  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 {\em 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 {\em degree} of the real interpretation of the given equation system $f_1,
>..., f_p$.
