# Topological noetherianity for cubic polynomials

## TL;DR

This paper proves that the space of complex cubic polynomials in infinitely many variables is noetherian under the action of the infinite general linear group, using a new invariant called q-rank, with implications for algebraic stability problems.

## Contribution

It introduces the concept of q-rank for cubic polynomials and demonstrates the noetherian property of the space under group action, advancing understanding in representation stability and algebraic geometry.

## Key findings

- Space of cubic polynomials is GL-infinity-noetherian
- Q-rank is a key invariant for analysis
- Results relate to stability problems like Stillman's conjecture

## Abstract

Let $P_3(\mathbf{C}^{\infty})$ be the space of complex cubic polynomials in infinitely many variables. We show that this space is $\mathbf{GL}_{\infty}$-noetherian, meaning that any $\mathbf{GL}_{\infty}$-stable Zariski closed subset is cut out by finitely many orbits of equations. Our method relies on a careful analysis of an invariant of cubics introduced here called q-rank. This result is motivated by recent work in representation stability, especially the theory of twisted commutative algebras. It is also connected to certain stability problems in commutative algebra, such as Stillman's conjecture.