Real and complex Waring rank of reducible cubic forms

TL;DR

This paper investigates the Waring rank of reducible cubic forms over real and complex fields, providing exact values for complex ranks and bounds for real ranks based on signature considerations.

Contribution

It computes the complex Waring rank for all reducible cubic forms and establishes bounds for their real ranks depending on the signature of the quadratic factor.

Findings

Exact complex Waring ranks for all reducible cubics.

Bounds on real Waring ranks based on quadratic signature.

Analysis of how reducible cubic forms' ranks vary over real and complex fields.

Abstract

In this paper, we study the real and the complex Waring rank of reducible cubic forms. In particular, we compute the complex rank of all reducible cubic forms. In the real case, for all reducible cubics, we either compute or bound the real rank depending on the signature of the degree two factor.