Stratification of the fourth secant variety of Veronese variety via the symmetric rank

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.

In Press: Advances in Pure and Applied Mathematics

References in corpus (5)

• Computing symmetric rank for symmetric tensors
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• Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture
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Cited by in corpus (12)

• The Hitchhiker guide to: Secant Varieties and Tensor Decomposition
• On the partially symmetric rank of tensor products of W-states and other symmetric tensors
• Waring, tangential and cactus decompositions
• A note on the gap between rank and border rank
• Generic Power Sum Decompositions and Bounds for the Waring Rank
• Decompositions and Terracini loci of cubic forms of low rank
• Linearly dependent and concise subsets of a Segre variety depending on k factors
• Partial stratification of secant varieties of Veronese varieties via curvilinear subschemes
• Curvilinear schemes and maximum rank of forms
• A uniqueness result on the decompositions of a bi-homogeneous polynomial
• The stratification by rank for homogeneous polynomials with border rank 5 which essentially depend on 5 variables
• On the typical rank of real polynomials (or symmetric tensors) with a fixed border rank