# On the topology of non-isolated real singularities

## TL;DR

This paper extends topological degree formulas for Euler characteristics from isolated to non-isolated real singularities, providing new algebraic and topological insights into real polynomial fibers.

## Contribution

It generalizes Khimshiashvili's degree formula to non-isolated singularities and introduces algebraic and topological formulas for real polynomial fibers.

## Key findings

- Generalized degree formula for non-isolated singularities
- Derived algebraic formula for Euler characteristic of weighted-homogeneous polynomial fibers
- Established a real version of the Lê-Iomdine formula

## Abstract

Khimshiashvili proved a topological degree formula for the Eu-ler characteristic of the Milnor fibres of a real function-germ with an isolated singularity. We give two generalizations of this result for non-isolated singularities. As corollaries we obtain an algebraic formula for the Euler characteristic of the fibres of a real weighted-homogeneous polynomial and a real version of the L{\^e}-Iomdine formula. We have also included some results of the same flavor on the local topology of locally closed definable sets.