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

TL;DR

This paper explores a new stratification approach for semi-algebraic sets that ensures shortest curves interact with each cell only finitely many times, enhancing understanding of geometric properties.

Contribution

It introduces a stratification method with an analytical property for semi-algebraic sets, focusing on shortest-length curves and their interactions with the stratification.

Findings

Stratification ensures finite interactions of shortest curves with cells.

Provides a new cell decomposition satisfying specific analytical properties.

Enhances understanding of geometric structure of semi-algebraic sets.

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.