Piecewise-linear (PL) Morse theory and discrete Morse theory are used in shape analysis tasks to investigate the topological features of discretized spaces. In spite of their common origin in smooth Morse theory, various notions of critical points have been given in the literature for the discrete setting, making a clear understanding of the relationships occurring between them not obvious. This paper aims at providing equivalence results about critical points of the two discretized Morse theories. First of all, we prove the equivalence of the existing notions of PL critical points. Next, under an optimality condition called relative perfectness, we show a dimension agnostic correspondence between the set of PL critical points and that of discrete critical simplices of the combinatorial approach. Finally, we show how a relatively perfect discrete gradient vector field can be algorithmically built up to dimension 3. This way, we guarantee a formal and operative connection between critical sets in the PL and discrete theories.

中文翻译：

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PL 和离散莫尔斯理论的临界集：一封信

分段线性 (PL) 莫尔斯理论和离散莫尔斯理论用于形状分析任务，以研究离散空间的拓扑特征。尽管它们共同起源于平滑莫尔斯理论，但文献中已经为离散设置给出了临界点的各种概念，使得对它们之间发生的关系的清晰理解并不明显。本文旨在提供两种离散莫尔斯理论临界点的等价结果。首先，我们证明了 PL 临界点的现有概念的等价性。接下来，在称为相对完美的最优性条件下，我们展示了 PL 临界点集与组合方法的离散临界单纯形集之间的维度不可知对应关系。最后，