×

Ranks and symmetric ranks of cubic surfaces. (English) Zbl 1444.14091

Author’s abstract: We study cubic surfaces as symmetric tensors of format \(4\times 4\times 4\). We consider the non-symmetric tensor rank and the symmetric Waring rank of cubic surfaces, and show that the two notions coincide over the complex numbers. The corresponding algebraic problem concerns border ranks. We show that the non-symmetric border rank coincides with the symmetric border rank for cubic surfaces. As part of our analysis, we obtain minimal ideal generators for the symmetric analogue to the secant variety from the salmon conjecture. We also give a test for symmetric rank given by the non-vanishing of certain discriminants. The results extend to order three tensors of all sizes, implying the equality of rank and symmetric rank when the symmetric rank is at most seven, and the equality of border rank and symmetric border rank when the symmetric border rank is at most five. We also study real ranks via the real substitution method.

MSC:

14N05 Projective techniques in algebraic geometry
14N07 Secant varieties, tensor rank, varieties of sums of powers
15A69 Multilinear algebra, tensor calculus
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alexander, J.; Hirschowitz, A., Polynomial interpolation in several variables, J. Algebraic Geom., 4, 201-222 (1995) · Zbl 0829.14002
[2] Ballico, E.; Bernardi, A.; Chiantini, L.; Guardo, E., Bounds on the tensor rank, Ann. Mat. Pura Appl., 197, 6, 1771-1785 (2018) · Zbl 1411.14058
[3] Banchi, M., Rank and border rank of real ternary cubics, Boll. Unione Mat. Ital., 8, 1, 65-80 (2015) · Zbl 1332.15058
[4] Bates, D.; Hauenstein, J.; Sommese, A.; Wampler, C., Bertini: software for numerical algebraic geometry (2006), Available at
[5] Bates, D.; Oeding, L., Toward a salmon conjecture, Exp. Math., 20, 3, 358-370 (2011) · Zbl 1262.14056
[6] Bernardi, A.; Blekherman, G.; Ottaviani, G., On real typical ranks, Boll. Unione Mat. Ital., 11, 3, 293-307 (2018) · Zbl 1403.15018
[7] Bosma, W.; Cannon, J.; Playoust, C., The Magma algebra system. I. The user language, J. Symb. Comput., 24, 235-265 (1997) · Zbl 0898.68039
[8] Brachat, J.; Comon, P.; Mourrain, B.; Tsigaridas, E., Symmetric tensor decomposition, Linear Algebra Appl., 433, 11-12, 1851-1872 (2010) · Zbl 1206.65141
[9] Bruce, J. W.; Wall, C. T.C., On the classification of cubic surfaces, J. Lond. Math. Soc. (2), 19, 2, 245-256 (1979) · Zbl 0393.14007
[10] Carlini, E.; Guo, C.; Ventura, E., Real and complex Waring rank of reducible cubic forms, J. Pure Appl. Algebra, 220, 11, 3692-3701 (2016) · Zbl 1338.11092
[11] Comon, P.; Golub, G.; Lim, L-H.; Mourrain, B., Symmetric tensors and symmetric tensor rank, SIAM J. Matrix Anal. Appl., 30, 3, 1254-1279 (2008) · Zbl 1181.15014
[12] Friedland, S., Remarks on the symmetric rank of symmetric tensors, SIAM J. Matrix Anal. Appl., 37, 1, 320-337 (2016) · Zbl 1382.15040
[13] Friedland, S.; Gross, E., A proof of the set-theoretic version of the salmon conjecture, J. Algebra, 356, 1, 374-379 (2012) · Zbl 1258.14001
[14] Gelfand, I.; Kapranov, M.; Zelevinsky, A., Discriminants, Resultants, and Multidimensional Determinants (1994), Birkhäuser · Zbl 0827.14036
[15] Grayson, D.R. Stillman, M.E., Macaulay2, a software system for research in algebraic geometry. Available at https://faculty.math.illinois.edu/Macaulay2/.
[16] Landsberg, J. M., Tensors: Geometry and Applications, Graduate Studies in Mathematics, 128 (2012), American Mathematical Society: American Mathematical Society Providence RI · Zbl 1238.15013
[17] Landsberg, J. M., Geometry and Complexity Theory, Cambridge Studies in Advanced Mathematics (2017), Cambridge University Press · Zbl 1387.68002
[18] Michalek, M.; Moon, H., Spaces of sums of powers and real rank boundaries, Beitr. Algebra Geom., 59, 4, 645-663 (2018) · Zbl 1403.14090
[19] Schmitt, A., Quaternary cubic forms and projective algebraic threefolds, Enseign. Math. (2), 43, 3-4, 253-270 (1997) · Zbl 0926.14017
[20] Segre, B., The Non-singular Cubic Surfaces (1942), Oxford University Press: Oxford University Press Oxford · JFM 68.0358.01
[21] Seigal, A.; Sturmfels, B., Real rank two geometry, J. Algebra, 484, 310-333 (2017) · Zbl 1401.14229
[22] Shitov, Y., A counterexample to Comon’s conjecture, SIAM J. Appl. Algebra Geom., 2, 3, 428-443 (2018) · Zbl 1401.15004
[23] Sturmfels, B., Solving Systems of Polynomial Equations, CBMS Regional Conference Series in Mathematics, vol. 97 (2002) · Zbl 1101.13040
[24] Sturmfels, B., The Hurwitz form of a projective variety, J. Symb. Comput., 79, 186-196 (2017) · Zbl 1359.14043
[25] The Sage Developers, SageMath, the Sage mathematics software system (version 7.4) (2016), Available at
[26] Vakil, R., The rising sea: foundations of algebraic geometry (2017)
[27] Zhang, X.; Huang, Z-H.; Qi, L., Comon’s conjecture, rank decomposition, and symmetric rank decomposition of symmetric tensors, SIAM J. Matrix Anal. Appl., 37, 4, 1719-1728 (2016) · Zbl 1349.15004
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.