Finite generation properties for various rings of integer-valued polynomials

For any subset E of a Dedekind domain D, we show the ring Int^{r} (E, D) of polynomials that are integer-valued on E together with all their divided differences of order up to r not to be a finitely generated D-algebra, contrary to the ring Int_x (E, D) of integer-valued polynomials on E having a given non-zero modulus x (which is hence Noetherian, since the domain D is so). Localization properties allow us to focus on valuation domains; furthermore, the consideration of precompact subsets allows us to consider valuation domains V of arbitrary dimension.