  In a previous paper, the third author proved that finite-degree polynomial
functors over infinite fields are topologically Noetherian. In this paper, we
prove that the same holds for polynomial functors from free $R$-modules to
finitely generated $R$-modules, for any commutative ring $R$ whose spectrum is
Noetherian. As Erman-Sam-Snowden pointed out, when applying this with $R =
\mathbb{Z}$ to direct sums of symmetric powers, one of their proofs of a
conjecture by Stillman becomes characteristic-independent.
  Our paper advertises and further develops the beautiful but not so well-known
machinery of polynomial laws. In particular, to any finitely generated R-module
M we associate a topological space, which we show is Noetherian when
$\operatorname{Spec}(R)$ is; this is the degree-zero case of our result on
polynomial functors.
