On the topology of semi-algebraic functions on closed semi-algebraic sets

2010/12/09 by Nicolas Dutertre, Dutertre, Nicolas
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #FOS: Mathematics #Functional Equations Stability Results #Mathematical and Theoretical Analysis #Nonlinear Differential Equations Analysis #math.AG

paper · pdf · doi:10.48550/arxiv.1012.2093

arxiv created 2010/12/09 · openalex publication_date 2010/12/09 · arxiv updated 2010/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a semi-algebraic function defined on a closed semi-algebraic set X. We give formulas relating the topology of X to the indices of the critical points of the function and to the topological behavior of the function at infinity. We give applications when X is Rⁿ and when the function is linear.

Related

• Stability of closedness of semi-algebraic sets under continuous semi-algebraic mappings
• Semi-algebraic geometry with rational continuous functions
• Topological equivalence of smooth functions with isolated critical points on a closed surfaces
• Critical Points of Functions on Singular Spaces
• Topological structure of functions with isolated critical points on a 3-manifold