### Abstract

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 - 13 Jun 2014 |

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### Keywords

- cylindrical algebraic decomposition
- symbolic computation
- real algebraic geometry
- equational constraints
- computer algebra

### Cite this

*Mathematics in Computer Science*,

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

**Cylindrical algebraic sub-decompositions.** / Wilson, D.; Bradford, R.; Davenport, J. H.; England, M.

Research output: Contribution to journal › Article

*Mathematics in Computer Science*, vol. 8, no. 2, pp. 263-288. https://doi.org/10.1007/s11786-014-0191-z

}

TY - JOUR

T1 - Cylindrical algebraic sub-decompositions

AU - Wilson, D.

AU - Bradford, R.

AU - Davenport, J. H.

AU - England, M.

PY - 2014/6/13

Y1 - 2014/6/13

N2 - 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.

AB - 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.

KW - cylindrical algebraic decomposition

KW - symbolic computation

KW - real algebraic geometry

KW - equational constraints

KW - computer algebra

U2 - 10.1007/s11786-014-0191-z

DO - 10.1007/s11786-014-0191-z

M3 - Article

VL - 8

SP - 263

EP - 288

JO - Mathematics in Computer Science

JF - Mathematics in Computer Science

SN - 1661-8270

IS - 2

ER -