Geometric interpretations of a counterexample to Hilbert's 14th problem, and rings of bounded polynomials on semialgebraic sets

We interpret a counterexample to Hilbert’s 14th problem by S. Kuroda geometrically in two ways: As ring of regular functions on a smooth rational quasiprojective variety over any field K of characteristic 0, and, in the special case where K are the real numbers R, as the ring of bounded polynomials on a regular semialgebraic subset of R^3. One motivation for this was to find a regular semialgebraic subset of a real vectorspace, such that the ring of bounded polynomials on it is not finitely generated as an R-algebra. In an appendix we prove some general properties of rings of bounded polynomials on regular semialgebraic subsets of normal R-varieties.