Cylindrical Algebraic Sub-Decompositions

D. J. Wilson, R. J. Bradford, J. H. Davenport, M. England

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Abstract

Cylindrical algebraic decompositions (CADs) are a key tool in real algebraic geometry, used primarily for eliminating quantifiers over the reals and studying semi-algebraic sets. In this paper we introduce cylindrical algebraic sub-decompositions (sub-CADs), which are subsets of CADs containing all the information needed to specify a solution for a given problem. We define two new types of sub-CAD: variety sub-CADs which are those cells in a CAD lying on a designated variety; and layered sub-CADs which have only those cells of dimension higher than a specified value. We present algorithms to produce these and describe how the two approaches may be combined with each other and the recent theory of truth-table invariant CAD. We give a complexity analysis showing that these techniques can offer substantial theoretical savings, which is supported by experimentation using an implementation in Maple.


Original languageEnglish
Pages (from-to)263-288
Number of pages26
JournalMathematics in Computer Science
Volume8
Issue number2
Early online date13 Jun 2014
DOIs
Publication statusPublished - 2014
Externally publishedYes

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Bibliographical note

Publisher Statement: The final publication is available at Springer via http://dx.doi.org/10.1007/s11786-014-0191-z

Keywords

  • Computer algebra
  • Cylindrical algebraic decomposition
  • Equational constraints
  • Real algebraic geometry
  • Symbolic computation

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Computational Theory and Mathematics

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