On the practical computation of one point in each connected component of a semi-algebraic set defined by a polynomial system of equations and non-strict inequalities

Colas Le Guernic Mohab Safey El Din 1, 2
1 SPACES - Solving problems through algebraic computation and efficient software
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : Given polynomials f_1, , f_k, g_1, , g_s in , we consider the semi-algebraic set defined by: \begin{array}l f_1== f_k=0 g_10, , g_s0 \end{array} and focus on the problem of computing at least one point in each connected component of . We first study how to solve this problem by considering as the union of solutions sets of polynomial systems of equations and strict inequalities and proceed to the complexity analysis of the underlying algorithm. Then, we improve this approach by proving that computing at least one point in each connected component of can be done by computing at least one point in each connected component of real algebraic sets defined by vanishing the polynomials f_1, , f_k and some of the polynomials g_1, , g_s. The complexity analysis shows that this latter approach is better than the former one. Finally, we present our implementation and use it to solve an application in Pattern Matching.
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https://hal.inria.fr/inria-00071504
Contributor : Rapport de Recherche Inria <>
Submitted on : Tuesday, May 23, 2006 - 5:47:31 PM
Last modification on : Thursday, March 21, 2019 - 1:00:36 PM
Long-term archiving on : Sunday, April 4, 2010 - 10:20:26 PM

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  • HAL Id : inria-00071504, version 1

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Colas Le Guernic, Mohab Safey El Din. On the practical computation of one point in each connected component of a semi-algebraic set defined by a polynomial system of equations and non-strict inequalities. [Research Report] RR-5079, INRIA. 2004. ⟨inria-00071504⟩

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