Decomposition of homogeneous polynomials with low rank

Ballico, Edoardo ; Bernardi, Alessandra (2010) Decomposition of homogeneous polynomials with low rank. [Preprint]

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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 a 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 made by linear forms that can be written using only two variables, the other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of $F$ if the rank is at most $d$ and a mild condition is satisfied.

Document type
Ballico, Edoardo
Bernardi, Alessandra
Waring problem, Polynomial decomposition, Symmetric rank, Symmetric tensors, Veronese varieties, Secant varieties.
Deposit date
01 Sep 2010 08:21
Last modified
16 May 2011 12:14

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