Abstract
The symmetric rank problem of 2 × 2 × ⋯ × 2 symmetric tensors are related to Waring’s problem of binary forms. In this paper, we characterize the symmetric rank of 2 × 2 × ⋯ × 2 symmetric tensors when the symmetric rank is 1 and 2 in arbitrary characteristic (either zero or strictly larger than the order of a tensor). Moreover, we characterize the symmetric rank for 2 × 2 × 2 symmetric tensors over the fields where the characteristic is zero or larger than three or \(\mathbb{F}_{2}\) or \(\mathbb{F}_{3}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bender M.R., Faugère J.-C., Perret L., Tsigaridas E., A superfast randomized algorithm to decompose binary forms. In: Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation, New York: ACM, 2016, 79–86
Bernardi A., Gimigliano A., Ida M., Computing symmetric rank for symmetric tensors. J. Symbolic Comput., 2011, 46(1): 34–53
Beylkin G., Mohlenkamp M.J., Algorithms for numerical analysis in high dimensions. SIAM J. Sci. Comput., 2005, 26(6): 2133–2159
Brachat J., Comon P., Mourrain B., Tsigaridas E., Symmetric tensor decomposition. Linear Algebra Appl., 2010, 433(11–12): 1851–1872
Comas G., Seiguer M., On the rank of a binary form. Found. Comput. Math., 2011, 11(1): 65–78
Comon P., Tensor Decompositions—State of the Art and Applications. Oxford: Oxford Univ. Press, 2002
Comon P., Golub G., Lim L-H., Mourrain B., Symmetric tensors and symmetric tensor rank. SIAM J. Matrix Anal. Appl., 2008, 30(3): 1254–1279
Comon P., Rajih M., Blind identification of under-determined mixtures based on the characteristic function. Signal Process, 2006, 86(9): 2271–2281
De Silva V., Lim L.-H., Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl., 2008, 31(3): 1084–1127
Gelfand I.M., Kapranov M.M., Zelevinsky A.V., Discriminants, Resultants, and Multidimensional Determinants. Boston, MA: Birkhäuser Boston, Inc., 2008
Hackbusch W., Khoromskij B.N., Tyrtyshnikov E.E., Hierarchical Kronecker tensor-product approximations. J. Numer. Math., 2005, 13(2): 119–156
Helmke U., Waring’s problem for binary forms. J. Pure Appl. Algebra, 1992, 80(1): 29–45
Hitchcock F.L., The expression of a tensor or a polyadic as a sum of products. J. Math. Phys., 1927, 6(1–4): 164–189
Hitchcock F.L., Multiple invariants and generalized rank of a P-way matrix or tensor. J. Math. Phys., 1928, 7(1–4): 39–79
Iarrobino A., Kanev V., Power Sums, Gorenstein Algebras, and Determinantal Loci. Lecture Notes in Math., 1721, Berlin: Springer-Verlag, 1999
Kolda T.G., Bader B.W., Tensor decompositions and applications. SIAM Rev., 2009, 51(3): 455–500
Landsberg J.M., Teitler Z., On the ranks of tensors and symmetric tensors. Found. Comput. Math., 2010, 10(3): 339–366
Reznick B., Sums of even powers of real linear forms. Mem. Amer. Math. Soc., 1992, 96(463): viii+155 pp.
Reznick B., On the Length of Binary Forms. Dev. Math., 31, New York: Springer, 2013
Stavrou S.G., Canonical forms of 2 × 2 × 2 and 2 × 2 × ⋯ × 2 symmetric tensors over prime fields. Linear Multilinear Algebra, 2014, 62(9): 1169–1186
Stavrou S.G., Canonical forms of order-k (k = 2, 3,4) symmetric tensors of format 3 × ⋯ × 3 over prime fields. Linear and Multilinear Algebra, 2015, 63(6): 1111–1124
Sylvester J.J., Sur une extension d’un théorème de Clebsch relatif aux courbes du quatrième degré. C. R. Math. Acad. Sci. Paris, 1886, 102: 1532–1534
Sylvester J.J., An essay on canonical forms, supplement to a sketch of a memoir on elimination, transformation and canonical forms. In: The Collected Mathematical Papers of James Joseph Sylvester, Vol. I (1837–1853), Cambridge: Cambridge University Press, 1904, 203–216
Sylvester J.J., On a remarkable discovery in the theory of canonical forms and of hyper-determinants. In: The Collected Mathematical Papers of James Joseph Sylvester, Vol. I (1837–1853), Cambridge: Cambridge University Press, 1904, 265–283
Tokcan N., On the Waring rank of binary forms. Linear Algebra Appl., 2017, 524: 250–262
Vasilescu M.A.O., Terzopoulos D., Multilinear analysis of image ensembles: TensorFaces. In: Computer Vision—ECCV 2002, Lecture Notes in Computer Sciences, 2350, Berlin: Springer, 2002, 447–460
Wang H.C., Ahuja N., Facial expression decomposition. In: Proceedings of the 9th IEEE International Conference on Computer Vision (ICCV), Piscataway, NJ: IEEE, 2003, 958–965
Zheng B.D., Huang R.G., Song X.Y., Xu J.L., On Comon’s conjecture over arbitrary fields. Linear Algebra Appl., 2020, 587: 228–242
Acknowledgements
This work was supported by National Natural Science Foundation of China (No. 11701075).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest The authors declare no conflict of interest.
Rights and permissions
About this article
Cite this article
Song, X., Zheng, B., Huang, R. et al. The Symmetric Rank of 2 × 2 × ⋯ × 2 Symmetric Tensors over an Arbitrary Field. Front. Math (2024). https://doi.org/10.1007/s11464-020-0165-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11464-020-0165-1