Selfintersections of a bézier bicubic surface

We present the computation of selfintersections as a major problem in Computer Aided Geometric Design (CAD) and Geometric Modeling, and particularly for patches of parametrized bicubic surfaces. Then we expose two complementary contributions on that subject with Computer Algebra tools: First, a specific sparse bivariate resultant adapted to the corresponding elimination problem, second a semi-numeric polynomial solver able to deal with large system of equations with floating point coefficients. Examples and timings are provided.