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

Mohab Safey El Din 1, 2 Eric Schost 3
1 SPACES - Solving problems through algebraic computation and efficient software
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
2 CALFOR - Calcul formel
LIP6 - Laboratoire d'Informatique de Paris 6
Abstract : Let $f_1, \ldots, f_s$ be polynomials in $\Q[X_1, \ldots, X_n]$ 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 $V\cap\R^n$. 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.
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Conference papers
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Submitted on : Tuesday, September 26, 2006 - 9:39:45 AM
Last modification on : Thursday, March 21, 2019 - 12:59:18 PM

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Mohab Safey El Din, Eric Schost. Polar varieties and computation of one point in each connected component of a smooth real algebraic set. International Symposium on Symbolic and Algebraic Computation 2003 - ISSAC'2003, Aug 2003, Philadelphie, PA, United States. pp.224-231, ⟨10.1145/860854.860901⟩. ⟨inria-00099649⟩

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