Constructing a single open cell in a cylindrical algebraic decomposition

Abstract

This paper presents an algorithm that, roughly speaking, constructs a single cell from a cylindrical algebraic decomposition (CAD). The algorithm takes as input a point and a set of polynomials, and computes a description of an cylindrical cell containing the point in which the input polynomials have constant non-zero sign, provided the point is sufficiently generic. The paper reports on a few example computations carried out by a test implementation of the algorithm, which demonstrate the functioning of the algorithm and illustrate the sense in which it is more efficient than following the usual open approach. Interest in the problem of computing a single cell from a CAD is motivated by a 2012 paper of Jovanovic and de Moura that require solving this problem repeatedly as a key step in NLSAT system. However, the example computations raise the possibility that repeated application of the new method may in fact be more efficient than the usual CAD approach, both in time and space, for a broad range of problems.