Topological noetherianity of polynomial functors

We prove that any finite-degree polynomial functor over an infinite field is topologically Noetherian. This theorem is motivated by the recent resolution, by Ananyan-Hochster, of Stillman's conjecture; and a recent Noetherianity proof by Derksen-Eggermont-Snowden for the space of cubics. Via work by Erman-Sam-Snowden, our theorem implies Stillman's conjecture and indeed boundedness of a wider class of invariants of ideals in polynomial rings with a fixed number of generators of prescribed degrees.