On the Waring Rank of Binary Forms

TL;DR

This paper investigates the Waring rank of binary forms, providing bounds based on factorization and classifying all forms with rank 3, advancing understanding of polynomial decompositions.

Contribution

It offers new lower bounds for complex and real Waring ranks and fully classifies binary forms with rank 3, enhancing the theoretical framework of polynomial decomposition.

Findings

Established lower bounds for Waring ranks based on factorization.

Classified all binary forms with Waring rank 3.

Provided insights into the structure of binary forms with specific ranks.

Abstract

The $K$-rank of a binary form $f$ in $K[x,y],~K\subseteq \mathbb{C},$ is the smallest number of $d$-th powers of linear forms over $K$ of which $f$ is a $K$-linear combination. We provide lower bounds for the $\mathbb{C}$-rank (Waring rank) and for the $\mathbb{R}$-rank (real Waring rank) of binary forms depending on their factorization. We completely classify binary forms of Waring rank 3.