Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Sur les integrales premieres
dans la classe de Nilsson
d'equations differentielles holomorphes

Author: Frédéric Touzet
Journal: Trans. Amer. Math. Soc. 352 (2000), 1601-1622
MSC (1991): Primary 32-XX
Published electronically: December 7, 1999
MathSciNet review: 1675242
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study two classes of holomorphic differential equations. The first one is constitued by elements admitting solutions defined in an algebraic way (the so called Liouville class) and the second of elements admitting solutions defined in an analytic way (the Nilsson class). We build up links between these two classes using special properties of the holonomy and its results on the monodromy.

References [Enhancements On Off] (What's this?)

  • [Ar] M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 277-291. MR 38:344
  • [Bj] J. E. Björk, Rings of differential operators, North-Holland, 1979, pp. 248-276. MR 82g:32013
  • [Br, Bo] C. Briot et J.-C. Bouquet, Recherches sur les fonctions définies par des équations définies par des équations différentielles (J. Ecole Polytechnique, Vol. XXI, 1856, pp. 134-198).
  • [Ca] C. Camacho, On the local structure of conformal mappings and holomorphic vector fields in $\mathbb{C}^2$, Astérisque 59-60 (1978), pp. 1-99. MR 81d:58016
  • [Ca, Ce] F. Cano, D. Cerveau, Desingularization of nondicritical holomorphic foliations and existence of separatrices, Acta Math. 169 (1992), 1-103. MR 93k:32069
  • [Ca, Sca] C. Camacho, B. Scardua, Liouvillian first integrals, solvable holonomy groups and Riccati foliations, Prépublication.
  • [Ce, Ma] D. Cerveau, J.-F. Mattéi, Formes intégrables holomorphes singulières, Astérisque, Vol. 97 (1982). MR 86f:58006
  • [Ce, Mo] D. Cerveau, R. Moussu, Groupes d'automorphismes de $\mathbb{C},0$ et équations différentielles $ydy+\dotsb=0$, Bull. Soc. Math. France 116 (1988), 459-488. MR 90m:58192
  • [Ka] I. Kaplansky, An introduction to differential algebra, Hermann, Paris, 1957. MR 20:177
  • [Lo] F. Loray, Feuilletages holomorphes à holonomie résoluble, Thèse de l'Université de Rennes I (1994).
  • [Ma, Mo] J.-F. Mattéi et R. Moussu, Holonomie et intégrales premières, Ann. Ecole Norm. Sup. 13 (1980), 469-523. MR 83b:58005
  • [Ma, Ra$_1$] J. Martinet, J.-P. Ramis, Problèmes de modules pour des équations différentielles non linéares du premier ordre, I.H.E.S. Publ. Math. 55 (1982), 63-164. MR 84k:34011
  • [Ma, Ra$_2$] J. Martinet, J.-P. Ramis, Classification analytique des équations différentielles non linéaires résonnantes du premier ordre, Ann. Sci. Ecole Norm. Sup. (4) 16 (1983), pp. 271-621. MR 86k:34034
  • [Ni, Pa] E. Paul et M. Nicolau, A geometric proof of differential Galois theory, Preprint.
  • [Nils] N. Nilsson, Some growth and ramification properties of certain integrals on algebraic manifolds, Ark. Mat. 5 (1965), 463-475. MR 31:180
  • [Pe, Yo] R. Pérez Marco, J.-C. Yoccoz, Germes de feuilletages holomorphes à holonomie prescrite, Astérisque 222 (1992), 345-371. MR 96b:58090
  • [Ro] M. Rosenlicht, On Liouville's theory of elementary functions, Pacific J. Math. 65 (1976), pp. 1-99. MR 56:5514
  • [Sca] B. Scardua, Transversely affine and transversely projective foliations on complex projective spaces, Thesis, IMPA, 1994.
  • [Sg] M.-F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992), 673-688. MR 92m:12014
  • [To] F. Touzet, Equations différentielles admettant des solutions Liouvilliennes, Thèse de l'Université de Rennes I, 1995.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 32-XX

Retrieve articles in all journals with MSC (1991): 32-XX

Additional Information

Frédéric Touzet
Affiliation: Institut de Recherche Mathématique de Rennes, Université de Rennes, 35042 Rennes Cedex, France

Received by editor(s): June 11, 1997
Received by editor(s) in revised form: March 3, 1998
Published electronically: December 7, 1999
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society