# Stability in the cohomology of the space of complex irreducible polynomials in several variables

## TL;DR

This paper proves that the cohomology of the space of complex irreducible polynomials stabilizes as the degree and number of variables increase, revealing deep topological stability properties inspired by finite field counting results.

## Contribution

It establishes two forms of homological stability for the space of complex irreducible polynomials, linking topology with finite field counting techniques.

## Key findings

- Cohomology stabilizes as degree increases
- Compactly supported cohomology stabilizes as variables increase
- Topological results are inspired by finite field counting

## Abstract

We prove that the space of complex irreducible polynomials of degree $d$ in $n$ variables satisfies two forms of homological stability: first, its cohomology stabilizes as $d$ increases, and second, its compactly supported cohomology stabilizes as $n$ increases. Our topological results are inspired by counting results over finite fields due to Carlitz and Hyde.