Consider $\mathbb{C}$-algebras
$$A = \mathbb{C}[t][x_1,\ldots, x_n]\subset\mathbb{C}(t)[x_1,\ldots, x_n] = B$$
Group $\operatorname{Aut}_{k(t)}(k(t)[x_1,\ldots, x_n])$ carry a power series topology (see https://arxiv.org/abs/1712.01490 p.2).
Fix $f\in B$. Is it true that $X =\{\pi\in\operatorname{Aut}_{\mathbb{C}(t)}(\mathbb{C}(t)[x_1,\ldots, x_n])|\pi(f)\in A\}$ is closed in $\operatorname{Aut}_{\mathbb{C}(t)}(\mathbb{C}(t)[x_1,\ldots, x_n])$ with respect to power series topology?
In fact, for my research it is more interesting to answer the following question
$$\mathcal{A} = \text{Der}_{\mathbb{C}[t]}(A) = A\partial_{x_1}\oplus\ldots\oplus A\partial_{x_n}\subset B\partial_{x_1}\oplus\ldots\oplus B\partial_{x_n} = \text{Der}_{\mathbb{C}(t)}(B) = \mathcal{B}$$
Fix $\partial\in\mathcal{B}$. Is it true that $\textbf{X} = \{\pi\in\text{Aut}_{\mathbb{C}(t)}(\mathbb{C}(t)[x_1,\ldots, x_n])|\pi(\partial)\in\mathcal{A}\}$ is closed in $\text{Aut}_{\mathbb{C}(t)}(\mathbb{C}(t)[x_1,\ldots, x_n])$ with respect to power series topology?
