Decomposition of homogeneous polynomials with low rank

Edoardo Ballico 1 Alessandra Bernardi 2
2 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
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 $ \PP {{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 $r$ is at most $d$ and a mild condition is satisfied.
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Edoardo Ballico, Alessandra Bernardi. Decomposition of homogeneous polynomials with low rank. Mathematische Zeitschrift, Springer, 2012, 271, pp.1141-1149. 〈http://link.springer.com/article/10.1007%2Fs00209-011-0907-6#page-1〉. 〈10.1007/s00209-011-0907-6〉. 〈hal-00645978〉

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