Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

On the characterization of algebraically integrable plane foliations

Author(s): C. Galindo; F. Monserrat
Journal: Trans. Amer. Math. Soc. 362 (2010), 4557-4568.
MSC (2010): Primary 32S65
Posted: April 27, 2010
MathSciNet review: 2645041
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We give a characterization theorem for non-degenerate plane foliations of degree different from 1 having a rational first integral. Moreover, we prove that the degree of a non-degenerate foliation as above provides the minimum number, , of points in the projective plane through which pass infinitely many algebraic leaves of the foliation.

References:

1.: L. Autonne, Sur la théorie des équations différentielles du premier ordre et du premier degré, J. École Polytech. 61 (1891), 35-122; 62 (1892), 47-180.
2.: A. Beauville, Complex algebraic surfaces, London Math. Soc. Student Texts 34, Cambridge University Press, 1996. MR 1406314 (97e:14045)
3.: M. Brunella, Birational Geometry of Foliations, Springer-Verlag, 2000. MR 1948251 (2004g:14018)
4.: C. Camacho and P. Sad, Pontos singulares de equações diferenciais analíticas, 16 Colóq. Bras. Mat., IMPA, 1987. MR 953780 (90a:58126)
5.: A. Campillo and M. Carnicer, Proximity inequalities and bounds for the degree of invariant curves by foliations of $\mathbb{P}_{\mathbb{C}}^2$ , Trans. Amer. Math. Soc. 349 (9) (1997), 2211-2228. MR 1407696 (97h:32051)
6.: A. Campillo and J. Olivares, Polarity with respect to a foliation and Cayley-Bacharach Theorems, J. Reine Angew. Math. 534 (2001), 95-118. MR 1831632 (2002c:32051)
7.: M. Carnicer, The Poincaré problem in the nondicritical case, Ann. of Math. (2) 140 (1994), 289-294. MR 1298714 (95k:32031)
8.: E. Casas-Alvero, Singularities of plane curves, London Math. Soc. Lecture Notes 276, Cambridge University Press, 2000. MR 1782072 (2003b:14035)
9.: G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. Math. 32 (1878), 60-96; 123-144; 151-200.
10.: A. A. du Plessis and C. T. C. Wall, Application of the theory of the discriminant to highly singular plane curves, Math. Proc. Cambridge Phil. Soc. 126 (1999), 259-266. MR 1670229 (99m:14053)
11.: E. Esteves and S. Kleiman, Bounds on leaves of one-dimensional foliations, Bull. Braz. Math. Soc. (N.S.) 34(1) (2003), 145-169. MR 1993042 (2004m:32060)
12.: C. Galindo and F. Monserrat, Algebraic integrability of foliations of the plane, J. Differential Equations 231 (2) (2006), 611-632. MR 2287899 (2008a:32028)
13.: J. García de la Fuente, Geometría de los sistemas lineales de series de potencias en dos variables, Ph.D. thesis, Valladolid University, 1989.
14.: G.M. Greuel, G. Pfister and H. Schöenemann, SINGULAR 3.0. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern, 2005. http://www.singular.uni-kl.de.
15.: P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley Interscience, 1978. MR 507725 (80b:14001)
16.: R. Hartshorne, Algebraic geometry, Springer-Verlag, 1977. MR 0463157 (57:3116)
17.: A. Lins-Neto, Some examples for the Poincaré and Painlevé problems, Ann. Sc. École Norm. Sup. (4) 35 (2002), 231-266. MR 1914932 (2003j:34009)
18.: J. Lipman, Proximity inequalities for complete ideals in two-dimensional regular local rings, Contemp. Math. 159 (1994), 293-306. MR 1266187 (95j:13018)
19.: P. Painlevé, Sur les intégrales algébriques des équations différentielles du premier ordre and Mémoire sur les équations différentielles du premier ordre in Oeuvres de Paul Painlevé, Tome II, Éditions du Centre National de la Recherche Scientifique 15, quai Anatole-France, Paris 1974.
20.: H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, J. Math. Pures Appl. 3 (7) (1881), 375-442; 3 (8) (1882), 251-296; 4 (1) (1885), 167-244; in Oeuvres de Henri Poincaré, vol. I, Gauthier-Villars, Paris, 1951, 3-84, 95-114.
21.: -, Sur l'intégration algébrique des équations différentielles du premier ordre et du premier degré (I), Rend. Circ. Mat. Palermo 5 (1891), 161-191.
22.: -, Sur l'intégration algébrique des équations différentielles du premier ordre et du premier degré (II), Rend. Circ. Mat. Palermo 11 (1897), 193-239.
23.: K. Saito, Quasihomogene isolierte singularitäten von hyperflächen, Invent. Math. 14 (1971), 123-142. MR 0294699 (45:3767)
24.: A. Seidenberg, Reduction of singularities of the differential equation , Amer. J. Math. 90 (1968), 248-269. MR 0220710 (36:3762)
25.: E. Yoshinaga and M. Suzuki, Topological types of quasihomogeneous singularities in $\mathbb{C}^2$ , Topology 18 (1979), 113-116. MR 544152 (80k:32017)
26.: O. Zariski and P. Samuel, Commutative algebra, Vol. II, Springer-Verlag, 1960. MR 0120249 (22:11006)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|