# The bottleneck degree of algebraic varieties

## TL;DR

This paper introduces the concept of bottleneck degree for algebraic varieties, linking it to classical invariants and providing formulas and algorithms for its computation, which measures the complexity of identifying all bottlenecks.

## Contribution

It defines the bottleneck degree for algebraic varieties, relates it to classical invariants, and offers explicit formulas and algorithms for its calculation.

## Key findings

- Bottleneck degree is expressed via Chern and polar classes.
- Explicit formulas provided for low-dimensional cases.
- An algorithm for computing bottleneck degree in general cases.

## Abstract

A bottleneck of a smooth algebraic variety $X \subset \mathbb{C}^n$ is a pair of distinct points $(x,y) \in X$ such that the Euclidean normal spaces at $x$ and $y$ contain the line spanned by $x$ and $y$. The narrowness of bottlenecks is a fundamental complexity measure in the algebraic geometry of data. In this paper we study the number of bottlenecks of affine and projective varieties, which we call the bottleneck degree. The bottleneck degree is a measure of the complexity of computing all bottlenecks of an algebraic variety, using for example numerical homotopy methods. We show that the bottleneck degree is a function of classical invariants such as Chern classes and polar classes. We give the formula explicitly in low dimension and provide an algorithm to compute it in the general case.