Unique decomposition for a polynomial of low rank

Edoardo Ballico¹ ; Alessandra Bernardi²

¹ Department of Mathematics University of Trento 38123 Povo (TN), Italy
² Dipartimento di Matematica “Giuseppe Peano” Università degli Studi di Torino Via Carlo Alberto 10 I-10123 Torino, Italy

Annales Polonici Mathematici, Tome 108 (2013) no. 3, pp. 219-224

Résumé

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 $\sum (d_i+1)=s-t$.

Détail

DOI : 10.4064/ap108-3-2

Keywords: homogeneous polynomial degree variables defined algebraically closed field characteristic suppose belongs sth secant variety d uple veronese embedding mathbb mathbb atopwithdelims its minimal decomposition sum dth powers linear forms requires summands leq uniquely written cdots q where ldots linear forms leq d binary form sum d d linear forms forms degree sum s t