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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| Vydáno v: | Mathematics in computer science Ročník 8; číslo 2; s. 263 - 288 |
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01.06.2014
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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
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| 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. |
| Author_xml | – sequence: 1 givenname: D. J. surname: Wilson fullname: Wilson, D. J. organization: Department of Computer Science, University of Bath – sequence: 2 givenname: R. J. surname: Bradford fullname: Bradford, R. J. organization: Department of Computer Science, University of Bath – sequence: 3 givenname: J. H. surname: Davenport fullname: Davenport, J. H. organization: Department of Computer Science, University of Bath – sequence: 4 givenname: M. surname: England fullname: England, M. email: M.England@bath.ac.uk organization: Department of Computer Science, University of Bath |
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| Keywords | Cylindrical algebraic decomposition Real algebraic geometry 68W30 (Symbolic Computation and Algebraic Computation) Symbolic computation Computer algebra Equational constraints |
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