank of homogeneous polynomials, revealing their relationship for low-degree forms using geometric invariant theory.

Contribution

It determines the asymptotic behavior of symmetric subrank and border subrank for forms as variables grow, and shows their equality for low-degree cases under certain conditions.

Findings

Asymptotic behavior of symmetric subrank and border subrank established.

Symmetric subrank and border subrank coincide for low-degree forms with small subrank.

Geometric invariant theory used to analyze subrank relationships.

Abstract

The symmetric subrank of homogeneous polynomial is the largest number of terms in a diagonal form to which it can be specialized by a (typically non-invertible) linear variable substitution. Building on earlier work by Derksen-Makam-Zuiddam and Biaggi-Chang-Draisma-Rupniewski for ordinary tensors, we determine the asymptotic behavior of symmetric subrank and symmetric border subrank of degree-d forms as the number of variables tends to infinity. Furthermore, by using results from geometric invariant theory we show that for cubic (resp. quartic) forms the symmetric subrank and symmetric border subrank coincide if the latter is at most three (resp. two).