Abstract
Let X be a polynomial vector field in ℂ2; then it defines an algebraic foliation \(\mathcal{F}\) on ℂP(2). If \(\mathcal{F}\) admits a Liouvillian first integral on ℂP(2), then it is transversely affine outside some algebraic invariant curve S⊂ ℂP(2). If, moreover, for some irreducible component S0 ⊂ S, the singularities q ∈ Sing \(\mathcal{F}\) ∪ S are generic, then either \(\mathcal{F}\) is given by a closed rational 1-form or it is a rational pull-back from a Bernoulli foliation \({\mathcal{R}}:p(x)dy - (y^2 a(x) + yb(x))dx = 0{ on }\bar {\mathbb{C}} \times \bar {\mathbb{C}}.\) This result has several applications such as the study of foliations with algebraic limit sets on ℂP(2)(2), the classification polynomial complete vector fields over ℂ2, and topological rigidity of foliations on ℂP(2). We also address the problem of moderate integration for germs of complex ordinary differential equations.
Similar content being viewed by others
References
J.E. Bjork, Rings of differential operators. North-Holland, Amsterdam, 1979, 248–276.
M. Berthier and F. Touzet, Sur l'intégration des equations différentielles holomorphes réduites en dimension deux. Prépublication Univ. Rennes I, 1996.
C. Camacho, A. Lins Neto, and P. Sad, Topological invariants and equidesingularization for holomorphic vector fields. J. Differ. Geometry 20 (1984), No. 1, 143–174.
C. Camacho, A. Lins Neto, and P. Sad, Foliations with algebraic limit sets. Ann. Math. 136 (1992), 429–446.
C. Camacho and B. Scárdua, Liouvillian first integrals, solvable holonomy groups and Bernoulli foliations. Preprint Inst. Mat. Pura Aplicada, July, 1995.
C. Camacho and B. Azevedo Scárdua, Complex foliations with algebraic limit sets. (to appear in: Astérisque), 1998.
D. Cerveau and J.-F. Mattei, Formes intégrables holomorphes singulières. Astérisque 97 (1982).
D. Cerveau and R. Moussu, Groupes d'automorphismes de (ℂ 0) et équations différentielles y dy + ··· = 0. Bull. Soc. Math. France 116 (1988), 459–488.
D. Cerveau and B. Azevedo Scárdua, On the integration of polynomial vector fields in dimension two. Preprint, IRMAR, April, 1996.
C. Godbillon, Feuilletages, Études Geométriques I. Université Louis Pasteur, Mai, 1985.
E. Ghys, Feuilletages holomorphes de codimension un sur les espaces homogènes complexes. Preprint, ENS, Lyon, 1995.
H. Hironaka and H. Matsumara, Formal functions and formal embeddings. J. Math. Soc. Japan 20 (1968), No. 1–2.
A. Lins Neto, Construction of singular holomorphic vector fields and foliations in dimension two. J. Differ. Geometry 26 (1987), 1–31.
A. Lins Neto, P. Sad, and B. Azevedo Scárdua, On topological Rigidity of Projective Foliations (to appear in: Bull. Soc. Math. France) (1998).
F. Loray, Feuilletages holomorphes à holonomie résoluble. These Univ. Rennes I, 1994.
J. Malmquist, Sur les functions à un nombre fini des branches définies par les équations différentielles du premier ordre. Acta Math. 36 (1913), 297–343.
J.F. Mattei and R. Moussu, Holonomie et intégrales premières. Ann. Ecole Norm. Super. 13 (1980), 469–523.
J. Martinet and J.-P. Ramis, Classification analytique des équations différentielles non lineaires resonnants du premier ordre. Ann. Sc. Ecole Norm. Super. 16 (1983), 571–621.
J. Martinet and J.-P. Ramis, Problème de modules pour des équations différentielles non lineaires du premier ordre. Publ. Math. Inst. Hautes Études Sci. 55 (1982), 63–124.
M. Nicolau and E. Paul, A geometric proof of a Galois differential theory theorem. Preprint, 1995.
N. Nilsson, Some growth and ramification properties of certain integrals on algebraic manifolds. Arkiv Mat. 5 (1965), 463–475.
P. Painlevé, Leçons sur la théorie analytique des équations différentielles. Librairie Scientifique A. Hermann, Paris, 1897.
E. Paul, Groupes de diffeomorphismes résolubles. Prépublication Univ. Paul Sabatier, Toulouse, 1995.
_____, Formes singuliéres à holonomie résoluble. Prépublication Univ. Paul Sabatier, Toulouse, 1995.
B. Azevedo Scárdua, Transversely affine and transversely projective foliations. Ann Sci. École Norm. Super. 4 sér. 30 (1997), 169–204.
_____, Parabolic uniform limits of holomorphic flows in ℂ2. Bol. Soc. Mat. Mexicana 4 (1998), No. 3, 123–130.
A. Seidenberg, Reduction of singularities of the differential equation A dy = B dx. Am. J. Math. 90 (1968), 248–269.
Bobo Seke, Sur les structures transversalement affines des feuilletages de codimension un. Ann. Inst. Fourier, Grenoble 30 (1980), No. 1, 1–29.
Y. Siu, Techniques of extension of analytic objects. Marcel Dekker, N.Y., 1974.
M.F. Singer, Liouvillian first integrals of differential equations. Trans. Am. Math. Soc. 333 (1992).
F. Touzet, Sur les intégrales premières dans la classe de Nilsson d'équations différentielles holomorphes. Pŕepublication Univ. Rennes I, Décembre, 1996.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Scárdua, B.A. Integration of Complex Differential Equations. Journal of Dynamical and Control Systems 5, 1–50 (1999). https://doi.org/10.1023/A:1021740700501
Issue Date:
DOI: https://doi.org/10.1023/A:1021740700501