In Artin1968 he considers $\underline{analytic}$ equations, but over the ring $R=k\{x_1,..,x_n\}$. In Artin1969 he works with $R=k\{x_1,..,x_n\}/I$, not necessarily regular, but considers $\underline{polynomial}$ equations.

Is there some version like this: "Let $R$ be a local Noetherian Henselian ring(not necessarily regular), over a normed field. Given an arbitrary (possibly countable) system of analytic equations over $R$, with a solution over the completion of $R$, there exists also a solution in $R$, sufficiently close to the formal solution" ??What is known for non-Noetherian rings? e.g. for $C^\infty$, $C^r$?

(Actually, for $C^\infty$ I learned about one approximation theorem, unpublished in the old USSR times..)

## 1 Answer

Concerning (2), here are some references:

For certain subrings of $R[[T_1,\dots,T_N]]$ where $R$ is a complete valuation ring of rank 1, see:

H. Schoutens: Approximation properties for some non-Noetherian local rings. Pac. J. Math. 131(2), 331–359 (1988).

For any henselian valuation ring, with fraction field $K$, such that the completion $\widehat{K}$ is separable over $K$, see my paper:

An extension of Greenberg’s theorem to general valuation rings, Manuscripta Math. 139, 153–166 (2012).

The case of a henselian valuation ring of rank 1 is already mentioned in Elkik's thesis:

R. Elkik: Solutions d’équations à coefficients dans un anneau hensélien. Ann. Sci. École Norm. Sup. (4) 6, 553–603 (1974) (see Remarque 2, p. 587),

and treated in more detail in chapter 1 of

A. Abbes: Éléments de géométrie rigide I. Progress in Mathematics. Birkhäuser, Boston (2011).

For rings of differentiable functions, perhaps you should look at Tougeron's papers.

finitely manyanalytic equations (since the local ring is noetherian, after all), and at least in the non-archimedean case I believe there is a paper of Siegfried Bosch on this generalization of Artin's result (i.e., considering analytic equations over $R$, not just polynomial equations over $R$). I don't remember the exact title, but if you search for papers of Bosch with "Artin" or "approximation" in the title then you should find it. $\endgroup$