Solving Schubert problems with Littlewood-Richardson homotopies. (English) Zbl 1321.68541
Watt, Stephen M. (ed.), Proceedings of the 35th international symposium on symbolic and algebraic computation, ISSAC 2010, Munich, Germany, July 25–28, 2010. New York, NY: Association for Computing Machinery (ACM) (ISBN 978-1-4503-0150-3). 179-186 (2010).
MSC:
| 68W30 | Symbolic computation and algebraic computation |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 14N15 | Classical problems, Schubert calculus |
Keywords:
Grassmannian; Schubert problems; continuation; geometric Littlewood-Richardson rule; homotopies; numerical Schubert calculus; path following; polynomial systemReferences:
| [1] | R. W. Brockett and C. I. Byrnes. Multivariate Nyquist criteria, root loci, and pole placement: a geometric viewpoint. IEEE Trans. Automat. Control, 26(1):271-284, 1981. · Zbl 0462.93026 |
| [2] | C. I. Byrnes. Pole assignment by output feedback. In H. Nijmacher and J. M. Schumacher, editors, Three Decades of Mathematical Systems Theory, volume 135 of Lecture Notes in Control and Inform. Sci., pages 13-78. Springer-Verlag, 1989. · Zbl 0701.93047 |
| [3] | A. Eremenko and A. Gabrielov. Pole placement by static output feedback for generic linear systems. SIAM J. Control Optim., 41(1):303-312, 2002. 10.1137/S0363012901391913 · Zbl 1031.93086 |
| [4] | W. Fulton. Young Tableau. With Applications to Representation Theory and Geometry. Cambridge University Press, 1997. · Zbl 0878.14034 |
| [5] | D. R. Grayson and M. E. Stillman. Macaulay 2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/. |
| [6] | C. Hillar, L. Garcia-Puente, A. Martin del Campo, J. Ruffo, Z. Teitler, S. L. Johnson, and F. Sottile. Experimentation at the frontiers of reality in Schubert calculus. Contemporary Mathematics, to appear. Preprint arXiv:0906.2497v2 {math.AG}. |
| [7] | B. Huber, F. Sottile, and B. Sturmfels. Numerical Schubert calculus. J. Symbolic Computation, 26(6):767-788, 1998. 10.1006/jsco.1998.0239 · Zbl 1064.14508 |
| [8] | B. Huber and J. Verschelde. Pieri homotopies for problems in enumerative geometry applied to pole placement in linear systems control. SIAM J. Control Optim., 38(4):1265-1287, 2000. 10.1137/S036301299935657X · Zbl 0955.14038 |
| [9] | A. Leykin and F. Sottile. Galois group of Schubert problems via homotopy continuation. Math. Comp., 78(267):1749-1765, 2009. · Zbl 1210.14064 |
| [10] | T. Y. Li, T. Sauer, and J. A. Yorke. The cheater’s homotopy: an efficient procedure for solving systems of polynomial equations. SIAM J. Numer. Anal., 26(5):1241-1251, 1989. 10.1137/0726069 · Zbl 0689.65032 |
| [11] | T. Y. Li, X. Wang, and M. Wu. Numerical Schubert calculus by the Pieri homotopy algorithm. SIAM J. Numer. Anal., 20(2):578-600, 2002. 10.1137/S003614290139175X · Zbl 1018.14018 |
| [12] | A. P. Morgan and A. J. Sommese. Coefficient-parameter polynomial continuation. Appl. Math. Comput., 29(2):123-160, 1989. Errata: Appl. Math. Comput. 51:207(1992). 10.1016/0096-3003(89)90099-4 · Zbl 0664.65049 |
| [13] | M. S. Ravi, J. Rosenthal, and X. Wang. Dynamic pole placement assignment and Schubert calculus. SIAM J. Control and Optimization, 34(3):813-832, 1996. 10.1137/S036301299325270X · Zbl 0856.93043 |
| [14] | H. Schubert. Anzahl-Bestimmungen für lineare Räume beliebiger Dimension. Acta. Math., 8:97-118, 1886. · JFM 18.0632.01 |
| [15] | A. J. Sommese and C. W. Wampler. The Numerical solution of systems of polynomials arising in engineering and science. World Scientific, 2005. · Zbl 1091.65049 |
| [16] | F. Sottile. Pieri’s formula via explicit rational equivalence. Canad. J. Math., 49(6):1281-1298, 1997. · Zbl 0933.14031 |
| [17] | F. Sottile. Real Schubert calculus: polynomial systems and a conjecture of Shapiro and Shapiro. Experiment. Math., 9(2):161-182, 2000. · Zbl 0997.14016 |
| [18] | F. Sottile. Enumerative real algebraic geometry. In S. Basu and L. Gonzalez-Vega, editors, Algorithmic and Quantitative Real Algebraic Geometry, pages 139-180. AMS, 2003. |
| [19] | F. Sottile. Frontiers of reality in Schubert calculus. Bulletin of the AMS, 47(1):31-71, 2010. · Zbl 1197.14052 |
| [20] | R. Vakil. A geometric Littlewood-Richardson rule. Annals of Math., 164(2):376-421, 2006. |
| [21] | J. Verschelde. Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Softw., 25(2):251-276, 1999. Software available at http://www.math.uic.edu/ jan. 10.1145/317275.317286 · Zbl 0961.65047 |
| [22] | J. Verschelde. Numerical evidence for a conjecture in real algebraic geometry. Experimental Mathematics, 9(2):183-196, 2000. · Zbl 1054.14080 |
| [23] | J. Verschelde and Y. Wang. Computing dynamic output feedback laws. IEEE Trans. Automat. Control., 49(8):1393-1397, 2004. · Zbl 1365.93198 |
| [24] | J. Verschelde and Y. Wang. Computing feedback laws for linear systems with a parallel Pieri homotopy. In Y. Yang, editor, Proceedings of the 2004 International Conference on Parallel Processing Workshops, 15-18 August 2004, Montreal, Quebec, Canada. High Performance Scientific and Engineering Computing, pages 222-229. IEEE Computer Society, 2004. 10.1109/ICPPW.2004.34 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
