Preprint A02/2001
Vector Fields, Invariant Varieties and Linear Systems

Jorge VitÃ³rio Pereira

**Keywords: **

We investigate the interplay between invariant varieties of vector fields and the inflection locus of linear systems with respect to the vector field. Among the consequences of such investigation we obtain a computational criteria for the existence of rational first integrals of a given degree, bounds for the number of first integrals on families of vector fields and a generalization of Darboux's criteria. We also provide a new proof of Gomez--Mont's result on foliations with all leaves algebraic.