# Topological Noetherianity of polynomial functors

## TL;DR

This paper proves that finite-degree polynomial functors are topologically Noetherian, leading to significant implications for algebraic invariants and resolving longstanding conjectures in commutative algebra.

## Contribution

It establishes the topological Noetherianity of polynomial functors, which implies Stillman's conjecture and broadens understanding of invariants in polynomial rings.

## Key findings

- Finite-degree polynomial functors are topologically Noetherian.
- Implication of the theorem confirms Stillman's conjecture.
- Boundedness of invariants of ideals in polynomial rings.

## Abstract

We prove that any finite-degree polynomial functor is topologically Noetherian. This theorem is motivated by the recent resolution of Stillman's conjecture and a recent Noetherianity proof for the space of cubics. Via work by Erman-Sam-Snowden, our theorem implies Stillman's conjecture and indeed boundedness of a wider class of invariants of ideals in polynomial rings with a fixed number of generators of prescribed degrees.