Abstract
Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering the singularities of the hypersurface defined by the polynomial. We obtain normal forms for polynomials of border rank up to five, and compute or bound the ranks of several classes of polynomials, including monomials, the determinant, and the permanent.
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References
J. Alexander, A. Hirschowitz, Polynomial interpolation in several variables, J. Algebr. Geom. 4(2), 201–222 (1995). MR 96f:14065.
M.C. Brambilla, G. Ottaviani, On the Alexander-Hirschowitz theorem, J. Pure Appl. Algebra 212(5), 1229–1251 (2008). MR MR2387598 (2008m:14104).
P. Bürgisser, M. Clausen, M.A. Shokrollahi, Algebraic Complexity Theory, Grundlehren der Mathematischen Wissenschaften. Fundamental Principles of Mathematical Sciences, vol. 315 (Springer, Berlin, 1997). With the collaboration of Thomas Lickteig. MR 99c:68002.
G. Comas, M. Seiguer, On the rank of a binary form. arXiv:math.AG/0112311 (2001).
P. Comon, B. Mourrain, Decomposition of quantics in sums of powers of linear forms, Signal Process. 53(2) (1996).
A. Geramita, Inverse systems of fat points: Waring’s problem, secant varieties of Veronese varieties and parameter spaces for Gorenstein ideals, in The Curves Seminar at Queen’s, Vol. X, Kingston, ON, 1995. Queen’s Papers in Pure and Appl. Math., vol. 102 (Queen’s University, Kingston, 1996), pp. 2–114.
P. Griffiths, J. Harris, Principles of Algebraic Geometry. Wiley Classics Library (Wiley, New York, 1994). Reprint of the 1978 original. MR 95d:14001.
L. Gurvits, Ryser (or polarization) formula for the permanent is essentially optimal: the Waring rank approach. Preprint, 2008.
J. Harris, Algebraic Geometry. Graduate Texts in Mathematics, vol. 133 (Springer, New York, 1995). A first course, Corrected reprint of the 1992 original. MR MR1416564 (97e:14001).
A. Iarrobino, V. Kanev, Power Sums, Gorenstein Algebras, and Determinantal loci. Lecture Notes in Mathematics, vol. 1721 (Springer, Berlin, 1999). Appendix C by Iarrobino and Steven L. Kleiman.
V. Kanev, Chordal varieties of Veronese varieties and catalecticant matrices, J. Math. Sci. 94(1), 1114–1125 (1999). Algebraic geometry, 9. MR MR1703911 (2001b:14078).
J.M. Landsberg, The Geometry of Tensors with Applications, book in preparation.
J.M. Landsberg, G. Ottaviani, Equations for secant varieties of Veronese varieties, in preparation.
L.-H. Lim, V. De Silva, Tensor rank and the ill-posedness of the best low-rank approximation problem, SIAM J. Matrix Anal. Appl. 31(3), 1084–1127 (2008).
F.S. Macaulay, The Algebraic Theory of Modular Systems. Cambridge Mathematical Library (Cambridge University Press, Cambridge, 1994). Revised reprint of the 1916 original, With an introduction by Paul Roberts.
S. Mukai, Fano 3-folds, in Complex Projective Geometry, Trieste, 1989/Bergen, 1989. London Math. Soc. Lecture Note Ser., vol. 179 (Cambridge University Press, Cambridge, 1992), pp. 255–263. MR MR1201387 (94a:14042).
G. Ottaviani, An invariant regarding Waring’s problem for cubic polynomials, Nagoya Math. J. 193, 95–110 (2009). MR MR2502909.
K. Ranestad, F.-O. Schreyer, Varieties of sums of powers, J. Reine Angew. Math. 525, 147–181 (2000). MR MR1780430 (2001m:14009).
H. Ryser, Combinatorial Mathematics. The Carus Mathematical Monographs, vol. 14 (Mathematical Association of America, Washington, 1963). MR MR0150048 (27 #51).
A. Schinzel, On a decomposition of polynomials in several variables, J. Théor. Nr. Bordx. 14(2), 647–666 (2002).
A. Schinzel, On a decomposition of polynomials in several variables. II, Colloq. Math. 92(1), 67–79 (2002).
R.P. Stanley, Hilbert functions of graded algebras, Adv. Math. 28(1), 57–83 (1978).
J. Weyman, Cohomology of Vector Bundles and Syzygies. Cambridge Tracts in Mathematics, vol. 149 (Cambridge University Press, Cambridge, 2003). MR MR1988690 (2004d:13020).
J. Yerushalmy, On the configuration of the nine base points of a pencil of equianharmonic cubics, Am. J. Math. 54(2), 279–284 (1932). MR MR1506892.
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Communicated by Peter Bürgisser.
J.M. Landsberg supported by NSF grant DMS-0805782.
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Landsberg, J.M., Teitler, Z. On the Ranks and Border Ranks of Symmetric Tensors. Found Comput Math 10, 339–366 (2010). https://doi.org/10.1007/s10208-009-9055-3
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DOI: https://doi.org/10.1007/s10208-009-9055-3