The McCallum Projection, Lifting, and Order-Invariance

Abstract

Abstract : 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 the 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 sign-invariant 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 neve 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 have 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 ought to be issued. Currently, QEPCAD may print warning messages that are not needed, and may fail to print warning messages when they are needed. Our implementation in QEPCAD ensures that warning messages are printed when needed, and reduces the number of times warning messages are printed when not needed. Neither McCallum's modified lifting method nor our improvement of it have been implemented in QEPCAD- the design of the system would make implementing such a feature quite difficult.