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Symmetric tensors and symmetric tensor rank
 Scientific Computing and Computational Mathematics (SCCM
, 2006
"... Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. An ..."
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Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1. Key words. Tensors, multiway arrays, outer product decomposition, symmetric outer product decomposition, candecomp, parafac, tensor rank, symmetric rank, symmetric tensor rank, generic symmetric rank, maximal symmetric rank, quantics AMS subject classifications. 15A03, 15A21, 15A72, 15A69, 15A18 1. Introduction. We
On the Alexander–Hirschowitz theorem
 J. Pure Appl. Algebra
, 2008
"... The AlexanderHirschowitz theorem says that a general collection of k double points in Pn imposes independent conditions on homogeneous polynomials of degree d with a well known list of exceptions. Alexander and Hirschowitz completed its proof in 1995, solving a long standing classical problem, con ..."
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The AlexanderHirschowitz theorem says that a general collection of k double points in Pn imposes independent conditions on homogeneous polynomials of degree d with a well known list of exceptions. Alexander and Hirschowitz completed its proof in 1995, solving a long standing classical problem, connected with the Waring problem for polynomials. We expose a selfcontained proof based mainly on previous works by Terracini, Hirschowitz, Alexander and Chandler, with a few simplifications. We claim originality only in the case d = 3, where our proof is shorter. We end with an account of the history of the work on this problem.
The geometric interpretation of Froberg–Iarrobino conjectures on the Hilbert functions of fat points
"... Abstract. The study of infinitesimal deformations of a variety embedded in projective space requires, at ground level, that of deformation of a collection of points, as specified by a zerodimensional scheme. Further, basic problems in infinitesimal interpolation correspond directly to the analysis ..."
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Abstract. The study of infinitesimal deformations of a variety embedded in projective space requires, at ground level, that of deformation of a collection of points, as specified by a zerodimensional scheme. Further, basic problems in infinitesimal interpolation correspond directly to the analysis of such schemes. An optimal Hilbert function of a collection of infinitesimal neighbourhoods of points in projective space is suggested by algebraic conjectures of R. Fröberg and A. Iarrobino. We discuss these conjectures from a geometric point of view. The conjectures give, for each such collection, a function (based on dimension, number of points, and order of each neighbourhood) which should serve as an upper bound to its Hilbert function (Weak Conjecture). The Strong Conjecture predicts when the upper bound is sharp, in the case of equal order throughout. In general we refer to the equality of the Hilbert function of a collection of infinitesimal neighbourhoods with that of the corresponding conjectural function as the Strong Hypothesis. We interpret these conjectures and hypotheses as accounting for the infinitesimal neighbourhoods of projective subspaces naturally occurring in the base locus of a linear system
A new proof of the AlexanderHirschowitz interpolation theorem
 Ann. Mat. Pura Appl
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Uniqueness of exceptional singular quartic
 Proc. Amer. Math. Soc
"... Abstract. We prove that given a general collection Γ of 14 points of P 4 = P 4 K (K an infinite field) there is a unique quartic hypersurface that is singular on Γ. This completes the solution to the open problem of the dimension of a linear system of hypersurfaces of P n that are singular on a coll ..."
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Abstract. We prove that given a general collection Γ of 14 points of P 4 = P 4 K (K an infinite field) there is a unique quartic hypersurface that is singular on Γ. This completes the solution to the open problem of the dimension of a linear system of hypersurfaces of P n that are singular on a collection of general points. 1.
Secant varieties to osculating varieties of Veronese embeddings of P n.
, 2011
"... Introduction. ABSTRACT: A well known theorem by AlexanderHirschowitz states that all the higher secant varieties of Vn,d (the duple embedding of P n) have the expected dimension, with few known exceptions. We study here the same problem for Tn,d, the tangential variety to Vn,d, and prove a conject ..."
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Introduction. ABSTRACT: A well known theorem by AlexanderHirschowitz states that all the higher secant varieties of Vn,d (the duple embedding of P n) have the expected dimension, with few known exceptions. We study here the same problem for Tn,d, the tangential variety to Vn,d, and prove a conjecture, which is the analogous of