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.
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.
@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},
volume = {57},
year = {2007},
pages = {2015-2030},
doi = {10.5802/aif.2321},
zbl = {1131.14059},
mrnumber = {2377894},
language = {en},
url = {http://dml.mathdoc.fr/item/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, Tome 57 (2007) pp. 2015-2030. doi : 10.5802/aif.2321. http://gdmltest.u-ga.fr/item/AIF_2007__57_6_2015_0/
[1] Compact Complex Surfaces, Springer-Verlag (1984) | MR 749574 | Zbl 0718.14023
[2] Éléments de mathématique. Fasc. XXXIV, Groupes et algèbres de Lie, Hermann, Paris (Actualités Scientifiques et Industrielles) Tome 1337 (1968) (Chap. 4-6) | MR 240238 | Zbl 0186.33001
[3] First steps towards total reality of meromorphic functions (submitted to Moscow Mathematical Journal) | Zbl 1126.14064
[4] Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry, Ann. of Math.(2), Tome 155 (2002) no. 1, pp. 105-129 | Article | MR 1888795 | Zbl 0997.14015
[5] Rational functions and real Schubert calculus (math.AG/0407408) | Zbl 1110.14052
[6] 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 726230
[7] Maximally inflected real rational curves, Mosc. Math. J. 3 (2003) no. 3, p. 947-987, 1199–1200 | MR 2078569 | Zbl 1052.14070
[8] The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz (preprint math.AG/0512299)
[9] Integer quadratic forms and some of their geometrical applications, Izv. Akad. Nauk SSSR, Ser. Mat, Tome 43 (1979) no. 1, pp. 111-177 (English transl. in Math. USSR–Izv. vol 43 (1979), 103–167) | MR 525944 | Zbl 0408.10011
[10] Linear series over real and -adic fields, Proc. AMS, Tome 134 (2005) no. 4, pp. 989-993 | Article | MR 2196029 | Zbl 1083.14034
[11] Experimentation and conjectures in the real Schubert calculus for flag manifolds (Preprint (2005), math.AG/0507377) | Zbl 1111.14049
[12] (website - www.expmath.org/extra/9.2/sottile)
[13] Enumerative geometry for real varieties, Proc. of Symp. Pur. Math., Tome 62 (1997) no. 1, pp. 435-447 | MR 1492531 | Zbl 0890.14030
[14] Enumerative geometry for the real Grassmannian of lines in projective space, Duke Math J., Tome 87 (1997), pp. 59-85 | Article | MR 1440063 | Zbl 0986.14033
[15] The special Schubert calculus is real, Electronic Res. Ann. of the AMS, Tome 5 (1999) no. 1, pp. 35-39 | MR 1679451 | Zbl 0921.14037
[16] Real Schubert calculus: polynomial systems and a conjecture of Shapiro and Shapiro, Experiment. Math., Tome 9 (2000) no. 2, pp. 161-182 | MR 1780204 | Zbl 0997.14016
[17] Numerical evidence for a conjecture in real algebraic geometry, Experiment. Math., Tome 9 (2000) no. 2, pp. 183-196 | MR 1780205 | Zbl 1054.14080
[18] Algebraic Curves, Princeton, N. J. (Princeton Mathematical Series) Tome 13 (1950), pp. x+201 | MR 33083 | Zbl 0039.37701