An Improved Projection Operation for Cylindrical Algebraic Decomposition

Abstract

A key component of the cylindrical algebraic decomposition (cad) algorithm is the projection (or elimination) operation: the projection of a set A of r-variate integral polynomials, where r ≥ 2 is defined to be a certain set PROJ(A) of (r − l)-variate integral polynomials. The property of the map PROJ of particular relevance to the cad algorithm is that, for any finite set A of r-variate integral polynomials, where r ≥ 2, if S is any connected subset of (r − 1)-dimensional real space ℝ^(r−1) in which every element of PROJ(A) is invariant in sign then the portion of the zero set of the product of those elements of A which do not vanish identically on S that lies in the cylinder S × ℝ over S consists of a number (possibly 0) of disjoint “layers” (or sections) over S in each of which every element of A is sign-invariant: that is, A is “delineable” on S. It follows from this property that, for any finite set A of r-variate integral polynomials, r ≥ 2, any decomposition of ℝ^(r−1) into connected regions such that every polynomial in PROJ (A) is invariant in sign throughout every region can be extended to a decomposition of ℝ^r (consisting of the union of all of the above-mentioned layers and the regions in between successive layers, for each region of ℝ^(r−1) such that every polynomial in A is invariant in sign throughout every region of ℝ^r.