  For planar polynomials systems the existence of an invariant algebraic curve
limits the number of limit cycles not contained in this curve. We present a
general approach to prove non existence of periodic orbits not contained in
this given algebraic curve. When the method is applied to parametric families
of polynomial systems that have limit cycles for some values of the parameters,
our result leads to effective algebraic conditions on the parameters that force
non existence of the periodic orbits. As applications we consider several
families of quadratic systems: the ones having some quadratic invariant
algebraic curve, the known ones having an algebraic limit cycle, a family
having a cubic invariant algebraic curve and other ones. For any quadratic
system with two invariant algebraic curves we prove a finiteness result for its
number of limit cycles that only depends on the degrees of these curves. We
also consider some families of cubic systems having either a quadratic or a
cubic invariant algebraic curve and a family of Lienard systems. We also give a
new and simple proof of the known fact that quadratic systems with an invariant
parabola have at most one limit cycle. In fact, what we show is that this
result is a consequence of the similar result for quadratic systems with an
invariant straight line.
