Polar Varieties and Efficient Real Elimination

Details Polar Varieties and Efficient Real Elimination Polar Varieties and Efficient Real Elimination

2000-05-04

arxiv.org/abs/math/0005041v1

MSRI Preprint N0. 1999-056

Mathematics

Algebraic Geometry

32 pages

Scientific paper

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 $f_1, >..., f_p$.

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