Decomposition of homogeneous polynomials with low rank

Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the s-th secant variety of the d-uple Veronese embedding of Pm into PN, N= \binom{m+d}{m}-1. but that its minimal decomposition as a sum of d-th powers of linear forms requires r>s d-powers of linear forms. We show that if s+r ≤ 2d+1, then such a decomposition of F can be split in two parts: one of them is made by linear forms that can be written using only two variables, the other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of F if r is at most d and a mild condition is satisfied.