Abstract
The Cylindrical Algebraic Decomposition (CAD) method of Collins [5] decomposes r-dimensional Euclidean space into regions over which a given set of polynomials have constant signs. An important component of the CAD method is the projection operation: given a set A of r-variate polynomials, the projection operation produces a set P of (r - 1)-variate polynomials such that a CAD of r-dimensional space for A can be constructed from a CAD of (r-1)-dimensional space for P. In this paper, we present an improvement to the projection operation. By generalizing a lemma on which the proof of the original projection operation is based, we are able to find another projection operation which produces a smaller number of polynomials. Let m be the number of polynomials contained in A, and let n be a bound for the degree of each polynomial in A in the projection variable. The number of polynomials produced by the original projection operation is dominated by m2n3 whereas the number of polynomials produced by our projection operation is dominated by m2n2.
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