Polar Varieties Real Equation Solving and Data Structures The Hypersurface Case B Bank

In this paper we apply for the rst time a new method for multivariate equation solving which was developed in for complex root determination to the real case Our main result concerns the problem of nding at least one representative point for each connected component of a real compact and smooth hypersurface The basic algorithm of yields a new method for symbolically solv ing zero dimensional polynomial equation systems over the complex numbers One feature of central importance of this algorithm is the use of a problem adapted data type represented by the data structures arithmetic network and straight line program arithmetic circuit The algorithm nds the complex solutions of any a ne zero dimensional equation system in non uniform sequential time that is polynomial in the length of the input given in straight line program representation and an adequately de ned geometric degree of the equation system Replacing the notion of geometric degree of the given polynomial equation system by a suitably de ned real or complex degree of certain polar varieties associated to Research partially supported by the Spanish government grant DGICT PB C Humboldt Universit at zu Berlin Untern den Linden D Berlin Germany bank mathematik hu berlin de mbakop mathematik hu berlin de GAGE Centre de Math ematiques Ecole Polytechnique F Palaiseau Cedex France giusti ariana polytechnique fr Dept de Matem aticas Estad stica y Computaci on Facultad de Ciencias Universidad de Cantabria E Santander Spain heintz matsun unican es Real equation solving the input equation of the real hypersurface under consideration we are able to nd for each connected component of the hypersurface a representative point this point will be given in a suitable encoding The input equation is supposed to be given by a straight line program and the sequential time complexity of the algorithm is polynomial in the input length and the degree of the polar varieties mentioned above