Algorithms to compute the topology of orientable real algebraic surfaces

We present constructive algorithms to determine the topological type of a non-singular orientable real algebraic projective surface S in the real projective space, starting from a polynomial equation with rational coefficients for S. We address this question when there exists a line in not intersecting the surface, which is a decidable problem; in the case of quartic surfaces, when this condition is always fulfilled, we give a procedure to find a line disjoint from the surface. Our algorithm computes the homology of the various connected components of the surface in a finite number of steps, using as a basic tool Morse theory. The entire procedure has been implemented in Axiom.