BINARY FORMS WITH THREE DIFFERENT RELATIVE RANKS

Suppose f(x, y) is a binary form of degree d with coefficients in a field K subset of 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 >= 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.