# Stationary points at infinity for analytic combinatorics

## TL;DR

This paper develops conditions to ignore complex behavior at infinity in algebraic varieties, enabling the application of Morse theory to analyze topological invariants in analytic combinatorics more effectively.

## Contribution

It establishes checkable conditions for when behavior at infinity can be disregarded, simplifying Morse theory applications in analytic combinatorics in several variables.

## Key findings

- Conditions for ignoring behavior at infinity are established.
- Simplified Morse theory arguments are enabled for complex algebraic varieties.
- New methods are developed for problems previously inaccessible.

## Abstract

On complex algebraic varieties, height functions arising in combinatorial applications fail to be proper. This complicates the description and computation via Morse theory of key topological invariants. Here we establish checkable conditions under which the behavior at infinity may be ignored, and the usual theorems of classical and stratified Morse theory may be applied. This allows for simplified arguments in the field of analytic combinatorics in several variables, and forms the basis for new methods applying to problems beyond the reach of previous techniques.