Computing with semi-algebraic sets represented by triangular decomposition

  • Authors:

  • Changbo Chen;James H. Davenport;Marc Moreno Maza;Bican Xia;Rong Xiao

  • Affiliations:

  • University of Western Ontario, London, Canada;University of Bath, Bath, United Kingdom;University of Western Ontario, London, Canada;Peking University, Beijing, China;UNIVERSITY OF WESTERN ONTARIO, London, Canada

  • Venue:
  • Proceedings of the 36th international symposium on Symbolic and algebraic computation
  • Year:
  • 2011

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Abstract

This article is a continuation of our earlier work [3], which introduced triangular decompositions of semi-algebraic systems and algorithms for computing them. Our new contributions include theoretical results based on which we obtain practical improvements for these decomposition algorithms. We exhibit new results on the theory of border polynomials of parametric semi-algebraic systems: in particular a geometric characterization of its "true boundary" (Definition 2). In order to optimize these algorithms, we also propose a technique, that we call relaxation, which can simplify the decomposition process and reduce the number of redundant components in the output. Moreover, we present procedures for basic set-theoretical operations on semi-algebraic sets represented by triangular decomposition. Experimentation confirms the effectiveness of our techniques.