Computing cylindrical algebraic decomposition via triangular decomposition

Abstract

Cylindrical algebraic decomposition is one of the most important tools for computing with semi-algebraic sets, while triangular decomposition is among the most important approaches for manipulating constructible sets. In this paper, for an arbitrary finite set F ⊂ R[y1, . . . , yn] we apply comprehensive triangular decomposition in order to obtain an F -invariant cylindrical decomposition of the n-dimensional complex space, from which we extract an F -invariant cylindrical algebraic decomposition of the n-dimensional real space. We report on an implementation of this new approach for constructing cylindrical algebraic decompositions.

Termination conditions for positivity proving procedures

“…All CAD computations carried out in this paper were performed with Mathematica's built-in implementation of CAD [24,25]. The computation time is negligible for all of them and could certainly be carried out with other implementations such as, e.g., [11,4,8] as well.…”

Cylindrical Algebraic Decomposition in the RegularChains Library

“…For each cell C of it, one evaluates the polynomials of A in n variables at a sample point of the cell and obtains a set of univariate polynomials. Isolating the real roots of them allows one to deduce all the cells of the CAD of R n whose projection are C. In [4], a different method for computing CADs was proposed. It first produces a cylindrical decomposition of the complex space (CCD) through the computation of regular GCDs, and then refines the CCD into a CAD of the real space by isolating real roots of univariate polynomials with real algebraic number coefficients encoded by regular chains and isolating boxes.…”

“…Both algorithms are based on triangular decomposition of polynomial systems and real root isolation of regular chains. For this reason, we call the CAD as computed in [4,3] RC-CAD. The algorithm of [4] was firstly implemented in the RegularChains library of Maple 14.…”

“…For this reason, we call the CAD as computed in [4,3] RC-CAD. The algorithm of [4] was firstly implemented in the RegularChains library of Maple 14. The implementation was revised in Maple 16 and has remained the same in the subsequent versions of Maple.…”

Cylindrical algebraic decomposition using local projections

“…Computation with semialgebraic sets is one of the core subjects in computer algebra and real algebraic geometry. A variety of algorithms have been developed for real system solving, satisfiability checking, quantifier elimination, optimization and other basic problems concerning semialgebraic sets [7,1,5,6,9,10,12,15,18,24,25]. Every semialgebraic set can be represented as a finite union of disjoint cells bounded by graphs of algebraic functions.…”