Ballico, Edoardo ; Bernardi, Alessandra
(2011)
*Unique decomposition for a polynomial of low rank.*
[Preprint]

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## 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 $\mathbb{P}^{{m+d\choose d}-1}$ but that its
minimal decomposition as a sum of $d$-th powers of linear forms requires more than $s$ addenda. We show that if $s\leq d$ 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 $Q$ 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 $\sum (d_i+1)=s-t$.

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 $\mathbb{P}^{{m+d\choose d}-1}$ but that its
minimal decomposition as a sum of $d$-th powers of linear forms requires more than $s$ addenda. We show that if $s\leq d$ 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 $Q$ 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 $\sum (d_i+1)=s-t$.

Document type

Preprint

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DOI

Deposit date

02 Aug 2011 16:26

Last modified

16 Sep 2011 12:17

URI

## Other metadata

Document type

Preprint

Creators

Subjects

DOI

Deposit date

02 Aug 2011 16:26

Last modified

16 Sep 2011 12:17

URI

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