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 Zénaïdi 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.

翻译：柱形代数分解（Cylindrical Algebraic Decomposition, CAD）算法通常生成适应于有限族半代数集合 $\mathcal{F}$ 的分解（即 $\mathcal{F}$ 中的每个成员均为单元之并）。不同算法可能产生不同输出，并引入不必要的单元划分。Michel、Mathonet 与 Zénaïdi 在 ISSAC 2024 的近期工作中，通过研究所有适应于 $\mathcal{F}$ 的 CAD 构成的集合上的细化序，并分析最小（最粗）适应 CAD 的存在性，形式化了该问题。结果表明，对于 $\mathbb{R}$ 和 $\mathbb{R}^2$ 的子集，此类最小适应 CAD 总是存在，但对 $\mathbb{R}^n$（$n \geqslant 3$）通常不成立。因此，自然需要寻找 $\mathbb{R}^n$ 中允许最小适应 CAD 存在的自然子集类。本文确定了 $\mathbb{R}^3$ 的一类子集，其中包含所有存在最小适应 CAD 的代数集。这为非平凡集合类的最小 CAD 存在性提供了首个正面存在性定理。