0 Polar Varieties and Efficient Real Elimination 1

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 .ICET 4571/96. The first two authors wish to thank the MSRI at Berkeley for its hospitality during their stay, fall 1998.