Enumerative Geometry versus Counting Problems on Low Degree Polynomial Vector Fields

Abstract

The purpose of this article is to introduce the perspective of
Enumerative Geometry in the theory of low degree polynomial
vector fields. Problems on these systems are easy to state and
hard to prove. Thus the second part of Hilbert's 16th problem, in
its original statement, is dauntingly difficult even for quadratic
vector fields. Asking questions from this perspective provides
us with the possibility of getting positive results in this
area of research. We give a list of such questions and indicate
references where solutions were found. These solutions also led
in some cases to the integrability of the systems via the theory of
Darboux. Extending the algebraic-geometric methods of enumerative
geometry by some analytic ones led us to obtain complete knowledge
of some classes of systems such as for example the Lotka-Volterra
systems, important for applications. A short description of
enumerative geometry and its history, several other counting problems for low degree
polynomial vector fields together with their solutions and some new problems are
presented. References as well as comments regarding the material
in the literature are also given.