Elementary first integrals and algebraic invariant curves of differential equations (Q1312855)

From author's summary: ``In this work we survey the relationship existing between existence of elementary first integrals of differential equations and invariant algebraic curves. We explain the method of integration of polynomial differential equations via invariant algebraic curves, method introduced by Darboux and developed later by Prelle and Singer. We indicate the importance of this method for the problem of the center for plane polynomial vector fields. We illustrate the method by exhibiting a simple and unifying proof of both the sufficiency of the algebraic conditions for the center and of the integrability of systems satisfying them in the case of quadratic systems, for all values of the parameters. Other applications and some open problems are mentioned''.