. Proof, As described in the previous sections, we are able to deduce the topology of the surface from the solution of system (1) and from the intersection points of the polar curve with planes V (x ? ?) where the ?'s lie in between the x-critical values of

L. Alberti, G. Comte, and B. Mourrain, Meshing implicit algebraic surfaces: the smooth case, Mathematical Methods for Curves and Surfaces: Tromso'04, pp.11-26, 2005.

L. Alberti and B. Mourrain, Visualisation of algebraic curves, The 15th Pacific Graphics, pp.303-312, 2007.
URL : https://hal.archives-ouvertes.fr/inria-00175062

J. G. Alcázar, J. R. Sendra, and J. Schicho, A delineability-based method for computing critical sets of algebraic surfaces, Journal of Symbolic Computation, vol.42, issue.6, pp.678-691, 2007.
DOI : 10.1016/j.jsc.2007.02.001

J. G. Alcázar and J. R. Sendra, Computation of the topology of real algebraic space curves, Journal of Symbolic Computation, vol.39, issue.6, pp.719-744, 2005.
DOI : 10.1016/j.jsc.2005.01.006

V. I. Arnol-'d, S. M. Guse?-in-zade, and A. N. Varchenko, Singularities of differentiable maps The classification of critical points, caustics and wave fronts, Monographs in Mathematics Birkhäuser Boston Inc, vol.82, 1985.

S. Basu, R. Pollack, and M. Roy, Algorithms in Real Algebraic Geometry, 2003.
DOI : 10.1007/978-3-662-05355-3
URL : https://hal.archives-ouvertes.fr/hal-01083587

E. Berberich, M. Kerber, and M. Sagraloff, Exact geometric-topological analysis of algebraic surfaces, Proceedings of the twenty-fourth annual symposium on Computational geometry , SCG '08, 2008.
DOI : 10.1145/1377676.1377703

J. Bochnak, M. Coste, and M. Roy, Géométrie Algébrique Réelle, 1987.

J. D. Boissonnat, D. Cohen-steiner, and G. Vegter, Isotopic implicit surface meshing, Proceedings of STOC 2004, pp.301-309, 2004.
DOI : 10.1007/978-0-387-87363-3_9
URL : https://hal.archives-ouvertes.fr/hal-00488819

L. Busé and B. Mourrain, Explicit factors of some iterated resultants and discriminants, Mathematics of Computation, vol.78, issue.265, 2008.
DOI : 10.1090/S0025-5718-08-02111-X

J. Canny, The Complexity of Robot Motion Planning, 1988.

G. E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decompostion, Proc. 2nd GI Conference on Automata Theory and Formal Languages, pp.134-183, 1975.
DOI : 10.1007/3-540-07407-4_17

M. Coste, An introduction to semi-algebraic geometry. RAAG network school, 2002.

A. Dimca, Singularities and topology of hypersurfaces. Universitext, 1992.

M. Elkadi and B. Mourrain, IntroductionàIntroduction`Introductionà la résolution des systèmes d'´ equations algébriques, of Mathématiques et Applications, 2007.

M. Kahoui, Topology of real algebraic space curves, Journal of Symbolic Computation, vol.43, issue.4, pp.235-258, 2008.
DOI : 10.1016/j.jsc.2007.10.008

E. Fortuna, P. M. Gianni, and D. Luminati, Algorithmical determination of the topology of a real algebraic surface, Journal of Symbolic Computation, vol.38, issue.6, pp.1551-1567, 2004.
DOI : 10.1016/j.jsc.2004.08.001

E. Fortuna, P. M. Gianni, P. Parenti, and C. Traverso, Algorithms to compute the topology of orientable real algebraic surfaces, Journal of Symbolic Computation, vol.36, issue.3-4, pp.3-4343, 2003.
DOI : 10.1016/S0747-7171(03)00085-3

G. Gatellier, A. Labrouzy, B. Mourrain, and J. Técourt, Computing the topology of 3-dimensional algebraic curves, Computational Methods for Algebraic Spline Surfaces, pp.27-44, 2004.

L. González-vega and I. Necula, Efficient topology determination of implicitly defined algebraic plane curves, Computer Aided Geometric Design, vol.19, issue.9, pp.719-743, 2002.
DOI : 10.1016/S0167-8396(02)00167-X

L. Gonzalez-vega, F. Rouillier, and M. F. Roy, Symbolic Recipes for Polynomial System Solving. Some Tapas of Computer Algebra, 1997.
DOI : 10.1007/978-3-662-03891-8_2
URL : https://hal.archives-ouvertes.fr/inria-00098570

R. M. Hardt, Triangulation of subanalytic sets and proper light subanalytic maps, Inventiones Mathematicae, vol.2, issue.no. 4, pp.207-21777, 1976.
DOI : 10.1007/BF01403128

H. Hironaka, Triangulations of algebraic sets, Algebraic geometry (Proc. Sympos. Pure Math, pp.165-185, 1974.
DOI : 10.1090/pspum/029/0374131

V. M. Kharlamov, S. Yu, and . Orevkov, The number of trees half of whose vertices are leaves and asymptotic enumeration of plane real algebraic curves, Journal of Combinatorial Theory, Series A, vol.105, issue.1
DOI : 10.1016/j.jcta.2003.10.007
URL : https://hal.archives-ouvertes.fr/hal-00013007

G. N. Khim?ia?vili, The local degree of a smooth mapping, Sakharth. SSR Mecn. Akad. Moambe, vol.85, issue.2, pp.309-312, 1977.

N. G. Lloyd, Degree Theory, 1978.

S. Mccallum and G. E. Collins, Local Box Adjacency Algorithms for Cylindrical Algebraic Decompositions, Journal of Symbolic Computation, vol.33, issue.3, pp.321-342, 2002.
DOI : 10.1006/jsco.2001.0499

D. N-'diatta, B. Mourrain, and O. Ruatta, On the computation of the topology of a non-reduced implicit space curve, 2008.

S. Yu, V. M. Orevkov, and . Kharlamov, Growth order of the number of classes of real plane algebraic curves as the degree grows, Zap. Nauchn. Sem. S.- Peterburg. Otdel. Mat. Inst. Steklov. (POMI)Teor. Predst. Din. Sist. Komb. i Algoritm. Metody, vol.266, issue.339, pp.218-233, 2000.

J. P. Speder, . Equisingularité, . Conditions, and . Whitney, Equisingularite et Conditions de Whitney, American Journal of Mathematics, vol.97, issue.3, pp.571-588, 1975.
DOI : 10.2307/2373766

D. Trotman, On Canny's roadmap algorithm: orienteering in semialgebraic sets (an application of singularity theory to theoretical robotics), Proceedings of the 1989 Warwick Singularity Theory Symposium, pp.320-339, 1991.
DOI : 10.1007/BFb0086391

Y. Yomdin and G. Comte, Tame geometry with applications in smooth analysis. LNM 1834, 2004.

O. Zariski, Studies in Equisingularity II. Equisingularity in Codimension 1 (and Characteristic Zero), American Journal of Mathematics, vol.87, issue.4, pp.972-1006, 1965.
DOI : 10.2307/2373257