On the isotopic meshing of an algebraic implicit surface

We present a new and complete algorithm for computing the topology of an algebraic surface S given by a square free polynomial in Q[X,Y,Z]. Our algorithm involves only subresultant computations and entirely relies on rational manipulation, which makes it direct to implement. We extend the work in Diatta et al. (2008), on the topology of non-reduced algebraic space curves, and apply it to the polar curve or apparent contour of the surface S. We exploit a simple algebraic criterion to certify the pseudo-genericity and genericity position of the surface. This gives us rational parametrizations of the components of the polar curve, which are used to lift the topology of the projection of the polar curve. We deduce the connection of the two-dimensional components above the cell defined by the projection of the polar curve. A complexity analysis of the algorithm is provided leading to a bound in O˜B(d21τ) for the complexity of the computation of the topology of an implicit algebraic surface defined by integer coefficient polynomial of degree d and coefficient size τ. Examples illustrate the implementation in Mathemagix of this first complete code for certified topology of algebraic surfaces.