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

Let P3(k) be the space of cubic polynomials in infinitely many variables over the algebraically closed field k (of characteristic 2,3). We show that this space is GL-noetherian, meaning that any GL-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.

noetherian, cubic, twisted commutative algebra
Mathematical Subject Classification 2010
Primary: 13A50
Secondary: 13E05
Received: 8 February 2017
Revised: 16 June 2017
Accepted: 20 June 2017
Published: 2 December 2017
Harm Derksen
Department of Mathematics
University of Michigan
Ann Arbor, MI
United States
Rob H. Eggermont
Faculteit Wiskunde en Informatica
Eindhoven University of Technology
The Netherlands
Andrew Snowden
Department of Mathematics
University of Michigan
Ann Arbor, MI
United States