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Properness defects of projections and computation of at least one point in each connected component of a real algebraic set

Mohab Safey El Din 1, 2 Eric Schost 3
2 SPACES - Solving problems through algebraic computation and efficient software
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : Computing at least one point in each connected component of a real algebraic set is a basic subroutine to decide emptiness of semi-algbraic sets, which is a fundamental algorithmic problem in effective real algebraic geometry. In this article, we propose a new algorithm for this task, which avoids a hypothesis of properness required in many of the previous methods. We show how studying the set of non-properness of a linear projection enables to detect connected components of a real algebraic set without critical points. Our algorithm is based on this result and its practical counterpoint, using the triangular representation of algebraic varieties. Our experiments show its efficiency on a family of examples.
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https://hal.inria.fr/inria-00099962
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Submitted on : Tuesday, September 26, 2006 - 10:12:57 AM
Last modification on : Thursday, March 5, 2020 - 6:30:49 PM

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Mohab Safey El Din, Eric Schost. Properness defects of projections and computation of at least one point in each connected component of a real algebraic set. Discrete and Computational Geometry, Springer Verlag, 2004, 32 (3), pp.417-430. ⟨10.1007/s00454-004-1107-5⟩. ⟨inria-00099962⟩

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