Lazard-style CAD and Equational Constraints

TL;DR

This paper explores Lazard-style CAD, addressing nullification issues when transporting equational constraints, and demonstrates how to achieve similar improvements as McCallum-style CAD without nullification failures.

Contribution

It adapts Lazard-style CAD to handle equational constraints effectively, solving nullification problems and extending improvements to multiple constraints.

Findings

Lazard-style CAD can match McCallum-style improvements with a single equational constraint.

The proposed approach avoids nullification failures present in McCallum-style CAD.

The method extends to multiple equational constraints, with some limitations.

Abstract

McCallum-style Cylindrical Algebra Decomposition (CAD) is a major improvement on the original Collins version, and has had many subsequent advances, notably for total or partial equational constraints. But it suffers from a problem with nullification. The recently-justified Lazard-style CAD does not have this problem. However, transporting the equational constraints work to Lazard-style does reintroduce nullification issues. This paper explains the problem, and the solutions to it, based on the second author's Ph.D. thesis and the Brown--McCallum improvement to Lazard. With a single equational constraint, we can gain the same improvements in Lazard-style as in McCallum-style CAD . Moreover, our approach does not fail where McCallum would due to nullification. Unsurprisingly, it does not achieve the same level of improvement as it does in the non-nullified cases. We also consider the case of multiple equational constraints.