Binary forms with three different relative ranks

Abstract

Suppose $f(x,y)$ is a binary form of degree $d$ with coefficients in a field $K\subseteq \mathbb{C}$. The $K$-rank of $f$ is the smallest number of $d$-th powers of linear forms over $K$ of which $f$ is a $K$-linear combination. We prove that for $d\ge 5$, there always exists a form of degree $d$ with at least three different ranks over various fields. The $K$-rank of a form $f$ (such as $x^3{y}^2$) may depend on whether -1 is a sum of two squares in $K$.