Polar Varieties and Eecient Real Equation Solving: the Hypersurface Case

The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo 10] can be applied to a case of real polynomial equation solving. Our main result concerns the problem of nding one representative point for each connected component of a real bounded smooth hypersurface. The algorithm in 10] yields a method for symbolically solving a zero-dimensional polynomial equation system in the aane (and toric) case. Its main feature is the use of adapted data structure: Arithmetical networks and straight-line programs. The algorithm solves any aane zero-dimensional equation system in non-uniform sequential time that is polynomial in the length of the input description and an adequately deened aane degree of the equation system. Replacing the aane degree of the equation system by a suitably deened real degree of certain polar varieties associated to the input equation, which describes the hypersurface under consideration, and using straight-line program codiication of the input and intermediate results, we obtain a method for the problem introduced above that is polynomial in the input length and the real degree.