Topological noetherianity for cubic polynomials

Derksen, Harm; Eggermont, Rob; Snowden, Andrew

doi:10.2140/ant.2017.11.2197

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Topological noetherianity for cubic polynomials

Harm Derksen, Rob H. Eggermont and Andrew Snowden

Vol. 11 (2017), No. 9, 2197–2212

DOI: 10.2140/ant.2017.11.2197

Abstract

Let $P_{3} (k^{\infty})$ be the space of cubic polynomials in infinitely many variables over the algebraically closed field $k$ (of characteristic $\neq 2, 3$ ). 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

Mathematical Subject Classification 2010

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

Vol. 11, No. 9, 2017