### 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 language | English |
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Pages (from-to) | 263-288 |

Number of pages | 26 |

Journal | Mathematics in Computer Science |

Volume | 8 |

Issue number | 2 |

Early online date | 13 Jun 2014 |

DOIs | |

Publication status | Published - 2014 |

Externally published | Yes |

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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

### Cite this

*Mathematics in Computer Science*,

*8*(2), 263-288. https://doi.org/10.1007/s11786-014-0191-z