Cylindrical Algebraic Sub-Decompositions

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...

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Published in:	Mathematics in computer science Vol. 8; no. 2; pp. 263 - 288
Main Authors:	Wilson, D. J., Bradford, R. J., Davenport, J. H., England, M.
Format:	Journal Article
Language:	English
Published:	Basel Springer Basel 01.06.2014
Subjects:	Computer Science Mathematics Mathematics and Statistics Cylindrical algebraic decomposition Real algebraic geometry 68W30 (Symbolic Computation and Algebraic Computation) Symbolic computation Computer algebra Equational constraints
ISSN:	1661-8270, 1661-8289
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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 .
AbstractList	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 .
Author	Davenport, J. H. Wilson, D. J. Bradford, R. J. England, M.
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Title	Cylindrical Algebraic Sub-Decompositions
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