On total reality of meromorphic functions

Alex Degtyarev[1]; Torsten Ekedahl[2]; Ilia Itenberg[3]; Boris Shapiro[2]; Michael Shapiro[4]

  • [1] Bilkent University Department of Mathematics Bilkent, Ankara 06533 (Turkey)
  • [2] Stockholm University Department of Mathematics SE-106 91 Stockholm (Sweden)
  • [3] Université Louis Pasteur IRMA 7 rue René Descartes 67084 Strasbourg cedex (France)
  • [4] Michigan State University Department of Mathematics East Lansing MI 48824-1027 (USA)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 6, page 2015-2030
  • ISSN: 0373-0956

Abstract

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We show that, if a meromorphic function of degree at most four on a real algebraic curve of an arbitrary genus has only real critical points, then it is conjugate to a real meromorphic function by a suitable projective automorphism of the image.

How to cite

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Degtyarev, Alex, et al. "On total reality of meromorphic functions." Annales de l’institut Fourier 57.6 (2007): 2015-2030. <http://eudml.org/doc/10285>.

@article{Degtyarev2007,
abstract = {We show that, if a meromorphic function of degree at most four on a real algebraic curve of an arbitrary genus has only real critical points, then it is conjugate to a real meromorphic function by a suitable projective automorphism of the image.},
affiliation = {Bilkent University Department of Mathematics Bilkent, Ankara 06533 (Turkey); Stockholm University Department of Mathematics SE-106 91 Stockholm (Sweden); Université Louis Pasteur IRMA 7 rue René Descartes 67084 Strasbourg cedex (France); Stockholm University Department of Mathematics SE-106 91 Stockholm (Sweden); Michigan State University Department of Mathematics East Lansing MI 48824-1027 (USA)},
author = {Degtyarev, Alex, Ekedahl, Torsten, Itenberg, Ilia, Shapiro, Boris, Shapiro, Michael},
journal = {Annales de l’institut Fourier},
keywords = {Total reality; meromorphic function; real curves on ellipsoid; K3-surface; total reality; real curves on ellipsoid, -surface},
language = {eng},
number = {6},
pages = {2015-2030},
publisher = {Association des Annales de l’institut Fourier},
title = {On total reality of meromorphic functions},
url = {http://eudml.org/doc/10285},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Degtyarev, Alex
AU - Ekedahl, Torsten
AU - Itenberg, Ilia
AU - Shapiro, Boris
AU - Shapiro, Michael
TI - On total reality of meromorphic functions
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 6
SP - 2015
EP - 2030
AB - We show that, if a meromorphic function of degree at most four on a real algebraic curve of an arbitrary genus has only real critical points, then it is conjugate to a real meromorphic function by a suitable projective automorphism of the image.
LA - eng
KW - Total reality; meromorphic function; real curves on ellipsoid; K3-surface; total reality; real curves on ellipsoid, -surface
UR - http://eudml.org/doc/10285
ER -

References

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  2. N. Bourbaki, Groupes et algèbres de Lie, 1337 (1968), Hermann, Paris Zbl0186.33001MR240238
  3. T. Ekedahl, B. Shapiro, M. Shapiro, First steps towards total reality of meromorphic functions Zbl1126.14064
  4. A. Eremenko, 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), 105-129 Zbl0997.14015MR1888795
  5. A. Eremenko, A. Gabrielov, M. Shapiro, A. Vainshtein, Rational functions and real Schubert calculus Zbl1110.14052
  6. D. Gudkov, E. Shustin, Classification of nonsingular eighth-order curves on an ellipsoid. (Russian), Methods of the qualitative theory of differential equations (1980), 104-107 MR726230
  7. V. Kharlamov, F. Sottile, Maximally inflected real rational curves, Mosc. Math. J. 3 (2003), 947-987, 1199–1200 Zbl1052.14070MR2078569
  8. E. Mukhin, V. Tarasov, A. Varchenko, The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz Zbl1213.14101
  9. V. V. Nikulin, Integer quadratic forms and some of their geometrical applications, Izv. Akad. Nauk SSSR, Ser. Mat 43 (1979), 111-177 Zbl0408.10011MR525944
  10. B. Osserman, Linear series over real and p -adic fields, Proc. AMS 134 (2005), 989-993 Zbl1083.14034MR2196029
  11. J. Ruffo, Y. Sivan, E. Soprunova, F. Sottile, Experimentation and conjectures in the real Schubert calculus for flag manifolds Zbl1111.14049
  12. F. Sottile 
  13. F. Sottile, Enumerative geometry for real varieties, Proc. of Symp. Pur. Math. 62 (1997), 435-447 Zbl0890.14030MR1492531
  14. F. Sottile, Enumerative geometry for the real Grassmannian of lines in projective space, Duke Math J. 87 (1997), 59-85 Zbl0986.14033MR1440063
  15. F. Sottile, The special Schubert calculus is real, Electronic Res. Ann. of the AMS 5 (1999), 35-39 Zbl0921.14037MR1679451
  16. F. Sottile, Real Schubert calculus: polynomial systems and a conjecture of Shapiro and Shapiro, Experiment. Math. 9 (2000), 161-182 Zbl0997.14016MR1780204
  17. J. Verschelde, Numerical evidence for a conjecture in real algebraic geometry, Experiment. Math. 9 (2000), 183-196 Zbl1054.14080MR1780205
  18. R. J. Walker, Algebraic Curves, 13 (1950), PressPrinceton UniversityP. U., Princeton, N. J. Zbl0039.37701MR33083

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