Binary forms with three different relative ranks

Source abstract

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