Geodesic
The Valuative Theory of Foliations
Canadian journal of mathematics, Tome 54 (2002) no. 5, pp. 897-915

Voir la notice de l'article provenant de la source Cambridge University Press

This paper gives a characterization of valuations that follow the singular infinitely near points of plane vector fields, using the notion of L'Hôpital valuation, which generalizes a well known classical condition. With that tool, we give a valuative description of vector fields with infinite solutions, singularities with rational quotient of eigenvalues in its linear part, and polynomial vector fields with transcendental solutions, among other results.
DOI : 10.4153/CJM-2002-033-x
Mots-clés : 12J20, 13F30, 16W60, 37F75, 34M25 
Ayuso, Pedro Fortuny. The Valuative Theory of Foliations. Canadian journal of mathematics, Tome 54 (2002) no. 5, pp. 897-915. doi: 10.4153/CJM-2002-033-x
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[1] [1] Camacho, C. and Sad, P., Invariant varieties through singularities of holomorphic vector fields. Ann. of Math. 115 (1982), 579–595. Google Scholar

[2] [2] Fortuny, P., L'H.opital. Ph.D. thesis, Universidad de Valladolid, 1999. Google Scholar

[3] [3] Kolchin, E. R., Rational approximation to solutions of algebraic differential equations. Proc. Amer. Math. Soc. 10 (1959), 238–244. Google Scholar

[4] [4] Morrison, S. D., Continuous derivations. J. Algebra 110 (1987), 468–479. Google Scholar

[5] [5] Rosenlicht, M., On the explicit solvability of certain transcendental equations. Inst. Hautes Études Sci. Publ. Math. 36 (1969), 15–22. Google Scholar

[6] [6] Rosenlicht, M., An analogue of L'Hospital's rule. Proc. Amer.Math. Soc. 37 (1973), 369–373. Google Scholar

[7] [7] Rosenlicht, M., Differential valuations. Pacific J. Math. 86 (1980), 301–309. Google Scholar

[8] [8] Seidenberg, A., Reduction of singularities of the differential equation ady = bdx. Amer. J. Math. 90 (1968), 248–269. Google Scholar

[9] [9] Seidenberg, A., Derivations and valuation rings. In: Contributions to algebra (eds. Bass, Cassidy, Kovacic), Academic Press, New York, 1977, 343–347. Google Scholar

[10] [10] Singer, M. F., Linear differential equations in function fields. Proc. Amer.Math. Soc. 54 (1976), 69–72. Google Scholar

[11] [11] Spivakovsky, M., Valuations in function fields of surfaces. Amer. J. Math. (1) 112 (1990), 107–156. Google Scholar

[12] [12] Vaquié, M., Valuations. In: Resolution of singularities (eds. H. Hauser et al.), Progr.Math. 181, Birkhäuser, 2000, 541–590. Google Scholar

[13] [13] Zariski, O., The reduction of singularities of an algebraic surface. Ann. of Math. 40 (1939), 639–689. Google Scholar

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