The McCallum Projection , Lifting , and Order - Invariance Brown , Christopher W . USNA - CS - TR - 2005

The McCallum Projection for Cylindrical Algebraic Decomposition (CAD) produces a smaller projection factor set than previous projections, however it does not always produce a sign-invariant CAD for the set of input polynomials. Problems may arise when a (k + 1)-level projection factor vanishes identically over a k-level cell. According to McCallum’s paper, when this happens (and k+1 is not the highest level in the CAD) we do not know whether the projection is valid, i.e. whether or not a signinvariant CAD for the set of input polynomials will be produced when lifting is performed in the usual way. When the k-level cell in question has dimension 0, McCallum suggests a modification of the lifting method that will ensure the validity of his projection, although to my knowledge this has never been implemented. In this paper we give easily computable criteria that often allow us to conclude that McCallum’s projection is valid even though a projection factor vanishes identically over a cell. We also improve on McCallum’s modified lifting method. We’ve incorporated the ideas contained in this paper into QEPCAD, the most complete implementation of CAD. When McCallum’s projection is invalid because of a projection factor not being order-invariant over a region on which it vanishes identically, at least a warning message ∗This article was originally posted on my website in September 2001 as MOTS2001.1.ps.gz. It is now being reformatted and republished as a USNA CS Department Technical Report. All references to “current” are to be taken as “current as of 1 September, 2001.