On the ultrametric Stone-Weierstrass theorem and Mahler's expansion

Cahen, Paul-Jean; Chabert, Jean-Luc

Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 43-57.

Nous explicitons une version ultramétrique du théorème de Stone-Weierstrass. Pour une partie $E$ d’un anneau de valuation $V$ de hauteur $1$, nous montrons, sans aucune hypothèse sur le corps résiduel, que l’ensemble des fonctions polynomiales est dense dans l’anneau des fonctions continues de $E$ dans $V$ si et seulement si la clôture topologique $\widehat{E}$ de $E$ dans le complété de $\widehat{V}$ est compacte. Nous explicitons ainsi le développement d’une fonction continue en série de fonctions polynomiales.

We describe an ultrametric version of the Stone-Weierstrass theorem, without any assumption on the residue field. If $E$ is a subset of a rank-one valuation domain $V$, we show that the ring of polynomial functions is dense in the ring of continuous functions from $E$ to $V$ if and only if the topological closure $\widehat{E}$ of $E$ in the completion $\widehat{V}$ of $V$ is compact. We then show how to expand continuous functions in sums of polynomials.

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