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

Topological Noetherianity of polynomial functors II: base rings with Noetherian spectrum