Stratification of the fourth secant variety of Veronese variety via the symmetric 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 : If $X\subset \mathbb{P}^n$ is a projective non degenerate variety, the $X$-rank of a point $P\in \mathbb{P}^n$ is defined to be the minimum integer $r$ such that $P$ belongs to the span of $r$ points of $X$. We describe the complete stratification of the fourth secant variety of any Veronese variety $X$ via the $X$-rank. This result has an equivalent translation in terms both of symmetric tensors and homogeneous polynomials. It allows to classify all the possible integers $r$ that can occur in the minimal decomposition of either a symmetric tensor or a homogeneous polynomial of $X$-border rank $4$ (i.e. contained in the fourth secant variety) as a linear combination of either completely decomposable tensors or powers of linear forms respectively.
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Submitted on : Monday, November 28, 2011 - 6:32:41 PM
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Edoardo Ballico, Alessandra Bernardi. Stratification of the fourth secant variety of Veronese variety via the symmetric rank. 2011. ⟨inria-00612460v2⟩

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