#### Vol. 11, No. 9, 2017

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
Topological noetherianity for cubic polynomials

### Harm Derksen, Rob H. Eggermont and Andrew Snowden

Vol. 11 (2017), No. 9, 2197–2212
##### Abstract

Let ${P}_{3}\left({k}^{\infty }\right)$ be the space of cubic polynomials in infinitely many variables over the algebraically closed field $k$ (of characteristic $\ne 2,3$). We show that this space is ${GL}_{\infty }$-noetherian, meaning that any ${GL}_{\infty }$-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 we introduce 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 uniformity problems in commutative algebra in the vein of Stillman’s conjecture.

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/ant

We have not been able to recognize your IP address 35.173.234.169 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

or by using our contact form.