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arXiv · 2011.14938

A triangulation of semi-algebraic sets concerning an analytical condition for shortest-length curves

Chengcheng Yang
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Abstract

This paper concerns an analytical stratification question of real algebraic and semi-algebraic sets. For Whitney's stratification in 1957, it partitions a real algebraic set into partial algebraic manifolds\cite{W}. In 1975 Hironaka reproved that a real algebraic set is triangulable and also generalized it to semi-algebraic sets, following the idea of Lojasiewicz's triangulation of semi-analytic sets in 1964. Following their examples and wondering how geometry looks like locally. this paper tries to come up with a stratification, in particular a cell decomposition, such that it satisfies the following analytical property. Given any shortest curve between two points in a real algebraic or semi-algebraic set, it interacts each cell (or simplex) at most finitely many times.

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Chengcheng Yang. 2020-11-30. A triangulation of semi-algebraic sets concerning an analytical condition for shortest-length curves. https://arxiv.org/abs/2011.14938

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