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

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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.

##### Keywords
noetherian, cubic, twisted commutative algebra
Primary: 13A50
Secondary: 13E05
##### Milestones
Received: 8 February 2017
Revised: 16 June 2017
Accepted: 20 June 2017
Published: 2 December 2017
##### Authors
 Harm Derksen Department of Mathematics University of Michigan Ann Arbor, MI United States Rob H. Eggermont Faculteit Wiskunde en Informatica Eindhoven University of Technology Eindhoven The Netherlands Andrew Snowden Department of Mathematics University of Michigan Ann Arbor, MI United States