Abstract: Recently V.I.Arnold introduced three new invariants
of a generic immersion of the circle to the plane. These invariants are
similar to Vassiliev invariants of classical knots. In a sense they are
of degree one. In this paper an investigation based on similar ideas is
done for real algebraic plane projective curves. In this more algebraic
setting Arnold's invariants have natural counter-parts, two of which admit
definitions in terms of the complexification of a curve. On the other hand,
the Rokhlin complex orientation formula for a real algebraic curve bounding
in its complexification suggests new combinatorial formulas for these two
Arnold's invariants. Using the formulas I prove Arnold's conjecture. Arnold's
invariants are generalized to generic collections of immersions of the
circle to the projective plane and other surfaces. Some invariants of high
degrees admitting similar formulas are discussed.