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Computing with SemiAlgebraic Sets: Relaxation Techniques and Effective Boundaries
"... We discuss parametric polynomial systems, with algorithms for real root classification and triangular decomposition of semialgebraic systems as our main applications. We exhibit new results in the theory of border polynomials of parametric semialgebraic systems: in particular a geometric character ..."
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We discuss parametric polynomial systems, with algorithms for real root classification and triangular decomposition of semialgebraic systems as our main applications. We exhibit new results in the theory of border polynomials of parametric semialgebraic systems: in particular a geometric characterization of its “true boundary ” (Definition 1). In order to optimize the corresponding decomposition algorithms, we also propose a technique, that we call relaxation, which can simplify the decomposition process and reduce the number of components in the output. This paper extends our earlier works [6, 7]. Key words: triangular decomposition, regular semialgebraic system, border polynomial, effective boundary, relaxation. 1.