Topological noetherianity for cubic polynomials

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.