Some examples for the Poincaré and Painlevé problems. (English) Zbl 1130.34301
Summary: In 1891, Poincaré started a series of three papers in which he tried to answer the following question (cf. [C. R. Acad. Sci. 102, 761–764 (1891; JFM 23.0319.01), Rend. Circ. Mat. Palermo 5, 161–191 (1891; JFM 23.0319.01), ibid. 11, 193–239 (1897;
JFM 28.0292.01)]): “Is it possible to decide if an algebraic differential equation in two variables is algebraically integrable?” (in the sense that it has a rational first integral).
More or less at the same time P. Painlevé asked the following question: “Is it possible to recognize the genus of the general solution of an algebraic differential equation in two variables which has a rational first integral?”.
In this paper we provide counterexamples to the above questions in the class of families of foliations where the analytic type of each singularity is constant.
With some of the families we can also answer a question posed by M. Brunella in [Enseign. Math. (2) 45, 195–216 (1999; Zbl 1004.32011)].
More or less at the same time P. Painlevé asked the following question: “Is it possible to recognize the genus of the general solution of an algebraic differential equation in two variables which has a rational first integral?”.
In this paper we provide counterexamples to the above questions in the class of families of foliations where the analytic type of each singularity is constant.
With some of the families we can also answer a question posed by M. Brunella in [Enseign. Math. (2) 45, 195–216 (1999; Zbl 1004.32011)].
MSC:
| 34A26 | Geometric methods in ordinary differential equations |
| 32S65 | Singularities of holomorphic vector fields and foliations |
| 34M35 | Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms |
| 37C10 | Dynamics induced by flows and semiflows |
| 37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |
| 37F75 | Dynamical aspects of holomorphic foliations and vector fields |
References:
| [1] | Ahlfors L.V. , Sario L. , Riemann Surfaces , Princeton , 1960 . MR 114911 | Zbl 0196.33801 · Zbl 0196.33801 |
| [2] | Arnold V. , Chapitres supplémentaires de la théorie des équations différentielles ordinaires , Éditions MIR , 1980 . MR 626685 | Zbl 0455.34001 · Zbl 0455.34001 |
| [3] | Baum P. , Bott R. , On the zeroes of meromorphic vector fields , in: Essais en l’honneur de De Rham , 1970 , pp. 29 - 47 . MR 261635 | Zbl 0193.52201 · Zbl 0193.52201 |
| [4] | Brunella M. , Feuilletages holomorphes sur les surfaces complexes compactes , Ann. Scient. Éc. Norm. Sup., 4e série 30 ( 1997 ) 569 - 594 . Numdam | MR 1474805 | Zbl 0893.32019 · Zbl 0893.32019 · doi:10.1016/S0012-9593(97)89932-6 |
| [5] | Brunella M. , Courbes entières et feuilletages holomorphes , Enseig. Math. 45 ( 1-2 ) ( 1999 ) 196 - 216 . MR 1703368 | Zbl 1004.32011 · Zbl 1004.32011 |
| [6] | Camacho C. , Lins Neto A. , Geometric Theory of Foliations , Birkhäuser , 1985 . MR 824240 | Zbl 0568.57002 · Zbl 0568.57002 |
| [7] | Campillo A. , Carnicer M. , Proximity inequalities and bounds for the degree of invariant curves by foliations in P 2 C , Trans. Amer. Math. Soc. 349 ( 9 ) ( 1997 ) 2211 - 2228 . MR 1407696 | Zbl 0873.32030 · Zbl 0873.32030 · doi:10.1090/S0002-9947-97-01898-9 |
| [8] | Carnicer M. , The Poincaré problem in the non-dicritical case , Ann. de Math. 140 ( 1994 ) 289 - 294 . MR 1298714 | Zbl 0821.32026 · Zbl 0821.32026 · doi:10.2307/2118601 |
| [9] | Cerveau D. , Lins Neto A. , Holomorphic foliations in CP(2) having an invariant algebraic curve , Ann. Inst. Fourier 41 ( 4 ) ( 1991 ) 883 - 903 . Numdam | MR 1150571 | Zbl 0734.34007 · Zbl 0734.34007 · doi:10.5802/aif.1278 |
| [10] | Cerveau D. , Mattei J.F. , Formes intégrables holomorphes singulières , Astérisque 97 ( 1982 ). MR 704017 | Zbl 0545.32006 · Zbl 0545.32006 |
| [11] | Darboux G. , Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges) , Bull. Sci. Math. (2) 2 ( 1878 ) 60 - 96 , 123-144, 151-200. Numdam | JFM 10.0214.01 · JFM 10.0214.01 |
| [12] | Ehresmann C. , Reeb G. , Les connexions infinitésimales dans un espace fibré différentiable , in: Colloque de topologie, Bruxelles , 1950 , pp. 29 - 55 . MR 42768 | Zbl 0054.07201 · Zbl 0054.07201 |
| [13] | Godbillon C. , Études géométriques I , Université Louis Pasteur , 1985 . |
| [14] | Inse E.L., Ordinary Differential Equations , Dover Books on Advanced Mathematics. Zbl 0063.02971 · Zbl 0063.02971 |
| [15] | Jouanolou J.-P. , Équations de Pfaff algébriques , Lecture Notes in Math. , 708 , Springer-Verlag , Berlin , 1979 . MR 537038 | Zbl 0477.58002 · Zbl 0477.58002 |
| [16] | Lima E.L. , Grupo Fundamental e espaços de recobrimento , Projeto Euclides, IMPA , 1993 . |
| [17] | Lins Neto A., Algebraic solutions of polynomial differential equations and foliations in dimension two, in: Lect. Notes Math., Vol. 1345 , pp. 192-231. Zbl 0677.58036 · Zbl 0677.58036 |
| [18] | Mattei J.F. , Moussu R. , Holonomie et intégrales premières , Ann. Sc. Ec. Norm. Sup. 13 ( 1980 ) 469 - 523 . Numdam | MR 608290 | Zbl 0458.32005 · Zbl 0458.32005 |
| [19] | Moulin Ollagnier J., About a conjecture on quadratic vector fields , Preprint École Polytechnique, Paris, 2000. |
| [20] | Painlevé P. , “Sur les intégrales algébriques des équations différentielles du premier ordre” and “Mémoire sur les équations différentielles du premier ordre” , Oeuvres de Paul Painlevé; Tome II , Éditions du Centre National de la Recherche Scientifique , 15, quai Anatole-France, 75700, Paris , 1974 . · JFM 22.0333.01 |
| [21] | Poincaré H. , Sur l’intégration algébrique des équations différentielles , C. R. Acad. Sci. 112 ( 1891 ) 761 - 764 . JFM 23.0319.01 · JFM 23.0319.01 |
| [22] | Poincaré H. , Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré , Rendiconti del Circolo Matemático di Palermo 5 ( 1891 ) 161 - 191 . JFM 23.0319.01 · JFM 23.0319.01 |
| [23] | Poincaré H. , Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré , Rendiconti del Circolo Matemático di Palermo 11 ( 1897 ) 193 - 239 . JFM 28.0292.01 · JFM 28.0292.01 |
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