Polar Varieties, Real Equation Solving, and Data Structures: The Hypersurface Case

In this paper we apply for the rst time a new method for multivariate equation solving which was developed in 18], 19], 20] 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 18], 19], 20] yields a new method for symbolically solving 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 aane 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 deened geometric degree of the equation system. Replacing the notion of geometric degree of the given polynomial equation system by a suitably deened real (or complex) degree of certain polar varieties associated to 1 2 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.