Binary forms with three different relative ranks

TL;DR

This paper investigates binary forms of degree at least five and demonstrates that such forms can have at least three different ranks over various fields, depending on properties like whether -1 is a sum of two squares in the field.

Contribution

The paper proves the existence of binary forms with at least three different ranks over different fields for degrees five and above, highlighting the dependence on field properties.

Findings

Existence of forms with multiple ranks over various fields.

The rank depends on whether -1 is a sum of two squares in the field.

For degree d ≥ 5, forms can have at least three different 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 $x^3y^2$) may depend on whether -1 is a sum of two squares in $K$.