# 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 }.

## Keywords

Power Series Expansion Common Zero Positive Degree Quantifier Elimination Projection Factor## Preview

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