On the length of binary forms

TL;DR

This paper investigates the concept of $K$-length of binary forms, providing new and existing results mainly for two-variable forms, and explores how the length varies over different fields.

Contribution

It offers new insights and results on the $K$-length of binary forms, including how it differs across various fields, with a focus on forms in two variables.

Findings

The $K$-length of specific forms varies with the field, e.g., three over $bq( ext{i} ext{i})$, four over $bq( ext{-}2)$, and five over $bq$.

Many results about $K$-length are presented, combining old and new findings, especially for two-variable forms.

The paper highlights the dependence of form length on the underlying field.

Abstract

The $K$-length of a form $f$ in $K[x_1,\dots,x_n]$, $K \subset \cc$, is the smallest number of $d$-th powers of linear forms of which $f$ is a $K$-linear combination. We present many results, old and new, about $K$-length, mainly in $n=2$, and often about the length of the same form over different fields. For example, the $K$-length of $3x^5 -20x^3y^2+10xy^4$ is three for $K = \qq(\sqrt{-1})$, four for $K = \qq(\sqrt{-2})$ and five for $K = \rr$.