Geometric conditions for strict submultiplicativity of rank and border rank

Abstract

The X-rank of a point p in projective space is the minimal number of points of an algebraic variety X whose linear span contains p. This notion is naturally submultiplicative under tensor product. We study geometric conditions that guarantee strict submultiplicativity. We prove that in the case of points of rank two, strict submultiplicativity is entirely characterized in terms of the trisecant lines to the variety. Moreover, we focus on the case of curves: we prove that for curves embedded in an even-dimensional projective space, there are always points for which strict submultiplicativity occurs, with the only exception of rational normal curves.

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References

  1. 1.

    Ådlandsvik, B.: Joins and higher secant varieties. Mathematica Scandinavica 61, 213–222 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Allman, E.S., Rhodes, J.A.: Phylogenetic ideals and varieties for the general Markov model. Adv. Appl. Math. 40(2), 127–148 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Le Barz, P.: Sur les espaces multisécants aux courbes algébriques. Manuscr. Math. 119(4), 433–452 (2006)

    MATH  Article  Google Scholar 

  4. 4.

    Buczyńska, W., Buczyński, J.: Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes. J. Algebr. Geom. 23(1), 63–90 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Ballico, E., Bernardi, A., Chiantini, L.: On the dimension of contact loci and the identifiability of tensors. Arkiv för Matematik 56(2), 265–283 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Ballico, E., Bernardi, A., Christandl, M., Gesmundo, F.: On the partially symmetric rank of tensor products of W-states and other symmetric tensors. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30, 93–124 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Bernardi, A., Brachat, J., Comon, P., Mourrain, B.: Multihomogeneous polynomial decomposition using moment matrices. In: Proc. 36th Int. Symp. Symb. Alg. Comp.—ISSAC’11. ACM Press (2011)

  8. 8.

    Ballico, E., Bernardi, A., Gesmundo, F.: A note on the cactus rank for Segre–Veronese varieties. J. Algebra 526, 6–11 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Bernardi, A., Carlini, E., Catalisano, M.V., Gimigliano, A., Oneto, A.: The Hitchhiker guide to: secant varieties and tensor decomposition. Mathematics 6(12), 314 (2018)

    MATH  Article  Google Scholar 

  10. 10.

    Beauville, A.: Complex Algebraic Surfaces, Student Texts, vol. 34. London Mathematical Society, London (1996)

    Google Scholar 

  11. 11.

    Bertin, M.-A.: On the singularities of the trisecant surface to a space curve. Le Matematiche (Catania) 53(3), 15–22 (1998)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Bernardi, A.: Ideals of varieties parameterized by certain symmetric tensors. J. Pure Appl. Alg. 212(6), 1542–1559 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Bernardi, A., Gimigliano, A., Idà, M.: Computing symmetric rank for symmetric tensors. J. Symb. Comp. 46, 34–53 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Buczyński, J., Ginensky, A., Landsberg, J.M.: Determinantal equations for secant varieties and the Eisenbud–Koh–Stillman conjecture. J. Lond. Math. Soc. 88(1), 1–24 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Buczyński, J., Postinghel, E., Rupniewski, F.: On Strassen’s rank additivity for small three-way tensors. SIAM J. Matrix Anal. Appl. 41(1), 106–133 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Bernardi, A., Ranestad, K.: On the cactus rank of cubic forms. J. Symb. Comp. 50, 291–297 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Chiantini, L., Ciliberto, C.: Weakly defective varieties. Trans. Am. Math. Soc. 454(1), 151–178 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Chiantini, L., Ciliberto, C.: On the concept of \(k\)-secant order of a variety. J. Lond. Math. Soc. 72(2), 151–178 (2006)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Carlini, E., Catalisano, M.V., Chiantini, L.: Progress on the symmetric Strassen conjecture. J. Pure Appl. Algebra 219(8), 3149–3157 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Carlini, E., Catalisano, M.V., Chiantini, L., Geramita, A.V., Woo, Y.: Symmetric tensors: rank, Strassen’s conjecture and e-computability. Ann. Sc. Norm. Super. Pisa Cl. Sci. XVIII(1), 363–390 (2018)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Carlini, E., Catalisano, M.V., Geramita, A.V.: The solution to the Waring problem for monomials and the sum of coprime monomials. J. Algebra 370, 5–14 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Carlini, E., Catalisano, M.V., Oneto, A.: Waring loci and the Strassen conjecture. Adv. Math. 314, 630–662 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Chen, L., Friedland, S.: The tensor rank of tensor product of two three-qubit W states is eight. Lin. Alg. Appl. 543, 1–16 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Christandl, M., Gesmundo, F., Jensen, A.K.: Border rank is not multiplicative under the tensor product. SIAM J. Appl. Alg. Geom. 3, 231–255 (2019)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Chiantini, L.: Quantum physics and geometry. In: Ballico, E., Bernardi, A., Carusotto, I., Mazzucchi, S., Moretti, V. (eds.) Hilbert Functions and Tensor Analysis, pp. 125–151. Springer, Cham (2019)

    Google Scholar 

  26. 26.

    Catalano-Johnson, M.L.: The homogeneous ideals of higher secant varieties. J. Pure Appl. Alg. 158(2–3), 123–129 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Christandl, M., Jensen, A.K., Zuiddam, J.: Tensor rank is not multiplicative under the tensor product. Lin. Alg. Appl. 543, 125–139 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Clebsch, A.: Uber Curven fierter Ordnung. J. Reine Angew. Math. 59, 125–145 (1861)

    MathSciNet  Google Scholar 

  29. 29.

    Comas, G., Seiguer, M.: On the rank of a binary form. Found. Comp. Math. 11(1), 65–78 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Dür, W., Vidal, G., Cirac, J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62(6), 062314, 12 (2000)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Eisenbud, D.: Commutative Algebra: with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)

    Google Scholar 

  32. 32.

    Gałązka, M.: Multigraded apolarity. arXiv:1601.06211 (2016)

  33. 33.

    Geramita, A.V.: Inverse systems of fat points: waring’s problem, secant varieties of Veronese varieties and parameter spaces for Gorenstein ideals. The curves seminar at Queen’s 10, 2–114 (1996)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Gallet, M., Ranestad, K., Villamizar, N.: Varieties of apolar subschemes of toric surfaces. Arkiv för Matematik 56(1), 73–99 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Hartshorne, R.: Algebraic Geometry, Graduate Texts in Mathematics. Springer, Berlin (1977)

    Google Scholar 

  36. 36.

    Harris, J.: Algebraic Geometry. A First Course, Graduate Texts in Mathematics, vol. 133. Springer, New York (1992)

    Google Scholar 

  37. 37.

    Iarrobino, A.: Power Sums, Gorenstein Algebras, and Determinantal Loci. Springer, Berlin (1999)

    Google Scholar 

  38. 38.

    Kaji, H.: On the tangentially degenerate curves. J. Lond. Math. Soc. (2) 33(3), 430–440 (1986)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Kogan, I.A., Maza, M.M.: Computation of canonical forms for ternary cubics. In: Proc. 2002 Int. Symp. Symb. Alg. Comp., ACM, pp. 151–160 (2002)

  40. 40.

    Landsberg, J.M.: Tensors: Geometry and Applications, vol. 128. American Mathematical Society, Providence (2012)

    Google Scholar 

  41. 41.

    Landsberg, J.M.M., Michałek, M.: Abelian tensors. J. de Mathématiques Pures et Appliquées 108(3), 333–371 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

    Landsberg, J.M., Ottaviani, G.: Equations for secant varieties of Veronese and other varieties. Ann. Mat. Pura Appl. 192(4), 569–606 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Landsberg, J.M., Teitler, Z.: On the ranks and border ranks of symmetric tensors. Found. Comp. Math. 10(3), 339–366 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Milne, J.S.: Abelian varieties. Course Notes. http://www.jmilne.org/math/CourseNotes/AV.pdf (2008). Accessed 21 May 2020

  45. 45.

    Palatini, F.: Sulla rappresentazione delle forme ed in particolare della cubica quinaria con la somme di potenze di forme lineari. Atti R. Accad. Sc. Torino 38, 43–50 (1902–1903)

  46. 46.

    Palatini, F.: Sulle superficie algebriche i cui \(S_h(h+1)\)-seganti non riempiono lo spazio ambiente. Atti Accad. Torino 41, 634–640 (1906)

    MATH  Google Scholar 

  47. 47.

    Ranestad, K., Schreyer, F.-O.: On the rank of a symmetric form. J. Algebra 346(1), 340–342 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    Russo, F.: On the Geometry of Some Special Projective Varieties. Lecture Notes of the Unione Matematica Italiana, vol. 18. Springer, Berlin (2016)

    Google Scholar 

  49. 49.

    Schönhage, A.: Partial and total matrix multiplication. SIAM J. Comput. 10(3), 434–455 (1981)

    MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Shitov, Y.: Counterexamples to Strassen’s direct sum conjecture. Acta Math. 222(2), 363–379 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Strassen, V.: Gaussian elimination is not optimal. Numer. Math. 13(4), 354–356 (1969)

    MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    Strassen, V.: Vermeidung von Divisionen. J. Reine Angew. Math. 264, 184–202 (1973)

    MathSciNet  MATH  Google Scholar 

  53. 53.

    Strassen, V.: Relative bilinear complexity and matrix multiplication. J. Reine Angew. Math. 375(376), 406–443 (1987)

    MathSciNet  MATH  Google Scholar 

  54. 54.

    Sidman, J., Van Tuyl, A.: Multigraded regularity: syzygies and fat points. Contrib. Algebra Geom. 47(1), 1–22 (2006)

    MathSciNet  MATH  Google Scholar 

  55. 55.

    Sylvester, J.J.: LX. On a remarkable discovery in the theory of canonical forms and of hyperdeterminants. Lond., Edinb., Dublin Philos. Mag. J. Sci. 2(12), 391–410 (1851)

    Article  Google Scholar 

  56. 56.

    Sylvester, J.J.: On the principles of the calculus of forms. Cambridge and Dublin Math. J., pp. 52–97 (1852)

  57. 57.

    Teitler, Z.: Sufficient conditions for Strassen’s additivity conjecture. Ill. J. Math. 59(4), 1071–1085 (2015)

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgements

E.B. and A.B. acknowledge financial support from GNSAGA of INDAM (Italy). F.G. acknowledges financial support from the VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059). A.O. acknowledges financial support from the Alexander von Humboldt-Stiftung via a Humboldt Research Fellowship for Postdoctoral Researchers (April 2019–March 2021) at OVGU Magdeburg (Germany). This collaboration started while F.G., A.O. and E.V. were visiting University of Trento for a Research in Pairs program at CIRM Trento in July 2018, continued while A.B. and F.G. were visiting the Institute for Computational and Experimental Research in Mathematics in Providence, RI, in Fall 2018, and was completed while A.B., F.G. and A.O. were visiting University of Pavia for the XXI Congresso dell’Unione Matematica Italiana, in September 2019. We thank CIRM, ICERM and UMI for providing good research environments to work on this Project.

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Correspondence to Emanuele Ventura.

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To Giorgio Ottaviani, on the occasion of his 60th birthday.

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Ballico, E., Bernardi, A., Gesmundo, F. et al. Geometric conditions for strict submultiplicativity of rank and border rank. Annali di Matematica (2020). https://doi.org/10.1007/s10231-020-00991-6

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Keywords

  • Rank
  • Border rank
  • Tensor product
  • Segre product
  • Secant variety

Mathematics Subject Classification

  • 15A69
  • 14N05
  • 14H99