Using the Regular Chains Library to build cylindrical algebraic decompositions by projecting and lifting

TL;DR

This paper presents ProjectionCAD, a Maple package that constructs cylindrical algebraic decompositions using projection and lifting methods based on the RegularChains Library, enhancing CAD construction techniques.

Contribution

It introduces a new implementation of CAD construction via projection and lifting within the RegularChains Library, combining complex space decomposition with real space refinement.

Findings

Efficient CAD construction using the ProjectionCAD package

Integration of complex space decomposition with real space CAD

Enhanced algorithms for projection and lifting in CAD

Abstract

Cylindrical algebraic decomposition (CAD) is an important tool, both for quantifier elimination over the reals and a range of other applications. Traditionally, a CAD is built through a process of projection and lifting to move the problem within Euclidean spaces of changing dimension. Recently, an alternative approach which first decomposes complex space using triangular decomposition before refining to real space has been introduced and implemented within the RegularChains Library of Maple. We here describe a freely available package ProjectionCAD which utilises the routines within the RegularChains Library to build CADs by projection and lifting. We detail how the projection and lifting algorithms were modified to allow this, discuss the motivation and survey the functionality of the package.