Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T20:04:20.789Z Has data issue: false hasContentIssue false
  • English
  • Français

Prolongations and Computational Algebra

Published online by Cambridge University Press:  20 November 2018

Jessica Sidman  and
Seth Sullivant
Jessica Sidman
Affiliation:
Department of Mathematics and Statistics, Mount Holyoke College, South Hadley, MA 01075, U.S.A., e-mail: jsidman@mtholyoke.edu
Seth Sullivant
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC, USA, e-mail: smsulli2@ncsu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We explore the geometric notion of prolongations in the setting of computational algebra, extending results of Landsberg and Manivel which relate prolongations to equations for secant varieties. We also develop methods for computing prolongations that are combinatorial in nature. As an application, we use prolongations to derive a new family of secant equations for the binary symmetric model in phylogenetics.

Keywords

13P1014M99
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 4 , 01 August 2009 , pp. 930 - 949
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Aoki, S. and Takemura, A., Minimal basis for a connected Markov chain over 3 × 3 × K contingency tables with fixed two-dimensional marginals. Aust. N. Z N. Z. J. Stat. 45(2003), no. 2, 229–249.Google Scholar
[2] Arbarello, E., Cornalba, M., Griffiths, P. A., and Harris, J., Geometry of Algebraic Curves. Springer-Verlag, New York, 1985.Google Scholar
[3] Bosma, W., Cannon, J., and Playoust, C.. The Magma algebra system. I. The user language. J. Symbolic Logic 24(1997), no. 3-4, 235–265.Google Scholar
[4] Bryant, R. L., Chern, S. S., Gardner, R. B., Goldschmidt, H. L., and Griffiths, P. A., Exterior differential systems. Mathematical Systems Research Institute Publications 18. Springer-Verlag, New York, 1991.Google Scholar
[5] Buczyńska, W. and Wiśniewski, J., On phylogenetic trees – a geometer's view. Preprint, 2006. arXiv:math.AG/0601357Google Scholar
[6] Catalano-Johnson, M., The homogeneous ideals of higher secant varieties. J. Pure Appl. Algebra 158(2001), no. 2-3, 123–129.Google Scholar
[7] Diaconis, P. and Sturmfels, B., Algebraic algorithms for sampling from conditional distributions. Ann. Statis. 26(1998), no. 1, 363–397.Google Scholar
[8] Eisenbud, D., Commutative Algebra: with a View Toward Algebraic Geometry. Graduate Texts in Mathematics 150. Springer-Verlag, New York, 1995.Google Scholar
[9] Fulton, W. and Harris, J., Representation Theory. Graduate Texts in Mathematics 129. Springer-Verlag, New York, 1991.Google Scholar
[10] Griffiths, P., Some aspects of exterior differential systems. Proc. Symnpos Pure Math. 53. American Mathematical Society, Providence, RI, 1991, pp. 151–173.Google Scholar
[11] Grayson, D. and Stillman, M., Macaulay 2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/ Macaulay2/.Google Scholar
[12] Ivey, T. and Landsberg, J. M., Cartan for Beginners: Differential Geometry Via Moving Frames and Exterior Differential Systems. Graduate Studies in Mathematics 61. American Mathematical Society, Providence, RI, 2003.Google Scholar
[13] Landsberg, J. M. and Manivel, L., On the projective geometry of rational homogeneous varieties. Comment. Math. Helv. 78(2003), no. 1, 65–100.Google Scholar
[14] Landsberg, J. M. and Manivel, L., On the ideals of secant varieties of Segre varieties. Found. Comput. Math. 4(2004), no. 4, 397–422.Google Scholar
[15] Simis, A. and Ulrich, B., On the ideal of an embedded join. J. Algebra 226(2000), no. 1, 1–14.Google Scholar
[16] Sturmfels, B., Gröbner Bases and Convex Polytopes. University Lecture Series 8. American Mathematical Society, Providence, RI, 1996.Google Scholar
[17] Sturmfels, B. and Sullivant, S., Toric ideals of phylogenetic invariants. J. Comput. Biology 12(2005), no. 2, 204–228 Google Scholar
[18] Sturmfels, B. and Sullivant, S., Combinatorial secant varieties. Pure Appl. Math. Q. 2(2006), no. 2, 285–309 Google Scholar
[19] Sullivant, S., Combinatorial symbolic powers. J. Algebra 319(2008), no. 1, 115–142.Google Scholar
[20] Vermeire, P., Some results on secant varieties leading to a geometric flip construction. Compositio Math. 125(2001), no. 3, 263–282.Google Scholar
[21] Weyman, J., Cohomology of Vector Bundles and Syzygies. Cambridge Tracts in Mathematics 149. Cambridge University Press, Cambridge, 2003.Google Scholar
[22] Weyl, H., Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, NJ, 1939.Google Scholar