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

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