# Global Euler obstruction, global Brasselet numbers and critical points

## TL;DR

This paper introduces global invariants called the global Brasselet number and the Brasselet number at infinity for polynomial functions on complex algebraic sets, linking them to the topology and critical points of the set.

## Contribution

It defines new global invariants for polynomial functions on complex algebraic sets and establishes formulas connecting these invariants to topology and critical points.

## Key findings

- Formulas relating global Brasselet numbers to topology
- Relations between critical points and Brasselet numbers
- Introduction of the Brasselet number at infinity

## Abstract

Let $X \subset \Bbb{C}^n$ be an equidimensional complex algebraic set and let $f: X \to \mathbb{C}$ be a polynomial function. For each $c \in \Bbb{C}$, we define the global Brasselet number of $f$ at $c$, a global counterpart of the Brasselet number defined by the authors in a previous work, and the Brasselet number at infinity of $f$ at $c$. Then we establish several formulas relating these numbers to the topology of $X$ and the critical points of $f$.