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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 sth secant variety of the d-uple Veronese embedding of ${\mathbb{P}}^m$ into ${\mathbb{P}}^{\genfrac{}{}{0pt}{}{m+d}{d}-1}$ but that its minimal decomposition as a sum of dth powers of linear forms requires more than s summands. We show that if s ≤ d then F can be uniquely written as $F=M{₁}^d+\cdots +M_t^d+Q$, where $M₁,...,M_t$ are linear forms with t ≤ (d-1)/2, and Q is a binary form such that $Q={\sum }_{i=1}^q{l}_i^{d-d_i}{m}_i$ with $l_i$’s linear forms and $m_i$’s forms...

We consider the k-osculating varietiesO to the Veronese d?uple embeddings of P. By studying the Hilbert function of certain zero-dimensional schemes Y ⊂ P, we find the dimension of O , the (s?1) secant varieties of O, for 3 ≤ s ≤ 6 and s = 9, and we determine whether those secant varieties are defective or not.