no. 6
On total reality of meromorphic functions
[Sur la réalité totale des fonctions méromorphes]
Degtyarev, Alex1 ; Ekedahl, Torsten2 ; Itenberg, Ilia3 ; Shapiro, Boris2 ; Shapiro, Michael4
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, Tome 57 (2007) no. 6, pp. 2015-2030.

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.

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) 
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Degtyarev, Alex; Ekedahl, Torsten; Itenberg, Ilia; Shapiro, Boris; Shapiro, Michael. On total reality of meromorphic functions. Annales de l'Institut Fourier, Tome 57 (2007) no. 6, pp. 2015-2030. doi : 10.5802/aif.2321. https://aif.centre-mersenne.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 (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 (1950), pp. x+201 | MR | Zbl

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