The Valuative Theory of Foliations

Abstract. This paper gives a characterization of valuations that follow the singular innitely near points of plane vector elds, using the notion of L'H opital valuation, which generalizes a well known classical condition. With that tool, we give a valuative description of vector elds with innite solu-tions, singularities with rational quotient of eigenvalues in its linear part, and polynomial vector elds with transcendental solutions, among other results.