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