Polar Varieties and Efficient Real Elimination
Bernd Bank, Marc Giusti, Joos Heintz, and Guy Merlin Mbakop
Let $S_0$ be a smooth and compact real variety given by a reduced regular sequence of polynomials $f_1 ,\ldots, f_p$. 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 $f_1 ,\ldots, f_p$ and in a suitably introduced, intrinsic geometric parameter, called the degree of the real interpretation of the given equation system $f_1 ,\ldots, f_p$.