Advanced search
Publication Cover
Experimental Mathematics Volume 15, 2006 - Issue 2
Submit an article Journal homepage
26
Views
12
CrossRef citations to date
0
Altmetric
Original Article

Experimentation and Conjectures in the Real Schubert Calculus for Flag Manifolds

Jim Ruffo Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
,
Yuval Sivan Department of Mathematics, University of Massachusetts, Lederle Graduate Research Tower, Amherst, MA, 01003, USA
,
Evgenia Soprunova Department of Mathematics, University of Massachusetts, Lederle Graduate Research Tower, Amherst, MA, 01003, USA
&
Frank Sottile Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
Pages 199-221 | Published online: 30 Jan 2011

  • 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) 155:1 (2002), 105–129.
  • A. Eremenko, A. Gabrielov, M. Shapiro, and A. Vainshtein. "Rational Functions and Real Schubert Calculus." To appear in Proc. AMS, 2006.
  • J.-C. Faugère, P. Gianni, D. Lazard, and T. Mora. "Efficient Computation of Zero-Dimensional Groebner Bases by Change of Ordering." J. Symb. Comp. 16 (1993), 329–344.
  • W. Fulton. Young Tableaux. Cambridge: Cambridge University Press, 1997.
  • G.-M. Greuel, G. Pfister, and H. Schönemann. "Singular 2.0: A Computer Algebra System for Polynomial Computations." Available online: (http://www.singular.uni-kl.de). http://www.singular.uni-kl.de
  • J. Harris. "Galois Groups of Enumerative Problems." Duke Math. J. 46 (1979), 685–724.
  • I. V. Itenberg, V.M. Kharlamov, and E. I. Shustin. "Logarithmic Equivalence of the Welschinger and the Gromov–Witten Invariants." Uspekhi Mat. Nauk 59:6 (360) (2004), 85–110.
  • V. Kharlamov and F. Sottile. "Maximally Inflected Real Rational Curves." Moscow Math. J. 3, 2003.
  • S. Kleiman. "The Transversality of a General Translate." Compositio Math. 28 (1974), 287–297.
  • Grigory Mikhalkin. "Enumerative Tropical Algebraic Geometry in ℝ2." J. Amer. Math. Soc. 18:2 (2005), 313–377 (electronic).
  • E. Mukhin and A. Varchenko. "Critical Points of Master Functions and Flag Varieties." Commun. Contemp. Math. 6:1 (2004), 111–163.
  • Kevin Purbhoo and Frank Sottile. The Recursive Nature of the Cominuscule Schubert Calculus. Manuscript, 2006.
  • J. Rosenthal and F. Sottile. "Some Remarks on Real and Complex Output Feedback." Systems & Control Lett. 33:2 (1998), 73–80.
  • Claus Scheiderer, "Sums of Squares of Regular Functions on Real Algebraic Varieties." Trans. Amer. Math. Soc. 352:3 (2000), 1039–1069.
  • V. Sedykh and B Shapiro. "On Two Conjectures Concerning Convex Curves." Internat. J. Math. 16:10 (2005), 1157–1173.
  • E. Soprunova and F. Sottile. "Lower Bounds for Real Solutions to Sparse Polynomial Systems." To appear in Adv. Math., 2006.
  • F. Sottile. "Pieri's Formula for Flag Manifolds and Schubert Polynomials." Ann. Inst. Fourier 46 (1996), 1–22.
  • F. Sottile. "Enumerative Geometry for Real Varieties." In Algebraic Geometry, Santa Cruz 1995, Proc. Sympos. Pure Math., 62, Part 1, edited by J. Kollár, R. Lazarsfeld, and D. Morrison, pp. 435–447. Providence: Amer. Math. Soc., 1997.
  • F. Sottile. "Enumerative Geometry for the Real Grassmannian of Lines in Projective Space." Duke Math. J. 87:1 (1997), 59–85.
  • F. Sottile. "The Special Schubert Calculus Is Real." ERA of the AMS 5 (1999), 35–39.
  • F. Sottile. "Real Schubert Calculus: Polynomial Systems and a Conjecture of Shapiro and Shapiro." Exper. Math. 9 (2000), 161–182.
  • F. Sottile. "Some Real and Unreal Enumerative Geometry for Flag Manifolds." Mich. Math. J. 48 (2000), 573–592.
  • F. Sottile. "From Enumerative Geometry to Solving Systems of Polynomial Equations." In Computations in Algebraic Geometry with Macaulay 2, Algorithms Comput. Math., 8, pp. 101–129. Berlin: Springer, 2002.
  • F. Sottile. "Enumerative Real Algebraic Geometry." In Algorithmic and Quantitative Real Algebraic Geometry, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 60, pp. 139–179. Providence: Amer. Math. Soc., 2003.
  • R. Vakil, "Schubert Induction." To appear in Annals of Math., 2006.
  • J. Verschelde, "Numerical Evidence of a Conjecture in Real Algebraic Geometry." Exper. Math. 9 (2000), 183–196.
  • Jean-Yves Welschinger, "Invariants of Real Rational Symplectic 4-Manifolds and Lower Bounds in Real Enumerative Geometry." C. R. Math. Acad. Sci. Paris 336:4 (2003), 341–344.
  • D. Eisenbud and J. Harris. "Divisors on General Curves and Cuspidal Rational Curves." Invent. Math. 74 (1983), 371–418.
  • D. Eisenbud and J. Harris. "When Ramification Points Meet." Invent. Math. 87 (1987), 485–493.
  • A. Eremenko and A. Gabrielov. "Degrees of Real Wronski Maps." Discrete Comput. Geom. 28:3 (2002), 331–347.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Order Reprints Request Corporate Permissions

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

Request Academic Permissions

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.

Related research

People also read lists articles that other readers of this article have read.

Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine.

Cited by lists all citing articles based on Crossref citations.
Articles with the Crossref icon will open in a new tab.