Abstract
Formulating a Schubert problem as solutions to a system of equations in either Plücker space or local coordinates of a Schubert cell typically involves more equations than variables. We present a novel primal-dual formulation of any Schubert problem on a Grassmannian or flag manifold as a system of bilinear equations with the same number of equations as variables. This formulation enables numerical computations in the Schubert calculus to be certified using algorithms based on Smale’s \(\alpha \)-theory.
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References
D. Armentano, Complexity of path-following methods for the eigenvalue problem, Found. Comput. Math. 14 (2014), no. 2, 185–236.
D.J. Bates, J.D. Hauenstein, A.J. Sommese, and C.W. Wampler, Bertini: Software for numerical algebraic geometry, Available at bertini.nd.edu. doi:10.7274/R0H41PB5.
C. Beltrán and A. Leykin, Certified numerical homotopy tracking, Exp. Math. 21 (2012), no. 1, 69–83.
C. Beltrán and A. Leykin, Robust certified numerical homotopy tracking, Found. Comput. Math. 13 (2013), no. 2, 253–295.
L. Blum, F. Cucker, M. Shub, and S. Smale, Complexity and real computation, Springer-Verlag, New York, 1998, With a foreword by Richard M. Karp
J.-P. Dedieu and M. Shub, Multihomogeneous newton methods, Math. Comp. 69 (2000), no. 231, 1071–1098.
A. Eremenko and A. Gabrielov, Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry, Ann. of Math. (2) 155 (2002), no. 1, 105–129.
W. Fulton, Young tableaux, London Mathematical Society Students Texts, 35, Cambridge University Press, Cambridge, 1997.
L. García-Puente, N. Hein, C. Hillar, A. Martín del Campo, J. Ruffo, F. Sottile, and Z. Teitler, The Secant Conjecture in the real Schubert Calculus, Exper. Math. 21 (2012), no. 3, 252–265.
J.D. Hauenstein, I. Haywood, and A.C. Liddell, Jr., An a posteriori certification algorithm for Newton homotopies, Proc. ISSAC 2014, ACM, 2014, pp. 248–255.
J.D. Hauenstein, N. Hein, C. Hillar, A. Martín del Campo, Frank Sottile, and Zach Teitler, The Monotone Secant Conjecture in the real Schubert calculus, MEGA11, Stockholm, 2011.
J.D. Hauenstein and F. Sottile, Algorithm 921: alphaCertified: Certifying solutions to polynomial systems, ACM Trans. Math. Softw. 38, (2012) no. 4, 28.
J.D. Hauenstein and F. Sottile, alphaCertified: Software for certifying numerical solutions to polynomial equations, Available at www.math.tamu.edu/~sottile/research/stories/alphaCertified.
N. Hein, C. Hillar, and F. Sottile, Lower bounds in real Schubert calculus, São Paulo Journal of Mathematics 7 (2013), no. 1, 33–58.
B. Huber, F. Sottile, and B. Sturmfels, Numerical Schubert calculus, J. Symb. Comp. 26 (1998), no. 6, 767–788.
S.L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287–297.
A. Leykin and F. Sottile, Galois groups of Schubert problems via homotopy computation, Math. Comp. 78 (2009), no. 267, 1749–1765.
A. Martín del Campo and F. Sottile, Experimentation in the Schubert calculus, arXiv:1308.3284, 2013.
E. Mukhin and V. Tarasov, Lower bounds for numbers of real solutions in problems of Schubert calculus, 2014, arXiv:1404.7194.
E. Mukhin, V. Tarasov, and A. Varchenko, The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, Ann. of Math. (2) 170 (2009), no. 2, 863–881.
E. Mukhin, V. Tarasov, and A. Varchenko, Schubert calculus and representations of the general linear group, J. Amer. Math. Soc. 22 (2009), no. 4, 909–940.
J. Ruffo, Y. Sivan, E. Soprunova, and F. Sottile, Experimentation and conjectures in the real Schubert calculus for flag manifolds, Exper. Math. 15 (2006), no. 2, 199–221.
H. Schubert, Kalkul der abzählenden Geometrie, Springer-Verlag, 1879, reprinted with an introduction by S. Kleiman, 1979.
H. Schubert, Anzahl-Bestimmungen für lineare Räume beliebiger Dimension, Acta. Math. 8 (1886), 97–118.
H. Schubert, Die \(n\) -dimensionalen Verallgemeinerungen der fundamentalen Anzahlen unseres Raums, Math. Ann. 26 (1886), 26–51, (dated 1884).
H. Schubert, Losüng des Charakteritiken-Problems für lineare Räume beliebiger Dimension, Mittheil. Math. Ges. Hamburg (1886), 135–155, (dated 1885).
M. Shub and S. Smale, Complexity of Bézout’s theorem. I. Geometric aspects, J. Amer. Math. Soc. 6 (1993), no. 2, 459–501.
S. Smale, Newton’s method estimates from data at one point, The merging of disciplines: new directions in pure, applied, and computational mathematics, Springer, New York, 1986, pp. 185–196.
A.J. Sommese and C. W. Wampler, II, The numerical solution of systems of polynomials, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.
F. Sottile, Real Schubert calculus: Polynomial systems and a conjecture of Shapiro and Shapiro, Exper. Math. 9 (2000), 161–182.
F. Sottile, Frontiers of reality in Schubert calculus, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 1, 31–71.
F. Sottile, Real solutions to equations from geometry, University Lecture Series, vol. 57, American Mathematical Society, 2011.
F. Sottile, R. Vakil, and J. Verschelde, Solving Schubert problems with Littlewood-Richardson homotopies, Proc. ISSAC 2010 (Stephen M. Watt, ed.), ACM, 2010, pp. 179–186.
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Communicated by Teresa Krick.
Research of Hauenstein supported in part by NSF Grant DMS-1262428 and DARPA YFA. Research of Hein and Sottile supported in part by NSF Grant DMS-0915211.
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Hauenstein, J.D., Hein, N. & Sottile, F. A Primal-Dual Formulation for Certifiable Computations in Schubert Calculus. Found Comput Math 16, 941–963 (2016). https://doi.org/10.1007/s10208-015-9270-z
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DOI: https://doi.org/10.1007/s10208-015-9270-z