Unique decomposition for a polynomial of low rank
Volume 108 / 2013
Annales Polonici Mathematici 108 (2013), 219-224
MSC: 15A21, 15A69, 14N15.
DOI: 10.4064/ap108-3-2
Abstract
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 $ \mathbb {P}^{{m+d\atopwithdelims ()d}-1}$ but that its minimal decomposition as a sum of $d$th powers of linear forms requires more than $s$ summands. We show that if $s\leq d$ then $F$ can be uniquely written as $F=M_1^d+\cdots + M_t^d+Q$, where $M_1, \ldots , M_t$ are linear forms with $t\leq (d-1)/2$, 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 of degree $d_i$ such that $\sum (d_i+1)=s-t$.
