On total reality of meromorphic functions
Annales de l'Institut Fourier, Volume 57 (2007) no. 6, pp. 2015-2030.

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.

On montre que, si tous les points critiques d’une fonction méromorphe de degré au plus quatre sur une courbe algébrique réelle de genre arbitraire sont réels, alors la fonction est conjugée à une fonction méromorphe réelle par un automorphisme projectif approprié de l’image.

DOI: 10.5802/aif.2321
Classification: 14P05, 14P25
Keywords: Total reality, meromorphic function, real curves on ellipsoid, K3-surface
Mot clés : réalité totale, fontion méromorphe, courbes réelles sur un ellipsoide, surface K3
Degtyarev, Alex 1; Ekedahl, Torsten 2; Itenberg, Ilia 3; Shapiro, Boris 2; Shapiro, Michael 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)
@article{AIF_2007__57_6_2015_0,
     author = {Degtyarev, Alex and Ekedahl, Torsten and Itenberg, Ilia and Shapiro, Boris and Shapiro, Michael},
     title = {On total reality of meromorphic functions},
     journal = {Annales de l'Institut Fourier},
     pages = {2015--2030},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {6},
     year = {2007},
     doi = {10.5802/aif.2321},
     zbl = {1131.14059},
     mrnumber = {2377894},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2321/}
}
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
SP  - 2015
EP  - 2030
VL  - 57
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2321/
DO  - 10.5802/aif.2321
LA  - en
ID  - AIF_2007__57_6_2015_0
ER  - 
%0 Journal Article
%A Degtyarev, Alex
%A Ekedahl, Torsten
%A Itenberg, Ilia
%A Shapiro, Boris
%A Shapiro, Michael
%T On total reality of meromorphic functions
%J Annales de l'Institut Fourier
%D 2007
%P 2015-2030
%V 57
%N 6
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2321/
%R 10.5802/aif.2321
%G en
%F AIF_2007__57_6_2015_0
Degtyarev, Alex; Ekedahl, Torsten; Itenberg, Ilia; Shapiro, Boris; Shapiro, Michael. On total reality of meromorphic functions. Annales de l'Institut Fourier, Volume 57 (2007) no. 6, pp. 2015-2030. doi : 10.5802/aif.2321. http://archive.numdam.org/articles/10.5802/aif.2321/

[1] Barth, W.; Peters, C.; de Ven, A. Van Compact Complex Surfaces, Springer-Verlag, 1984 | MR | Zbl

[2] Bourbaki, N. Éléments de mathématique. Fasc. XXXIV, Groupes et algèbres de Lie (Actualités Scientifiques et Industrielles), Volume 1337, Hermann, Paris (1968) (Chap. 4-6) | MR | Zbl

[3] Ekedahl, T.; Shapiro, B.; Shapiro, M. First steps towards total reality of meromorphic functions (submitted to Moscow Mathematical Journal) | Zbl

[4] Eremenko, A.; Gabrielov, A. Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry, Ann. of Math.(2), Volume 155 (2002) no. 1, pp. 105-129 | DOI | MR | Zbl

[5] Eremenko, A.; Gabrielov, A.; Shapiro, M.; Vainshtein, A. Rational functions and real Schubert calculus (math.AG/0407408) | Zbl

[6] Gudkov, D.; Shustin, E. Classification of nonsingular eighth-order curves on an ellipsoid. (Russian), Methods of the qualitative theory of differential equations (1980), pp. 104-107 (Gor’kov. Gos. Univ., Gorki.) | MR

[7] Kharlamov, V.; Sottile, F. Maximally inflected real rational curves, Mosc. Math. J. 3 (2003) no. 3, p. 947-987, 1199–1200 | MR | Zbl

[8] Mukhin, E.; Tarasov, V.; Varchenko, A. The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz (preprint math.AG/0512299)

[9] Nikulin, V. V. Integer quadratic forms and some of their geometrical applications, Izv. Akad. Nauk SSSR, Ser. Mat, Volume 43 (1979) no. 1, pp. 111-177 English transl. in Math. USSR–Izv. vol 43 (1979), 103–167 | MR | Zbl

[10] Osserman, B. Linear series over real and p-adic fields, Proc. AMS, Volume 134 (2005) no. 4, pp. 989-993 | DOI | MR | Zbl

[11] Ruffo, J.; Sivan, Y.; Soprunova, E.; Sottile, F. Experimentation and conjectures in the real Schubert calculus for flag manifolds Preprint (2005), math.AG/0507377 | Zbl

[12] Sottile, F. (website - www.expmath.org/extra/9.2/sottile)

[13] Sottile, F. Enumerative geometry for real varieties, Proc. of Symp. Pur. Math., Volume 62 (1997) no. 1, pp. 435-447 | MR | Zbl

[14] Sottile, F. Enumerative geometry for the real Grassmannian of lines in projective space, Duke Math J., Volume 87 (1997), pp. 59-85 | DOI | MR | Zbl

[15] Sottile, F. The special Schubert calculus is real, Electronic Res. Ann. of the AMS, Volume 5 (1999) no. 1, pp. 35-39 | MR | Zbl

[16] Sottile, F. Real Schubert calculus: polynomial systems and a conjecture of Shapiro and Shapiro, Experiment. Math., Volume 9 (2000) no. 2, pp. 161-182 | MR | Zbl

[17] Verschelde, J. Numerical evidence for a conjecture in real algebraic geometry, Experiment. Math., Volume 9 (2000) no. 2, pp. 183-196 | MR | Zbl

[18] Walker, R. J.; Press, Princeton University Algebraic Curves (Princeton Mathematical Series), Volume 13, Princeton, N. J. (1950), pp. x+201 | MR | Zbl

Cited by Sources: