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On the Rank of a Binary Form

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Abstract

We describe in the space of binary forms of degree d the strata of forms having a given rank. We also give a simple algorithm for determining the rank of a given form.

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References

  1. J. Alexander, A. Hirschowitz, Polynomial interpolation in several variables, J. Algebr. Geom. 4(2), 201–222 (1995).

    MATH  MathSciNet  Google Scholar 

  2. M. Boij, E. Carlini, A.V. Geramita, Monomials as sums of powers: the real binary case. arXiv:1005.3050v1, 2010.

  3. J. Buczynski, A. Ginesky, J.M. Landsberg, Determinantal equations for secant varieties and the Eisenbud–Koh–Stillman conjecture. arXiv:0909.4865, 2010.

  4. M.V. Catalisano, A.V. Geramita, A. Gimigliano, Ranks of tensors, secant varieties of Segre varieties and fat points, Linear Algebra Appl. 355, 263–285 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  5. A. Causa, R. Re, On the maximum rank of a real binary form. arXiv:1006.5127, 2010.

  6. G. Chen, R.K. Brylinski (eds.), Mathematics of Quantum Computation. Computational Mathematics Series (Chapman & Hall/CRC, Boca Raton, 2002).

    Google Scholar 

  7. C. Ciliberto, Geometric aspects of polynomial interpolation in more variables and of Waring’s problem, in European Congress of Mathematics, vol. I, Barcelona, 2000. Progr. Math., vol. 201 (Birkhäuser, Basel, 2001), pp. 289–316.

    Google Scholar 

  8. P. Common, G. Ottaviani, On the typical rank of real binary forms. arXiv:0909.4865, 2009.

  9. E.B. Elliott, An Introduction to the Algebra of Quantics (Clarendon Press, Oxford, 1895).

    Google Scholar 

  10. J.H. Grace, A. Young, The Algebra of Invariants (Univ. Press, Cambridge, 1903).

    MATH  Google Scholar 

  11. S. Gundelfinger, Zur théorie der binaren formen, J. Reine Angew. Math. 100, 413–424 (1886).

    Google Scholar 

  12. J. Harris, Algebraic Geometry, A First Course. Graduate Texts in Mathematics (Springer, Berlin, 1992).

    MATH  Google Scholar 

  13. 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.

    MATH  Google Scholar 

  14. J.P.S. Kung, Canonical forms for binary forms of even degree, in Invariant Theory. Lecture Notes in Math., vol. 1278 (Springer, Berlin, 1987), pp. 52–61.

    Chapter  Google Scholar 

  15. J.P.S. Kung, G.-C. Rota, The invariant theory of binary forms, Bull. Am. Math. Soc. (N.S.) 10(1), 27–85 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  16. J.M. Landsberg, Geometry and the complexity of matrix multiplication, Bull. Am. Math. Soc. (N.S.) 45(2), 247–284 (2008).

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. B. Reznick, Homogeneous polynomial solutions to constant coefficient PDE’s, Adv. Math. 117(2), 179–192 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  19. J.J. Sylvester, An essay on canonical forms, supplement to a sketch of a memoir on elimination, transformation and canonical forms, in Collected Works, vol. I (Cambridge University Press, Cambridge, 1904), pp. 203–216.

    Google Scholar 

  20. J.J. Sylvester, Sketch of a memoir on elimination, transformation and canonical forms, in Collected Works, vol. I (Cambridge University Press, Cambridge, 1904), pp. 184–197.

    Google Scholar 

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Correspondence to Gonzalo Comas.

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Communicated by Marie-Francoise Roy.

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Comas, G., Seiguer, M. On the Rank of a Binary Form. Found Comput Math 11, 65–78 (2011). https://doi.org/10.1007/s10208-010-9077-x

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  • DOI: https://doi.org/10.1007/s10208-010-9077-x

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