  We study the following question: given a set P of 3d-2 points and an immersed
curve G in the real plane R^2, all in general position, how many real rational
plane curves of degree d pass through these points and are tangent to this
curve. We count each such curve with a certain sign, and present an explicit
formula for their algebraic number. This number is preserved under small
regular homotopies of a pair (P, G), but jumps (in a well-controlled way) when
in the process of homotopy we pass a certain singular discriminant. We discuss
the relation of such enumerative problems with finite type invariants. Our
approach is based on maps of configuration spaces and the intersection theory
in the spirit of classical algebraic topology.

Geometric Topology;Algebraic Geometry