On the waring rank of binary forms

The K-rank of a binary form f in K[x,y],K⊆C, is the smallest number of d-th powers of linear forms over K of which f is a K-linear combination. We provide lower bounds for the C-rank (Waring rank) and for the R-rank (real Waring rank) of binary forms depending on their factorization. We study binary forms with unique C-minimal representation.