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Example to “Determinantal equations for secant varieties and the EisenbudKohStillman conjecture
, 2012
"... Abstract. We address special cases of a question of Eisenbud on the ideals of secant varieties of Veronese reembeddings of arbitrary varieties. Eisenbud’s question generalizes a conjecture of Eisenbud, Koh and Stillman (EKS) for curves. We prove that settheoretic equations of small secant varietie ..."
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Abstract. We address special cases of a question of Eisenbud on the ideals of secant varieties of Veronese reembeddings of arbitrary varieties. Eisenbud’s question generalizes a conjecture of Eisenbud, Koh and Stillman (EKS) for curves. We prove that settheoretic equations of small secant varieties to a high degree Veronese reembedding of a smooth variety are determined by equations of the ambient Veronese variety and linear equations. However this is false for singular varieties, and we give explicit counterexamples to the EKS conjecture for singular curves. The techniques we use also allow us to prove a gap and uniqueness theorem for symmetric tensor rank. We put Eisenbud’s question in a more general context about the behaviour of border rank under specialisation to a linear subspace, and provide an overview of conjectures coming from signal processing and complexity theory in this context. 1.
THE CACTUS RANK OF CUBIC FORMS
, 1110
"... Abstract. We prove that the smallest degree of an apolar 0dimensional scheme of a general cubic form in n+1 variables is at most 2n+2, when n ≥ 8, and therefore smaller than the rank of the form. For the general reducible cubic form the smallest degree of an apolar subscheme is n+2, while the rank ..."
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Abstract. We prove that the smallest degree of an apolar 0dimensional scheme of a general cubic form in n+1 variables is at most 2n+2, when n ≥ 8, and therefore smaller than the rank of the form. For the general reducible cubic form the smallest degree of an apolar subscheme is n+2, while the rank is at least 2n.
Mourrain, A comparison of different notions of ranks of symmetric tensors, Preprint: http://hal.inria.fr/hal00746967
"... Abstract. We introduce various notions of rank for a high order symmetric tensor,namely: rank,borderrank,catalecticantrank,generalizedrank,scheme length, borderschemelength, extensionrankandsmoothablerank. Weanalyze the stratification induced by these ranks. The mutual relations between these strati ..."
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Cited by 1 (1 self)
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Abstract. We introduce various notions of rank for a high order symmetric tensor,namely: rank,borderrank,catalecticantrank,generalizedrank,scheme length, borderschemelength, extensionrankandsmoothablerank. Weanalyze the stratification induced by these ranks. The mutual relations between these stratifications, allowus to describethe hierarchyamongall the ranks. We show that strict inequalities are possible between rank, border rank, extension rank and catalecticant rank. Moreover we show that scheme length, generalized rank and extension rank coincide. hal00746967, version 2 26 Nov 2012
Author manuscript, published in "Mathematische Zeitschrift (2011)" DOI: 10.1007/s0020901109076 DECOMPOSITION OF HOMOGENEOUS POLYNOMIALS WITH LOW RANK
, 2011
"... 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 sth secant variety of the duple Veronese embedding of Pm into P (m+d d)−1 but that its minimal decomposition as a sum of dth ..."
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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 sth secant variety of the duple Veronese embedding of Pm into P (m+d d)−1 but that its minimal decomposition as a sum of dth powers of linear forms M1,..., Mr is F = M d 1 + · · ·+M d r with r> s. We show that if s+r ≤ 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.
GENERAL TENSOR DECOMPOSITION, MOMENT MATRICES AND APPLICATIONS
, 2011
"... Abstract. The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for fl ..."
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Abstract. The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for flat extension of QuasiHankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algorithm is described. It applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester to binary forms. An example illustrates the algebraic operations involved in this approach and how the decomposition can be recovered from eigenvector computation.
Decomposition of homogeneous polynomials with low rank
"... 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 sth secant variety of the duple Veronese embedding of Pm () m + d d −1 into P but that its minimal decomposition as a sum of d ..."
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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 sth secant variety of the duple Veronese embedding of Pm () m + d d −1 into P but that its minimal decomposition as a sum of dth powers of linear forms M1,...,Mr is F = Md 1 +···+Md r with r> s. We show that if s +r ≤ 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.
RESEARCH ARTICLE On the typical rank of real binary forms
, 2011
"... We determine the rank of a general real binary form of degree d = 4 or d = 5. In the case d = 5, the possible values of the rank of such general forms are 3,4,5. This is the first reported case, to our knowledge, where more than two typical ranks have been found. We prove that a real binary form of ..."
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We determine the rank of a general real binary form of degree d = 4 or d = 5. In the case d = 5, the possible values of the rank of such general forms are 3,4,5. This is the first reported case, to our knowledge, where more than two typical ranks have been found. We prove that a real binary form of degree d with d real roots has rank d.
CURVILINEAR SCHEMES AND MAXIMUM RANK OF FORMS
, 2012
"... Abstract. We define the curvilinear rank of a degree d form P in n+1 variables as the minimum length of a curvilinear scheme, contained in the dth Veronese embedding of P n, whose span contains the projective class of P. Then, we give a bound for rank of any homogenous polynomial, in dependance on ..."
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Abstract. We define the curvilinear rank of a degree d form P in n+1 variables as the minimum length of a curvilinear scheme, contained in the dth Veronese embedding of P n, whose span contains the projective class of P. Then, we give a bound for rank of any homogenous polynomial, in dependance on its curvilinear rank.
A REMARK ON WARING DECOMPOSITIONS OF SOME SPECIAL PLANE QUARTICS ∗
"... Abstract. Motivated by questions on tensor rank, this work concerns the following unexpected result concerning Waring decompositions of plane quartics containing a double line, along with some preparatory and additional remarks. Let x,l1,...,l7 be linear forms and q a quadratic form on a vector spac ..."
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Abstract. Motivated by questions on tensor rank, this work concerns the following unexpected result concerning Waring decompositions of plane quartics containing a double line, along with some preparatory and additional remarks. Let x,l1,...,l7 be linear forms and q a quadratic form on a vector space of dimension 3. If x 2 q = l 4 1 + ·· · + l4 7 and the lines l1 = 0,..., l7 = 0 in P 2 intersect x = 0 at seven distinct points, then the line x = 0 is (possibly improperly) tangent to the conic q = 0.