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Abstract The Witt algebra  $${\mathfrak{W}}_{n}$$ is the Lie algebra of all derivations of the $$n$$-variable polynomial ring $$\textbf{V}_{n}=\textbf{C}[x_{1}, \ldots , x_{n}]$$ (or of algebraic vector fields on $$\textbf{A}^{n}$$). A representation of $${\mathfrak{W}}_{n}$$ is polynomial if it arises as a subquotient of a sum of tensor powers of $$\textbf{V}_{n}$$. Our main theorems assert that finitely generated polynomial representations of $${\mathfrak{W}}_{n}$$ are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of $$\textbf{Fin}^{\textrm{op}}$$, where $$\textbf{Fin}$$ is the category of finite sets. We also show that polynomial representations of $${\mathfrak{W}}_{n}$$ are equivalent to polynomial representations of the endomorphism monoid of $$\textbf{A}^{n}$$. These equivalences are a special case of an operadic version of Schur–Weyl duality, which we establish.