Unique decomposition for a polynomial of 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 ${\mathbb{P}}^m$ into $\text{PP}\left(\genfrac{}{}{0px}{}{m+d}{d}\right)-1$ but that its minimal decomposition as a sum of $d$-th powers of linear forms requires more than $s$ addenda. We show that if $s\le d$ then $F$ can be uniquely written as $F=M_1^d+\cdots +M_t^d+Q$, where $M_1,\dots ,M_t$ are linear forms with $t\le (d-1)\mathrm{/}2$, and $Q$ a binary form such that $Q=\sum_{i=1}^q l_i^{d-d_i}m_i$with$l_i${}^{\mathrm{\prime }}slinearformsand$m_i${}^{\mathrm{\prime }}sformsofdegree$d_i$suchthat$\sum (d_i+1)=s-t$.