Regular cylindrical algebraic decomposition

TL;DR

This paper proves that certain well-structured cylindrical algebraic decompositions of bounded semi-algebraic sets are regular cell decompositions, with conditions applicable to many existing algorithms and specific low-dimensional cases.

Contribution

It establishes that strong well-based CADs are regular cell decompositions regardless of construction method, and identifies local boundary simple connectivity as a sufficient condition in low dimensions.

Findings

Strong well-based CADs are regular cell decompositions.

The result applies to many widely used algorithms.

In dimensions up to 3, local boundary simple connectivity suffices.

Abstract

We show that a strong well-based cylindrical algebraic decomposition P of a bounded semi-algebraic set is a regular cell decomposition, in any dimension and independently of the method by which P is constructed. Being well-based is a global condition on P that holds for the output of many widely used algorithms. We also show the same for S of dimension at most 3 and P a strong cylindrical algebraic decomposition that is locally boundary simply connected: this is a purely local extra condition.