Improved Projection for Cylindrical Algebraic Decomposition

Cylindrical algebraic decomposition (CAD) is an important tool for the investigation of real algebraic and semi-algebraic sets. Introduced by Collins in the early 1970s (Collins, 1975) as the basis of his quantifier elimination method, the algorithm for CAD construction has been steadily improved, and has found application in many areas including stability analysis (Hong et al., 1997) and numerical integration (Strzebonski, 2000). Given a set A ⊂ R[x1, . . . , xk], the CAD algorithm constructs a decomposition of R into cylindrically arranged cells such that the signs of the elements of A are constant inside any given cell. This cylindrical arrangement means that the projections onto Rk−1 of any two cells are either identical or disjoint. CAD construction proceeds in two phases, projection and lifting. The projection phase, which is the focus of this paper, computes a set of polynomials called the projection factor set. The projection factor set contains the irreducible factors of the set A, and, in general, other polynomials as well. The maximal connected regions in which the projection factors have invariant signs are the cells of the CAD that is to be constructed. Thus, the projection factor set provides an implicit representation of the CAD. The lifting phase then constructs an explicit representation of this CAD. General descriptions of CAD construction may be found in Collins and Hong (1991), Arnon et al. (1984), and Collins (1975). The projection phase is typically determined by a projection operator which, from the set A, defines a set A′ ⊂ R[x1, . . . , xk−1] such that if c ⊆ Rk−1 is a cell in a CAD produced from A′, then the maximal connected regions in c × R in which the elements of A have invariant sign are cylindrically arranged. Thus, after applying the projection operator to A to produce A′, we are left with the problem of producing a CAD in whose cells