The Valuative Theory of Foliations
P. Fortuny Ayuso
This paper gives a characterization of valuations which follow the singular infinitely near points of plane vector fields, using the notion of \lh 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.
00father.tex
% este fichero es el padre de la tesis
\documentclass{amsart}
%\newcounter{chapter}
\usepackage{amssymb}  
\usepackage{amsmath}  
\usepackage{verbatim}
%\usepackage{ite}
%\usepackage[english]{babel} 
\usepackage{bm} 
%\usepackage{spanish} 
\input{0999com.ltx} 
%\input{diagram.tex}
\title{The Valuative Theory of Foliations}
\author{P. Fortuny Ayuso} 
\thanks{Partially supported by the European Commission: TMR Network ``Singularidades de Ecuaciones
  Diferenciales y Foliaciones'', ERBFMRXCT960040.\\
The author wishes to thank Prof. J. M. Aroca for his constant support during the preparation of his Ph. D. Thesis.}
\date{\today}
%\makeindex
\begin{document}   
\vfuzz4pt %d\'ejame poner acentos en l's encabezamientos 
\hfuzz3pt %no me des la brasa con overfulls muy peque\~nos 
\begin{abstract}
  This paper gives a characterization of valuations which follow the singular infinitely near points of
  plane vector fields, using the notion of \lh 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.  
\end{abstract}
\maketitle
\input{introduction.ltx} 
\input{03dimension-2.ltx} 
%\input{03lacking-separatrices.ltx}
\input{03irrational.ltx}
\input{03rank1-trascendence.ltx}
\input{03lacking-separatrices.ltx}
%\input{04riemann-surface.ltx} 
%\input{031singular.tex.ltx}    
%\input{05higher-dim.ltx}   
%\input{06oscilation.ltx} 
%\input{06ordenmayor.tex} 
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\begin{thebibliography}{1}

\bibitem{CS}
C.~Camacho and P.~Sad, \emph{Invariant varieties through singularities of
  holomorphic vector fields}, Ann. of Math. \textbf{115} (1982), 579--595.

\bibitem{tesis}
P.~Fortuny, \emph{L'{H{\^o}pital}}, Ph.D. thesis, Universidad de Valladolid,
  1999.

\bibitem{Ros}
M.~Rosenlicht, \emph{On the explicit solvability of certain transcendental
  equations}, Inst. Haute {\'E}tudes Sci. Publ. Math. \textbf{36} (1969),
  15--22.

\bibitem{Sei}
A.~Seidenberg, \emph{Reduction of singularities of the differential equation
  $ady=bdx$}, Amer. J. Math. (1968), 248--269.

\bibitem{Sei2}
\bysame, \emph{Contributions to algebra (eds. {B}ass, {C}assidy, {K}ovacic)},
  ch.~Derivations and valuation rings, pp.~343--347, Academic Press, New York,
  1977.

\bibitem{Sp}
M.~Spivakovsky, \emph{Valuations in function fields of surfaces}, Amer. J.
  Math. \textbf{112} (1990), no.~1, 107--156.

\bibitem{Zar4}
O.~Zariski, \emph{The reduction of singularities of an algebraic surface}, Ann.
  of Math. \textbf{40} (1939), 639--689.

\end{thebibliography}
\end{document} 

math0103123.extracted_bib
\begin{thebibliography}{1}

\bibitem{CS}
C.~Camacho and P.~Sad, \emph{Invariant varieties through singularities of
  holomorphic vector fields}, Ann. of Math. \textbf{115} (1982), 579--595.

\bibitem{tesis}
P.~Fortuny, \emph{L'{H{\^o}pital}}, Ph.D. thesis, Universidad de Valladolid,
  1999.

\bibitem{Ros}
M.~Rosenlicht, \emph{On the explicit solvability of certain transcendental
  equations}, Inst. Haute {\'E}tudes Sci. Publ. Math. \textbf{36} (1969),
  15--22.

\bibitem{Sei}
A.~Seidenberg, \emph{Reduction of singularities of the differential equation
  $ady=bdx$}, Amer. J. Math. (1968), 248--269.

\bibitem{Sei2}
\bysame, \emph{Contributions to algebra (eds. {B}ass, {C}assidy, {K}ovacic)},
  ch.~Derivations and valuation rings, pp.~343--347, Academic Press, New York,
  1977.

\bibitem{Sp}
M.~Spivakovsky, \emph{Valuations in function fields of surfaces}, Amer. J.
  Math. \textbf{112} (1990), no.~1, 107--156.

\bibitem{Zar4}
O.~Zariski, \emph{The reduction of singularities of an algebraic surface}, Ann.
  of Math. \textbf{40} (1939), 639--689.

\end{thebibliography}