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.
Highlights
1.1 MotivationA cylindrical algebraic decomposition (CAD) is a decomposition of Rn into cells arranged cylindrically each of which is a semialgebraic set
In this paper we focus on improvements to the lifting phase, but if only a 1-layered CAD is required this further saving in the projection phase is available
We provide a complexity analysis of the algorithms to compute sign-invariant variety sub-CADs and 1-layered variety sub-CADs in the case where the equational constraint has all factors with main variable xn
Summary
A cylindrical algebraic decomposition (CAD) is a decomposition of Rn into cells arranged cylindrically (meaning the projections of any pair of cells onto the first k coordinates are either equal or disjoint) each of which is a semialgebraic set (and so may be described by polynomial relations). Often a problem will be represented by a formula and we require a CAD such that the formula has constant truth value on each cell This can be achieved by building a CAD sign-invariant for the polynomials in the formula, but that may introduce cell divisions not relevant to the formula itself. The projection operator used must ensure that over each cell of a sign-invariant CAD for the projection polynomials in r variables, the polynomials in r + 1 variables are delineable
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