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

TL;DR

This paper provides geometric interpretations of a counterexample to Hilbert's 14th problem, linking it to rings of regular functions on varieties and bounded polynomials on semialgebraic sets, revealing non-finite generation properties.

Contribution

It offers new geometric perspectives on Kuroda's counterexample, connecting algebraic and semialgebraic structures, and explores properties of bounded polynomial rings on semialgebraic sets.

Findings

Counterexample interpreted as rings of regular functions on varieties

Ring of bounded polynomials on certain semialgebraic sets is not finitely generated

General properties of bounded polynomial rings on normal R-varieties proved

Abstract

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.