Computing the topology of real algebraic surfaces

Abstract

We present constructive algorithms to determine the topological type of a non-singular real algebraic projective surface S in the real projective space; we address this question when there exists a line in RP3 not intersecting the surface. Starting from a polynomial equation with rational coefficients for S, our algorithm computes the homology of the various connected components of the surface. We reconstruct the homology in a finite number of steps, using as a basic tool Morse theory. The entire procedure has been implemented in Axiom.