Rational functions and real Schubert calculus
Authors:
A. Eremenko, A. Gabrielov, M. Shapiro and A. Vainshtein
Journal:
Proc. Amer. Math. Soc. 134 (2006), 949-957
MSC (2000):
Primary 14P05; Secondary 26C15
DOI:
https://doi.org/10.1090/S0002-9939-05-08048-2
Published electronically:
July 25, 2005
MathSciNet review:
2196025
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We single out some problems of Schubert calculus of subspaces of codimension 
that have the property that all their solutions are real whenever the data are real. Our arguments explore the connection between subspaces of codimension 
and rational functions of one variable.
- 1. 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. MR 1888795, https://doi.org/10.2307/3062151
- 2. Alexandre Eremenko and Andrei Gabrielov, The Wronski map and Grassmannians of real codimension 2 subspaces, Comput. Methods Funct. Theory 1 (2001), no. 1, [On table of contents: 2002], 1–25. MR 1931599, https://doi.org/10.1007/BF03320973
- 3. Ph. Griffiths and J. Harris, Principles of algebraic geometry, Willey-Interscience, NY, 1978. MR 0507725 (80b:14001)
- 4. Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
- 5. Laurent Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs, vol. 6, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001. Translated from the 1998 French original by John R. Swallow; Cours Spécialisés [Specialized Courses], 3. MR 1852463
- 6. J. Ruffo, Y. Sivan, E. Soprounova and F. Sottile, Experimentation and conjectures in the real Schubert calculus, work in progress, www.math.umass.edu/~ sottile/pages/Flags.
- 7. Frank Sottile, The special Schubert calculus is real, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 35–39. MR 1679451, https://doi.org/10.1090/S1079-6762-99-00058-X
- 8. Frank Sottile, Real Schubert calculus: polynomial systems and a conjecture of Shapiro and Shapiro, Experiment. Math. 9 (2000), no. 2, 161–182. MR 1780204
- 9. Frank Sottile, Enumerative real algebraic geometry, Algorithmic and quantitative real algebraic geometry (Piscataway, NJ, 2001) DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 60, Amer. Math. Soc., Providence, RI, 2003, pp. 139–179. MR 1995019
- 10. Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 14P05, 26C15
Retrieve articles in all journals with MSC (2000): 14P05, 26C15
Additional Information
A. Eremenko
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-2067
Email:
eremenko@math.purdue.edu
A. Gabrielov
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-2067
Email:
agabriel@math.purdue.edu
M. Shapiro
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email:
mshapiro@math.msu.edu
A. Vainshtein
Affiliation:
Department of Computer Science, University of Haifa, Mount Carmel, 31905 Haifa, Israel
Email:
alek@cs.haifa.ac.il
DOI:
https://doi.org/10.1090/S0002-9939-05-08048-2
Received by editor(s):
August 25, 2004
Received by editor(s) in revised form:
October 29, 2004
Published electronically:
July 25, 2005
Additional Notes:
The authors were supported by NSF grants DMS-0100512 and DMS-0244421 (A.E.), DMS-0200861 and DMS-0245628 (A.G.), and DMS-0401178 (M.S.); and by the BSF grant 2002375 (M.S. and A.V.) and by the Institute of Quantum Science, MSU (M.S.).
Communicated by:
John R. Stembridge
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
