On Minimum CADs for Algebraic Sets in Dimension Three

TL;DR

This paper identifies a class of algebraic sets in three-dimensional space for which minimum cylindrical algebraic decompositions (CADs) exist, addressing a key open problem in CAD theory.

Contribution

It introduces the first positive existence theorem for minimum CADs in a non-trivial class of algebraic sets in three dimensions.

Findings

Minimum adapted CADs exist for a specific class of algebraic sets in R^3.

Such sets include all algebraic sets within this class.

This provides a new understanding of CAD minimality in three dimensions.

Abstract

Cylindrical Algebraic Decomposition (CAD) algorithms typically produce a decomposition adapted to a finite family of semi-algebraic sets $\mathcal{F}$ (i.e. every member of $\mathcal{F}$ is a union of cells). Different algorithms may produce different outputs, and introduce unnecessary cell divisions. Recent work by Michel, Mathonet, and Z\'ena\"idi in ISSAC 2024 formalised this issue by studying the refinement order on the set of all CADs adapted to $\mathcal{F}$ and analysing the existence of a minimum (coarsest) adapted CAD. It was shown that such a minimum adapted CAD always exists for subsets of $\mathbb{R}$ and $\mathbb{R}^2$, but not of $\mathbb{R}^n$ ($n \geqslant 3$) in general. It is natural to seek natural classes of subsets of $\mathbb{R}^n$ that admit a minimum adapted CAD. In this paper, we identify a class of subsets of $\mathbb{R}^3$ that contains all algebraic sets for which minimum adapted CADs do exist. This provides the first positive existence theorem for minimum CAD for a non-trivial class of sets.