Improvements in cad-based quantifier elimination

Many important mathematical and applied mathematical problems can be formulated as quantifier elimination problems (QE) in elementary algebra and geometry. Among several proposed QE methods, the best one is the CAD-based method due to G. E. Collins. The major contributions of this thesis are centered around improving each phase of the CAD-based method: the projection phase, the truth-invariant CAD construction phase, and the solution formula construction phase. Improvements in the projection phase. By generalizing a lemma on which the proof of the original projection operation is based, we are able to reduce the size of the projection set significantly, and thus speed up the whole QE process. Improvements in the truth-invariant CAD construction phase. By intermingling the stack constructions with the truth evaluations, we are usually able to complete quantifier elimination with only a partially built CAD, resulting in significant speedup. Improvements in the solution formula construction phase. By constructing defining formulas for a collection of cells instead of individual cells, we are often able to produce simple solution formulas, without incurring the enormous cost of augmented projection. The new CAD-based QE algorithm, which integrates all the improvements above, has been implemented and tested on ten QE problems from diverse application areas. The overall speedups range from 2 to perhaps 300,000 times at least. We also implemented D. Lazard's recent improvement on the projection phase. Tests on the ten QE problems show additional speedups by up to 1.8 times.