A note on the Waring ranks of reducible cubic forms

TL;DR

This paper investigates the Waring ranks of reducible cubic forms in multiple variables, establishing an upper bound for these ranks and contributing to the understanding of their algebraic complexity.

Contribution

The paper proves that the Waring ranks of reducible cubic forms in n+1 variables are contained within the set {1, ..., 2n+1}, providing a new bound.

Findings

Waring ranks of reducible cubic forms are bounded by 2n+1.

The set of possible ranks is contained within the first 2n+1 positive integers.

The result advances understanding of the algebraic structure of reducible cubics.

Abstract

Let $W_3(n)$ be the set of Waring ranks of reducible cubic forms in $n+1$ variables. We prove that $W_3(n)\subseteq \lbrace 1,..., 2n+1\rbrace$.