Numerical evidence for a conjecture in real algebraic geometry

Experimental Mathematics

Volumn 9, Issue 2, 2000, Pages 183-196

  Verschelde, Jan ^a  

^a UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

Author keywords

Cheater's homotopy; Control theory; Flat deformation; Numerical Schubert calculus; Pole placement problem; Polyhedral continuation; Polynomial system; SAGBI homotopy

Indexed keywords

EID: 0034348609     PISSN: 10586458     EISSN: 1944950X     Source Type: Journal    
DOI: 10.1080/10586458.2000.10504645     Document Type: Article

References (50)

1. E. L. Allgower and K. Georg, Numerical continuation methods: an introduction, Springer Ser. Comput. Math. 13, Springer, Berlin, 1990.
2. E. L. Allgower and K. Georg, “Numerical path following”, pp. 3-207 in Handbook of numerical analysis, V: Techniques of scientific computing (Part 2), edited by P. G. Ciarlet and J. L. Lions, North-Holland, Amsterdam, 1997.
3. T. Ando, “Totally positive matrices”, Linear Algebra Appl. 90 (1987), 165-219.
4. D. N. Bernshtem, “The number of roots of a system of equations”, Funktsional. Anal, i Prilozhen. 9: 3 (1975), 1-4. In Russian; translated in Functional Anal. Appl., 9: 3 (1975), 183-185.
5. A. T. Bharucha- Reid and M. Sambandham, Random polynomials, Academic Press Inc., Orlando, FL, 1986.
6. L. Blum, F. Cucker, M. Shub, and S. Smale, Complexity and real computation, Springer, New York, 1998.
7. R. W. Brockett and C. I. Byrnes, “Multivariable Nyquist criteria, root loci, and pole placement: a geometric viewpoint”, IEEE Trans. Automat. Control 26: 1 (1981), 271-284.
8. C. I. Byrnes, “Pole assignment by output feedback”, pp. 31-78 in Three decades of mathematical system theory, edited by H. Nijmacher and J. M. Schumacher, Lecture Notes in Control and Inform. Sci. 135, Springer, Berlin, 1989.
9. D. Cox, J. Little, and D. O’Shea, Ideals, varieties, and algorithms: An introduction to computational algebraic geometry and commutative algebra, 2nd ed., Undergraduate Texts in Math., Springer, New York, 1997.
10. D. Cox, J. Little, and D. O’Shea, Using algebraic geometry, Graduate Texts in Math. 185, Springer, New York, 1998.
11. F. Cucker and M.-F. Roy, “A theorem on random polynomials and some consequences in average complexity”, J. Symbolic Comput. 10: 5 (1990), 405-409.
12. J. P. Dedieu and M. Shub, “Newton’s method for overdetermined systems of equations”, Math. Comp. 69: 231 (2000), 1099-1115.
13. A. Edelman, T. A. Arias, and S. T. Smith, “The geometry of algorithms with orthogonality constraints”, SIAM J. Matrix Anal. Appl. 20: 2 (1999), 303-353.
14. D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Math. 150, Springer, New York, 1995.
15. J.-C. Faugere, F. Rouillier, and P. Zimmerman, 1998. Personal communication.
16. W. Fulton, Young tableaux, London Math. Soc. Student Texts 35, Cambridge University Press, Cambridge, 1997.
17. C. F. Gerald, Applied numerical analysis, 2nd ed., Addison-Wesley, Reading, MA, 1978.
18. G. H. Golub and C. F. Van Loan, Matrix computations, 3rd ed., Johns Hopkins Univ. Press, Baltimore, 1996.
19. B. Huber and B. Sturmfels, “A polyhedral method for solving sparse polynomial systems”, Math. Comp. 64: 212 (1995), 1541-1555.
20. B. Huber and J. Verschelde, “Polyhedral end games for polynomial continuation”, Numer. Algorithms 18: 1 (1998), 91-108.
21. 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 (2000), 1265-1287.
22. B. Huber, F. Sottile, and B. Sturmfels, “Numerical Schubert calculus”, J. Symbolic Comput. 26: 6 (1998), 767-788.
23. T. Kailath, Linear systems, Prentice-Hall Inc., Englewood Cliffs, NJ, 1980.
24. A. G. Kushnirenko, “Newton polytopes and the Bezout theorem”, Funktsional. Anal, i Prilozhen. 10: 3 (1976), 82-83. In Russian; translated in Functional Anal. Appl. 10: 3 (1976), 233-235.
25. T. Y. Li, “Numerical solution of multivariate polynomial systems by homotopy continuation methods”, pp. 399-436 in Acta numerica, 1997, Cambridge Univ. Press, Cambridge, 1997.
26. T. Y. Li and X. S. Wang, “Nonlinear homotopies for solving deficient polynomial systems with parameters”, SIAM J. Numer. Anal. 29: 4 (1992), 1104-1118.
27. 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 (1989), 1241-1251.
28. C. Loewner, “On totally positive matrices”, Math. Z. 63 (1955), 338-340.
29. A. Morgan, Solving polynomial systems using continuation for engineering and scientific problems, Prentice-Hall Inc., Englewood Cliffs, NJ, 1987.
30. A. Morgan and A. Sommese, “A homotopy for solving general polynomial systems that respects m-homogeneous structures”, Appl. Math. Comput. 24: 2 (1987), 101-113.
31. A. P. Morgan and A. J. Sommese, “Coefficient-parameter polynomial continuation”, Appl. Math. Comput. 29: 2(11) (1989), 123-160. Errata in 51: 2-3 (1992), 207.
32. J. Rosenthal, “On dynamic feedback compensation and compactification of systems”, SIAM J. Control Optim. 32: 1 (1994), 279-296.
33. J. Rosenthal and J. M. Schumacher, “Realization by inspection”, IEEE Trans. Automat. Control 42: 9 (1997), 1257-1263.
34. J. Rosenthal and F. Sottile, “Some remarks on real and complex output feedback”, Systems Control Lett. 33: 2 (1998), 73-80. See http: //www.nd.edu/-rosen/pole for a description of computational aspects of the paper.
35. J. Rosenthal and J. C. Willems, “Open problems in the area of pole placement”, pp. 181-191 in Open problems in mathematical systems and control theory, edited by V. D. Blon- del et al., Comm, and Control Engin. Series, Springer, London, 1999.
36. H. Schubert, “Beziehungen zwischen den linearen Raumen auferlegbaren charakteristischen Bedingungen”, Math. Ann. 38 (1891), 588-602.
37. F. Sottile, “The special Schubert calculus is real”, Electron. Res. Announc. Amer. Math. Soc. 5, 35-39. See http: //www.ams.org/era.
38. F. Sottile, “Real rational curves in Grassmannians”, J. Amer. Math. Soc. 13: 2 (2000), 333-341.
39. F. Sottile, “Real Schubert calculus: polynomial systems and a conjecture of Shapiro 182 and Shapiro", Experiment. Math. 9: 2 (2000), 161-182.
40. H. J. Stetter, “Numerical polynomial algebra”, Technical report, 1998. Notes for the tutorial Interaction between Numerical Analysis and Computer Algebra, held at ISSAC’98, Rostock, Germany.
41. B. Sturmfels, Algorithms in invariant theory, Springer, Vienna, 1993.
42. B. Sturmfels, Grobner bases and convex polytopes, Univ. Lecture Ser. 8, Amer. Math. Soc., Providence, RI, 1996.
43. B. Sturmfels, “Polynomial equations and convex polytopes”, Amer. Math. Monthly 105: 10(1998), 907-922.
44. J. Verschelde, “Algorithm 795: PHC- pack: A general-purpose solver for polynomial systems by homotopy continuation”, ACM Trans. Math. Software 25: 2 (1999), 251-276. See http: //www.math.msu.edu/~jan.
45. J. Verschelde, P. Verlinden, and R. Cools, “Homotopies exploiting Newton polytopes for solving sparse polynomial systems”, SIAM J. Numer. Anal. 31: 3 (1994), 915-930.
46. J. Verschelde, K. Gatermann, and R. Cools, “Mixed-volume computation by dynamic lifting applied to polynomial system solving”, Discrete Comput. Geom. 16: 1 (1996), 69-112.
47. C. W. Wampler, “Bezout number calculations for multi-homogeneous polynomial systems”, Appl. Math. Comput. 51: 2-3 (1992), 143-157.
48. C. W. Wampler, A. P. Morgan, and A. J. Sommese, “Numerical continuation methods for solving polynomial systems arising in kinematics”, ASME J. of Mechanical Design 112: 1 (1990), 59-68.
49. C. W. Wampler, A. P. Morgan, and A. J. Sommese, “Complete solution of the nine-point path synthesis problem for four-bar linkages”, ASME J. of Mechanical Design 114: 1 (1992), 153-159.
50. A. M. Whitney, “A reduction theorem for totally positive matrices”, J. Analyse Math. 2 (1952), 88-92.