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.
Keywords: Total reality, meromorphic function, real curves on ellipsoid, K3-surface
Mot clés : réalité totale, fontion méromorphe, courbes réelles sur un ellipsoide, surface K3
@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://www.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://www.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://www.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, Tome 57 (2007) no. 6, pp. 2015-2030. doi : 10.5802/aif.2321. http://www.numdam.org/articles/10.5802/aif.2321/
[1] Compact Complex Surfaces, Springer-Verlag, 1984 | MR | Zbl
[2] 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] First steps towards total reality of meromorphic functions (submitted to Moscow Mathematical Journal) | Zbl
[4] 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] Rational functions and real Schubert calculus (math.AG/0407408) | Zbl
[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
[7] Maximally inflected real rational curves, Mosc. Math. J. 3 (2003) no. 3, p. 947-987, 1199–1200 | MR | Zbl
[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, Volume 43 (1979) no. 1, pp. 111-177 English transl. in Math. USSR–Izv. vol 43 (1979), 103–167 | MR | Zbl
[10] Linear series over real and -adic fields, Proc. AMS, Volume 134 (2005) no. 4, pp. 989-993 | DOI | MR | Zbl
[11] Experimentation and conjectures in the real Schubert calculus for flag manifolds Preprint (2005), math.AG/0507377 | Zbl
[12]
(website - www.expmath.org/extra/9.2/sottile)[13] Enumerative geometry for real varieties, Proc. of Symp. Pur. Math., Volume 62 (1997) no. 1, pp. 435-447 | MR | Zbl
[14] Enumerative geometry for the real Grassmannian of lines in projective space, Duke Math J., Volume 87 (1997), pp. 59-85 | DOI | MR | Zbl
[15] The special Schubert calculus is real, Electronic Res. Ann. of the AMS, Volume 5 (1999) no. 1, pp. 35-39 | MR | Zbl
[16] 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] Numerical evidence for a conjecture in real algebraic geometry, Experiment. Math., Volume 9 (2000) no. 2, pp. 183-196 | MR | Zbl
[18] Algebraic Curves (Princeton Mathematical Series), Volume 13, Princeton, N. J. (1950), pp. x+201 | MR | Zbl
Cité par Sources :