Border Waring Rank via Asymptotic Rank

TL;DR

This paper extends a lower bound on the Waring (cactus) rank of homogeneous forms to border rank for specific classes, utilizing recent tensor asymptotic rank results to deepen understanding of polynomial decompositions.

Contribution

It generalizes a known lower bound on Waring rank to border rank for certain forms using asymptotic tensor rank techniques.

Findings

Lower bound extends to border rank for specific forms

Utilizes recent advances in tensor asymptotic rank

Provides new insights into polynomial decomposition complexity

Abstract

We investigate an extension of a lower bound on the Waring (cactus) rank of homogeneous forms due to Ranestad and Schreyer. We show that for particular classes of homogeneous forms, for which a generalization of this method applies, the lower bound extends to the level of border (cactus) rank. The approach is based on recent results on tensor asymptotic rank.