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Let S 0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f 1 , . . . , f p . This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S 0 . For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of S 0 . This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations f 1 , . . . , f p and in a suitably introduced, intrinsic geometric parameter, called the degree of the real interpretation of the given equation system f 1 , . . . , f p .

Let V be a closed algebraic subvariety of the n-dimensional projective space over the complex or real numbers and suppose that V is non-empty and equidimensional. The classic notion of a polar variety of V associated with a given linear subvariety of the ambient space of V was generalized and motivated in Bank et al. (Kybernetika 40 (2004), to appear). As particular instances of this notion of a generalized polar variety one reobtains the classic one and an alternative type of a polar variety, called dual. As main result of the present paper we show that for a generic choice of their parameters the generalized polar varieties of V are empty or equidimensional and smooth in any regular point of V. In the case that the variety V is affine and smooth and has a complete intersection ideal of definition, we are able, for a generic parameter choice, to describe locally the generalized polar varieties of V by explicit equations. Finally, we indicate how this description may be used in order to design in the context of algorithmic elimination theory a highly efficient, probabilistic elimination procedure for the following task: In case, that the variety V is Q-definable and affine, having a complete intersection ଁ ideal of definition, and that the real trace of V is non-empty and smooth, find for each connected component of the real trace of V an algebraic sample point.

DEDICATED TO SHMUEL WINOGRADIn this paper we apply for the first time a new method for multivariate equation solving which was developed for complex root determination to the real case. Our main result concerns the problem of finding at least one representative point for each connected component of a real compact and smooth hypersurface. The basic algorithm 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 finds the complex solutions

We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are necessary to prove the correctness and complexity estimates of these algorithms. Our results form also the geometrical main ingredients for the computational treatment of singular hypersurfaces.In particular, we show the non-emptiness of suitable generic dual polar varieties of (possibly singular) real varieties, show that generic polar varieties may become singular at smooth points of the original variety and exhibit a sufficient criterion when this is not the case. Further, we introduce the new concept of meagerly generic polar varieties and give a degree estimate for them in terms of the degrees of generic polar varieties. The statements are illustrated by examples and a computer experiment.