Cylindrical Algebraic Sub-Decompositions

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

Research output: Contribution to journalArticlepeer-review

17 Citations (Scopus)
27 Downloads (Pure)


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
Issue number2
Early online date13 Jun 2014
Publication statusPublished - 2014
Externally publishedYes

Bibliographical note

Publisher Statement: The final publication is available at Springer via


  • 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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