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

TL;DR

This paper explores the relationship between the topology of semi-algebraic sets and the critical points of semi-algebraic functions, providing formulas and applications for linear functions and Euclidean spaces.

Contribution

It introduces formulas linking the topology of semi-algebraic sets to critical point indices and behavior at infinity, with specific applications to linear functions and Euclidean spaces.

Findings

Formulas relating topology and critical points of semi-algebraic functions

Analysis of the function's behavior at infinity

Applications to R^n and linear functions

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 R^n and when the function is linear.