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Relevant bibliographies by topics / Secant varieties, rank of symmetric tensors

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Academic literature on the topic 'Secant varieties, rank of symmetric tensors'

Author: Grafiati

Published: 14 March 2023

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Journal articles on the topic "Secant varieties, rank of symmetric tensors"

1

Casarotti, Alex, Alex Massarenti, and Massimiliano Mella. "On Comon’s and Strassen’s Conjectures." Mathematics 6, no. 11 (October 25, 2018): 217. http://dx.doi.org/10.3390/math6110217.

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Comon’s conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen’s conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon’s conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties.

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2

Bernardi, Alessandra, Enrico Carlini, Maria Catalisano, Alessandro Gimigliano, and Alessandro Oneto. "The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition." Mathematics 6, no. 12 (December 8, 2018): 314. http://dx.doi.org/10.3390/math6120314.

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We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject.

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3

Ballico, E. "Symmetric Tensor Rank and Scheme Rank: An Upper Bound in terms of Secant Varieties." Geometry 2013 (September 8, 2013): 1–3. http://dx.doi.org/10.1155/2013/614195.

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Let X⊂ℙr be an integral and nondegenerate variety. Let c be the minimal integer such that ℙr is the c-secant variety of X, that is, the minimal integer c such that for a general O∈ℙr there is S⊂X with #(S)=c and O∈〈S〉, where 〈 〉 is the linear span. Here we prove that for every P∈ℙr there is a zero-dimensional scheme Z⊂X such that P∈〈Z〉 and deg(Z)≤2c; we may take Z as union of points and tangent vectors of Xreg.

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4

Cartwright, Dustin, and Melody Chan. "Three notions of tropical rank for symmetric matrices." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AN,..., Proceedings (January 1, 2010). http://dx.doi.org/10.46298/dmtcs.2865.

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International audience We introduce and study three different notions of tropical rank for symmetric matrices and dissimilarity matrices in terms of minimal decompositions into rank 1 symmetric matrices, star tree matrices, and tree matrices. Our results provide a close study of the tropical secant sets of certain nice tropical varieties, including the tropical Grassmannian. In particular, we determine the dimension of each secant set, the convex hull of the variety, and in most cases, the smallest secant set which is equal to the convex hull. Nous introduisons et étudions trois notions différentes de rang tropical pour des matrices symétriques et des matrices de dissimilarité, en utilisant des décompositions minimales en matrices symétriques de rang 1, en matrices d'arbres étoiles, et en matrices d'arbres. Nos résultats donnent lieu à une étude détaillée des ensembles des sécantes tropicales de certaines jolies variétés tropicales, y compris la grassmannienne tropicale. En particulier, nous déterminons la dimension de chaque ensemble des sécantes, l'enveloppe convexe de la variété, ainsi que, dans la plupart des cas, le plus petit ensemble des sécantes qui est égal à l'enveloppe convexe.

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5

Casarotti, Alex, Elsa Corniani, and Alex Massarenti. "Complete Singular Collineations and Quadrics." International Mathematics Research Notices, October 5, 2022. http://dx.doi.org/10.1093/imrn/rnac271.

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Abstract We construct wonderful compactifications of the spaces of linear maps and symmetric linear maps of a given rank as blowups of secant varieties of Segre and Veronese varieties. Furthermore, we investigate their birational geometry and their relations with some spaces of degree two stable maps.

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6

Kummer, Mario, and Rainer Sinn. "Hyperbolic secant varieties of M-curves." Journal für die reine und angewandte Mathematik (Crelles Journal), April 20, 2022. http://dx.doi.org/10.1515/crelle-2022-0012.

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Abstract We relate the geometry of curves to the notion of hyperbolicity in real algebraic geometry. A hyperbolic variety is a real algebraic variety that (in particular) admits a real fibered morphism to a projective space whose dimension is equal to the dimension of the variety. We study hyperbolic varieties with a special interest in the case of hypersurfaces that admit a real algebraic ruling. The central part of the paper is concerned with secant varieties of real algebraic curves where the real locus has the maximal number of connected components, which is determined by the genus of the curve. For elliptic normal curves, we further obtain definite symmetric determinantal representations for the hyperbolic secant hypersurfaces, which implies the existence of symmetric Ulrich sheaves of rank one on these hypersurfaces. We also use this to derive better bounds on the size of semidefinite representations for convex hulls of real algebraic curves of genus 1.

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7

Casarotti, Alex, and Massimiliano Mella. "Tangential Weak Defectiveness and Generic Identifiability." International Mathematics Research Notices, June 14, 2021. http://dx.doi.org/10.1093/imrn/rnab091.

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Abstract We investigate the uniqueness of decomposition of general tensors $T\in{\mathbb C}^{n_1+1}\otimes \cdots \otimes{\mathbb C}^{n_r+1}$ as a sum of tensors of rank $1$. This is done extending the theory developed in [ 28] to the framework of non-twd varieties. In this way, we are able to prove the non-generic identifiability of infinitely many partially symmetric tensors.

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8

Ballico, Edoardo, and Alessandra Bernardi. "Stratification of the fourth secant variety of Veronese varieties via the symmetric rank." Advances in Pure and Applied Mathematics 4, no. 2 (January 1, 2013). http://dx.doi.org/10.1515/apam-2013-0015.

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Dissertations / Theses on the topic "Secant varieties, rank of symmetric tensors"

1

mazzon, andrea. "Hilbert functions and symmetric tensors identifiability." Doctoral thesis, Università di Siena, 2021. http://hdl.handle.net/11365/1133145.

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We study the Waring decompositions of a given symmetric tensor using tools of algebraic geometry for the study of finite sets of points. In particular we use the properties of the Hilbert functions and the Cayley-Bacharach property to study the uniqueness of a given decomposition (the identifiability problem), and its minimality, and show how, in some cases, one can effectively determine the uniqueness even in some range in which the Kruskal's criterion does not apply. We give also a more efficient algorithm that, under some hypothesis, certify the identifiability of a given symmetric tensor.

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