Computing with semi-algebraic sets: Relaxation techniques and effective boundaries

Authors:
Changbo Chen;James H. Davenport;Marc Moreno Maza;Bican Xia;Rong Xiao
Affiliations:
University of Western Ontario, London, Ontario, Canada N6A 5B7;University of Bath, Bath BA2 7AY, United Kingdom;University of Western Ontario, London, Ontario, Canada N6A 5B7;Peking University, Beijing 100871, China;University of Western Ontario, London, Ontario, Canada N6A 5B7
Venue:
Journal of Symbolic Computation
Year:
2013

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Abstract

We discuss parametric polynomial systems, with algorithms for real root classification and triangular decomposition of semi-algebraic systems as our main applications. We exhibit new results in the theory of border polynomials of parametric semi-algebraic 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 (Chen et al., 2010, 2011).