The special Schubert calculus is real

Author:
Frank Sottile

Journal:
Electron. Res. Announc. Amer. Math. Soc. **5** (1999), 35-39

MSC (1991):
Primary 14P99, 14N10, 14M15, 14Q20; Secondary 93B55

DOI:
https://doi.org/10.1090/S1079-6762-99-00058-X

Published electronically:
April 1, 1999

MathSciNet review:
1679451

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Abstract: We show that the Schubert calculus of enumerative geometry is real, for special Schubert conditions. That is, for any such enumerative problem, there exist real conditions for which all the *a priori* complex solutions are real.

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*Pole assignment by output feedback*, in Three Decades of Mathematical Systems Theory, H. Nijmeijer and J. M. Schumacher, eds., vol. 135 of Lecture Notes in Control and Inform. Sci., Springer-Verlag, Berlin, 1989, pp. 31–78.
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*The Stewart-Gough platform of general geometry can have 40 real postures*, in Advances in Robot Kinematics: Analysis and Control, Kluwer Academic Publishers, 1998, pp. 1–10.
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*Divisors on general curves and cuspidal rational curves*, Invent. Math., 74 (1983), pp. 371–418.
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*Private communication*. 1998.
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*Introduction to Intersection Theory in Algebraic Geometry*, CBMS 54, AMS, 1984.
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*Young Tableaux*, Cambridge University Press, 1997.
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*Principles of Algebraic Geometry*, J. Wiley and Sons, 1978.
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*Methods of Algebraic Geometry*, vol. II, Cambridge University Press, 1952.
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*Numerical Schubert calculus*. J. Symb. Comp., 26 (1998), pp. 767–788.
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*The number of conics tangent to 5 given conics: the real case*, Rev. Mat. Univ. Complut. Madrid, 10 (1997), pp. 391–421.
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*Some remarks on real and complex output feedback*, Systems & Control Lett., 33 (1998), pp. 73–80. . For a description of the computational aspects, see http://www.nd.edu/~rosen/pole/.
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*Enumerative geometry for real varieties*, in Algebraic Geometry, Santa Cruz 1995, J. Kollár, R. Lazarsfeld, and D. Morrison, eds., vol. 62, Part 1 of Proc. Sympos. Pure Math., Amer. Math. Soc., 1997, pp. 435–447.
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*Enumerative geometry for the real Grassmannian of lines in projective space*, Duke Math. J., 87 (1997), pp. 59–85.
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*Real enumerative geometry and effective algebraic equivalence*, J. Pure Appl. Alg., 117 & 118 (1997), pp. 601–615. Proc., MEGA’96.
- F. Sottile,
*Real Schubert calculus: Polynomial systems and a conjecture of Shapiro and Shapiro*. MSRI preprint # 1998-066. For an archive of computations and computer algebra scripts, see http://www.math.wisc.edu/~sottile/pages/shapiro/index.html, 1998.
- J. Verschelde,
*Numerical evidence of a conjecture in real algebraic geometry*. MSRI preprint # 1998-064, 1998.

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Additional Information

**Frank Sottile**

Affiliation:
Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720

Address at time of publication:
Department of Mathematics, University of Wisconsin, Van Vleck Hall, 480 Lincoln Drive, Madison, Wisconsin 53706-1388

MR Author ID:
355336

ORCID:
0000-0003-0087-7120

Keywords:
Schubert calculus,
enumerative geometry,
Grassmannian,
pole placement problem

Received by editor(s):
December 20, 1998

Published electronically:
April 1, 1999

Additional Notes:
MSRI preprint # 1998-067.

Research supported by NSF grant DMS-9701755.

Communicated by:
Robert Lazarsfeld

Article copyright:
© Copyright 1999
American Mathematical Society