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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
Published electronically:
July 25, 2005
MathSciNet review:
2196025
Full-text PDF Free Access
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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
(2003c:58028), http://dx.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, 1–25. MR 1931599
(2003h:26022)
- 3.
Phillip
Griffiths and Joseph
Harris, Principles of algebraic geometry, Wiley-Interscience
[John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725
(80b:14001)
- 4.
Witold
Hurewicz and Henry
Wallman, Dimension Theory, Princeton Mathematical Series, v.
4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
(3,312b)
- 5.
Laurent
Manivel, Symmetric functions, Schubert polynomials and degeneracy
loci, SMF/AMS Texts and Monographs, vol. 6, American Mathematical
Society, Providence, RI, 2001. Translated from the 1998 French original by
John R. Swallow; Cours Spécialisés [Specialized Courses], 3.
MR
1852463 (2002h:05161)
- 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 (electronic). MR 1679451
(2000c:14074), http://dx.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
(2001e:14054)
- 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
(2004j:14065)
- 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 (2000k:05026)
- 1.
- A. Eremenko and A. Gabrielov, Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry, Ann. Math., 155 (2002), 105-129. MR 1888795 (2003c:58028)
- 2.
- A. Eremenko and A. Gabrielov, Wronski map and Grassmannians of real codimension 2 subspaces, Computational Methods and Function Theory, 1 (2001) 1-25. MR 1931599 (2003h:26022)
- 3.
- Ph. Griffiths and J. Harris, Principles of algebraic geometry, Willey-Interscience, NY, 1978. MR 0507725 (80b:14001)
- 4.
- W. Hurewicz and H. Wallman, Dimension theory, Princeton Univ. Press, 1948. MR 0006493 (3,312b)
- 5.
- L. Manivel, Symmetric functions, Schubert Polynomials and degeneracy loci, AMS, Providence, RI, 2001. MR 1852463 (2002h:05161)
- 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.
- F. Sottile, The special Schubert calculus is real, Electron. Res. Announc. AMS, 5 (1999) 35-39. MR 1679451 (2000c:14074)
- 8.
- F. Sottile, Real Schubert calculus: polynomial systems and a conjecture of Shapiro and Shapiro, Experimental Math., 9 (2000) 161-182. MR 1780204 (2001e:14054)
- 9.
- F. Sottile, Enumerative real algebraic geometry, DIMACS Series in Discrete Math. and Computer Sci., vol. 60, AMS, Providence, RI (2003), 139-179. MR 1995019 (2004j:14065)
- 10.
- R. Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, 1999. MR 1676282 (2000k:05026)
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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:
http://dx.doi.org/10.1090/S0002-9939-05-08048-2
PII:
S 0002-9939(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.
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