Abstract

Let $F$ be a homogeneous polynomial of degree $d$ in $m+1$ variables defined over an algebraically closed field of characteristic zero and suppose that $F$ belongs to the $s$-th secant varieties of the standard Veronese variety $X_{m,d}\subset \mathbb{P}^{{m+d\choose d}-1}$ but that its minimal decomposition as sum of $d$-th powers of linear forms $M_1, \ldots , M_r$ is $F=M_1^d+\cdots + M_r^d$ with $r>s$. We show that if $s+r\leq 2d+1$ then such a decomposition of $F$ can be split in two parts: one of them is uniquely determined by linear forms that can be written using only two variables, the other part is algorithmically computable. We also show that the $0$-dimensional scheme $\mathcal{Z}$ of degree $s$ that is contained in $X_{m,d}$ and such that $F\in \langle \mathcal{Z}\rangle$ is uniquely determined by $F$ itself.