Complete algebraic vector fields on affine surfaces, Part I

Abstract

In this work we study the singularities at infinity of algebraic vector fields in dimension 2.These singularities will be classified under a mild assumption. The general problem is also reduced to the study of the combinatorics of certain resolutions which will be developed in Part II. Our main results are local and therefore can be carried over more general surfaces. Whereas we deal with ℂ -complete vector fields, the results also apply to ℝ-complete ones thanks to a theorem of Forstneric [7].