On the length of binary forms

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 $K = \rr$.