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 $C^r$ ($r \in \mathbb{N} \cup \{\infty, ω\}$) 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 $C^r$.

翻译：本文考虑将柱形代数分解（CAD）作为表示 $\mathbb{R}^n$ 中半代数子集的工具。在此框架下，若集合 $S$ 是 CAD $\mathscr{C}$ 中若干单元的并集，则称 $\mathscr{C}$ 适配于 $S$。计算适配 CAD 的不同算法可能产生不同的输出，通常包含冗余的单元划分。本文分析消除冗余数据的可能性，为此考察 $C^r$ 类（$r \in \mathbb{N} \cup \{\infty, ω\}$）CAD 的集合 $\text{CAD}^r(\mathcal{F})$，其中每个 CAD 均适配于 $\mathbb{R}^n$ 中有限半代数集族 $\mathcal{F}$，该集合配备细化偏序关系，我们研究 $\text{CAD}^r(\mathcal{F})$ 中极小元与最小元的存在性。证明对任意这样的 $\mathcal{F}$ 和任意 $\mathscr{C} \in \text{CAD}^r(\mathcal{F})$，总存在 $C^r$ 类的极小 CAD 适配于 $\mathcal{F}$，且小于（即粗于）或等于 $\mathscr{C}$。在维度 $n=1$ 或 $n=2$ 时，通过证明 $\text{CAD}^r(\mathcal{F})$ 存在最小元强化了该结论。相反地，对任意 $n \geq 3$，我们给出半代数集的显式例子，其适配 CAD 的偏序集不存在最小元。随后在 $\text{CAD}^r(\mathcal{F})$ 上引入约化关系以定义极小 CAD 的计算算法，并通过关联约化系统的合流性来刻画使 $\text{CAD}^r(\mathcal{F})$ 存在最小元的半代数集 $\mathcal{F}$ 的特征。最后提出判定半代数集是否容许最小 CAD 的实用准则，并应用于描述若干具体半代数集实例及其对应的 $C^r$ 类最小 CAD。