Further results on Minimal and Minimum Cylindrical Algebraic Decompositions

TL;DR

This paper investigates the structure of cylindrical algebraic decompositions (CADs) adapted to semi-algebraic sets, analyzing minimal and minimum CADs, their existence, and algorithms for computing them, with results varying by dimension.

Contribution

It introduces a framework for understanding minimal and minimum CADs, establishes their existence in low dimensions, and develops algorithms and criteria for their computation and identification.

Findings

Existence of minimal CADs for all finite families of semi-algebraic sets.

In dimensions 1 and 2, minimum CADs always exist.

For dimensions 3 and higher, minimum CADs may not exist, with explicit counterexamples.

Abstract

We consider cylindrical algebraic decompositions (CADs) as a tool for representing semi-algebraic subsets of $\mathbb{R}^n$. In this framework, a CAD $\mathscr{C}$ is adapted to a given set $S$ if $S$ is a union of cells of $\mathscr{C}$. Different algorithms computing an adapted CAD may produce different outputs, usually with redundant cell divisions. In this paper we analyse the possibility to remove the superfluous data. We thus consider the set $\text{CAD}^r(\mathcal{F})$ of CADs of class $C^r$ ($r \in \mathbb{N} \cup \{\infty, \omega\}$) that are adapted to a finite family $\mathcal{F}$ of semi-algebraic sets of $\mathbb{R}^n$, endowed with the refinement partial order and we study the existence of minimal and minimum element in $\text{CAD}^r(\mathcal{F})$. We show that for every such $\mathcal{F}$ and every $\mathscr{C} \in \text{CAD}^r(\mathcal{F})$, there is a minimal CAD of class $C^r$ adapted to $\mathcal{F}$ and smaller (i.e. coarser) than or equal to $\mathscr{C}$. In dimension $n=1$ or $n=2$, this result is strengthened by proving the existence of a minimum element in $\text{CAD}^r(\mathcal{F})$. In contrast, for any $n \geq 3$, we provide explicit examples of semi-algebraic sets whose associated poset of adapted CADs does not admit a minimum. We then introduce a reduction relation on $\text{CAD}^r(\mathcal{F})$ in order to define an algorithm for the computation of minimal CADs and we characterise those semi-algebraic sets $\mathcal{F}$ for which $\text{CAD}^r(\mathcal{F})$ has a minimum by means of confluence of the associated reduction system. We finally provide practical criteria for deciding if a semi-algebraic set does admit a minimum CAD and apply them to describe various concrete examples of semi-algebraic sets, along with their minimum CAD of class $C^r$.