Polar varieties and computation of one point in each connected component of a smooth real algebraic set

Authors:
Mohab Safey El Din;Éric Schost
Affiliations:
Université Paris, Paris, France;École polytechnique, Palaiseau, France
Venue:
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Year:
2003

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Abstract

Let f1, ldots, fs be polynomials in Q[X1, ..., Xn] that generate a radical ideal and let V be their complex zero-set. Suppose that V is smooth and equidimensional; then we show that computing suitable sections of the polar varieties associated to generic projections of V gives at least one point in each connected component of V ∩ Rn. We deduce an algorithm that extends that of Bank, Giusti, Heintz and Mbakop to non-compact situations. Its arithmetic complexity is polynomial in the complexity of evaluation of the input system, an intrinsic algebraic quantity and a combinatorial quantity.