On the Length of Binary Forms

Abstract

The K-length of a form f in K[x 1 ,. .. , x n ], K ⊂ C, 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 3 y 2 + 10xy 4 is three for K = Q(√ −1), four for K = Q(√ −2) and five for K = R.