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For any continuous function $f:\mathbb{Z}_p \to \mathbb{Q}_p$, Mahler's theorem provides us with a relatively explicit series of polynomials converging uniformly to $f$. Is there any analogue for other non archimedean fields? In particular, what about the completion $\mathbb{C}_p$ of the algebraic closure of $\mathbb{Q}_p$?

ReLonzo
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  • I think the same proof should work for the map $\mathcal{O}K \to K$ where $K$ is a local field. But for $\mathcal{O}{\mathbb{C}_p} \to \mathbb{C}_p$, I am not sure. – Offlaw Dec 11 '23 at 19:13
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    @Offlaw no, it doesn't work, e.g., $\binom{x}{n}$ is not $\mathcal O_K$-valued when the residue field of $\mathcal O_K$ is not represented by integers. – KCd Dec 11 '23 at 22:18
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    This has been asked and answered already on MO: see https://mathoverflow.net/questions/15209/when-does-a-p-adic-function-have-a-mahler-expansion. – KCd Dec 11 '23 at 22:20

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