Algorithms to compute the topology of orientable real algebraic surfaces

Authors:
E. Fortuna;P. Gianni;P. Parenti;C. Traverso
Affiliations:
Dipartimento di Matematica, Università di Pisa, via Buonarroti 2, I-56127 Pisa, Italy;Dipartimento di Matematica, Università di Pisa, via Buonarroti 2, I-56127 Pisa, Italy;Dipartimento di Matematica, Università di Pisa, via Buonarroti 2, I-56127 Pisa, Italy;Dipartimento di Matematica, Università di Pisa, via Buonarroti 2, I-56127 Pisa, Italy
Venue:
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
Year:
2003

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Abstract

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 RP3 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.