There is a classical method started by Darboux
in 1878 about the computation of a special class
of first integrals(called Darboux first integrals) of
polynomial differential systems. These Darboux first
integrals are the products of powers of polynomi-
als (often called Darboux polynomials). Note that
the zero set of the Darboux polynomials define in-
variant algebraic curves of the corresponding vector
field. So, algebraic curves and its multiplicity are the
key points in order to construct the...
There is a classical method started by Darboux
in 1878 about the computation of a special class
of first integrals(called Darboux first integrals) of
polynomial differential systems. These Darboux first
integrals are the products of powers of polynomi-
als (often called Darboux polynomials). Note that
the zero set of the Darboux polynomials define in-
variant algebraic curves of the corresponding vector
field. So, algebraic curves and its multiplicity are the
key points in order to construct the Darboux first
integrals. More recently this method has been gen-
eralized by several authors like Jouanolou, Singer,
Schlomiuk, Llibre, Christopher, Zhang among oth-
ers and is related with some aplications concerning
limit cicles and bifurcation problems. Several inverse
problems in dimension two have also been consid-
ered. In this talk we present necessary and su
ficient conditions for the existence of Darboux first inte-
grals and we also present a generalization for the
nonautonomous case.