×

Degrees of Kalman varieties of tensors. (English) Zbl 1497.14104

Kalman varieties are defined as varieties which parametrize all tensors of a given shape, having a singular t-uple of vectors \((x_1,\dots,x_t)\) in which \(x_1\) parametrizes a point lying in a fixed linear subspace. The authors generalize the notion by requiring that for all \(i=1,\dots,t\) the entry \(x_i\) of the singular t-uple parametrizes a point sitting in a fixed projective subvariety \(Z_i\). The authors determine some invariants of generalized Kalman varieties of symmetric tensors. They prove that when each \(Z_i\) is irreducible and not contained in the isotropic quadric, then the corresponding generalized Kalman variety is connected of codimension \(\sum_{i=1}^t \delta_i\), where \(\delta_i\) is the codimension of \(Z_i\). The authors also determine a formula for the degree of generalized Kalman varieties, which depend on the degrees of the \(Z_i\)’s. The coefficients in the formula are defined implicitly. The authors study their asymptotic behavior, in some numerical case. When the codimensions \(\delta_i\)’s are \(0\) for all \(i>1\), then the authors provide generating functions for the coefficients in the formula of the degree.

MSC:

14N07 Secant varieties, tensor rank, varieties of sums of powers
15A18 Eigenvalues, singular values, and eigenvectors
15A69 Multilinear algebra, tensor calculus
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Beorchia, Valentina; Galuppi, Francesco; Venturello, Lorenzo, Eigenschemes of ternary tensors, SIAM J. Appl. Algebra Geom., 5, 4, 620-650 (2021) · Zbl 1478.13047
[2] Draisma, Jan; Horobeţ, Emil; Ottaviani, Giorgio; Sturmfels, Bernd; Thomas, Rekha R., The Euclidean distance degree of an algebraic variety, Found. Comput. Math., 16, 1, 99-149 (2016) · Zbl 1370.51020
[3] Ekhad, Shalosh B.; Zeilberger, Doron, On the number of singular vector tuples of hyper-cubical tensors, Pers. J. Shalosh B. Ekhad and Doron Zeilberger (2016) · Zbl 0851.15010
[4] Friedland, Shmuel; Ottaviani, Giorgio, The number of singular vector tuples and uniqueness of best rank-one approximation of tensors, Found. Comput. Math., 14, 6, 1209-1242 (2014) · Zbl 1326.15036
[5] Gel’fand, Israel M.; Kapranov, Mikhail M.; Zelevinsky, Andrey V., Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory & Applications (1994), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 0827.14036
[6] Grayson, Daniel, Stillman, Michael, 1997. Macaulay 2-a system for computation in algebraic geometry and commutative algebra.
[7] Hartshorne, Robin, Algebraic Geometry, Graduate Texts in Mathematics, vol. 52 (1977), Springer-Verlag: Springer-Verlag New York-Heidelberg · Zbl 0367.14001
[8] (Hogben, Leslie, Handbook of Linear Algebra. Handbook of Linear Algebra, Discrete Mathematics and Its Applications (Boca Raton) (2007), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL) · Zbl 1122.15001
[9] Holme, Audun, The geometric and numerical properties of duality in projective algebraic geometry, Manuscr. Math., 61, 2, 145-162 (1988) · Zbl 0657.14033
[10] Huang, Hang, Equations of Kalman varieties, Proc. Am. Math. Soc. (2020)
[11] MacMahon, Percy Alexander, Combinatory Analysis, vol. 1 (1915), Cambridge University Press · JFM 45.1271.01
[12] Ottaviani, Giorgio; Sturmfels, Bernd, Matrices with eigenvectors in a given subspace, Proc. Am. Math. Soc., 141, 4, 1219-1232 (2013) · Zbl 1272.15010
[13] Ottaviani, Giorgio; Shahidi, Zahra, Tensors with eigenvectors in a given subspace, Rend. Circ. Mat. Palermo, II. Ser., 1-12 (2021) · Zbl 1486.14072
[14] Ottaviani, Giorgio; Sodomaco, Luca; Ventura, Emanuele, Asymptotics of degrees and ED degrees of Segre products, Adv. Appl. Math., 130, Article 102242 pp. (2021) · Zbl 1473.14100
[15] Pantone, Jay, The asymptotic number of simple singular vector tuples of a cubical tensor, Online J. Anal. Comb., 12, 11 (2017) · Zbl 1376.65054
[16] Pólya, George, Induction and Analogy in Mathematics. Mathematics and Plausible Reasoning, vol. I (1954), Princeton University Press: Princeton University Press Princeton, N. J. · Zbl 0056.24101
[17] Qi, Liqun; Luo, Ziyan, Tensor Analysis. Spectral Theory and Special Tensors (2017), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, PA · Zbl 1370.15001
[18] Raichev, Alexander; Wilson, Mark C., Asymptotics of coefficients of multivariate generating functions: improvements for smooth points, Electron. J. Comb., 15, 1, Article 89 pp. (2008) · Zbl 1165.05309
[19] Sam, Steven V., Equations and syzygies of some Kalman varieties, Proc. Am. Math. Soc., 140, 12, 4153-4166 (2012) · Zbl 1281.14041
[20] Sodomaco, Luca, The product of the eigenvalues of a symmetric tensor, Linear Algebra Appl., 554, 224-248 (2018) · Zbl 1401.15026
[21] Sodomaco, Luca, The Distance Function from the Variety of partially symmetric rank-one Tensors (2020), Università degli Studi di Firenze, Available at · Zbl 1453.65046
[22] Stanley, Richard P., Enumerative Combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics, vol. 49 (2012), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1247.05003
[23] Teixera Turatti, Ettore, On tensors that are determined by their singular tuples (2021)
[24] Vakil, Ravi, The rising sea. Foundations of algebraic geometry (2017)
[25] Wilf, Herbert S., generatingfunctionology (1990), Academic Press, Inc.: Academic Press, Inc. Boston, MA · Zbl 0689.05001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.