Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T20:14:57.596Z Has data issue: false hasContentIssue false
  • English
  • Français

The Valuative Theory of Foliations

Published online by Cambridge University Press:  20 November 2018

Pedro Fortuny Ayuso
Pedro Fortuny Ayuso*
Affiliation:
School of Mathematical Sciences, Queen Mary College, University of London, Mile End Road, London E1 4NS UK, email: pf@maths.qmul.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

12J2013F3016W6037F7534M25
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 5 , 01 October 2002 , pp. 897 - 915
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] Camacho, C. and Sad, P., Invariant varieties through singularities of holomorphic vector fields. Ann. of Math. 115 (1982), 579–595.Google Scholar
[2] Fortuny, P., L'H.opital. Ph.D. thesis, Universidad de Valladolid, 1999.Google Scholar
[3] Kolchin, E. R., Rational approximation to solutions of algebraic differential equations. Proc. Amer. Math. Soc. 10 (1959), 238–244.Google Scholar
[4] Morrison, S. D., Continuous derivations. J. Algebra 110 (1987), 468–479.Google Scholar
[5] Rosenlicht, M., On the explicit solvability of certain transcendental equations. Inst. Hautes Études Sci. Publ. Math. 36 (1969), 15–22.Google Scholar
[6] Rosenlicht, M., An analogue of L'Hospital's rule. Proc. Amer.Math. Soc. 37 (1973), 369–373.Google Scholar
[7] Rosenlicht, M., Differential valuations. Pacific J. Math. 86 (1980), 301–309.Google Scholar
[8] Seidenberg, A., Reduction of singularities of the differential equation ady = bdx. Amer. J. Math. 90 (1968), 248–269.Google Scholar
[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] Singer, M. F., Linear differential equations in function fields. Proc. Amer.Math. Soc. 54 (1976), 69–72.Google Scholar
[11] Spivakovsky, M., Valuations in function fields of surfaces. Amer. J. Math. (1) 112 (1990), 107–156.Google Scholar
[12] Vaquié, M., Valuations. In: Resolution of singularities (eds. H. Hauser et al.), Progr.Math. 181, Birkhäuser, 2000, 541–590.Google Scholar
[13] Zariski, O., The reduction of singularities of an algebraic surface. Ann. of Math. 40 (1939), 639–689.Google Scholar