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

中文翻译：

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正则圆柱代数分解

我们证明了一个强大的，基于良好的圆柱代数分解 $\mathcal{P}$ 有界半代数集 $\mathrm{小号}$ 是规则的细胞分解，在任何维度上均与方法无关 $\mathcal{P}$被建造。良好的基础是全球的条件$\mathcal{P}$这适用于许多广泛使用的算法的输出。我们也显示相同的$\mathrm{小号}$ 最多尺寸 $3$ 和 $\mathcal{P}$ 在局部边界简单连接的强圆柱代数分解：这是纯粹的局部额外条件。