Solving the likelihood equations to compute Euler obstruction functions

TL;DR

This paper introduces algorithms, both symbolic and numerical, for computing the Euler obstruction function of algebraic varieties, utilizing maximum likelihood degrees and implementations in Macaulay2 and Bertini.

Contribution

It presents new algorithms for computing the Euler obstruction function at a point, combining symbolic and numerical methods with software implementations.

Findings

Algorithms successfully compute Euler obstruction functions.

Implementation in Macaulay2 and Bertini demonstrates practical applicability.

Provides insights into singular algebraic varieties through Euler obstruction calculations.

Abstract

Macpherson defined Chern-Schwartz-Macpherson (CSM) classes by introducing the (local) Euler obstruction function, which is an integer valued function on the variety that is constant on each stratum of a Whitney stratification of an algebraic variety. By understanding the Euler obstruction function, one gains insights about a singular algebraic variety. It was recently shown by the author and B. Wang, how to compute these functions using maximum likelihood degrees. This paper discusses a symbolic and a numerical implementation of algorithms to compute the Euler obstruction at a point. Macaulay2 and Bertini are used in the implementations.