When is the intersection of two finitely generated subalgebras of a polynomial ring also finitely generated?

TL;DR

This paper investigates when the intersection of two finitely generated subalgebras of polynomial rings remains finitely generated, identifying minimal dimensions for counterexamples and connecting the problem to algebraic compactifications and the moment problem.

Contribution

It determines the minimal dimension where counterexamples exist for the finite generation of intersections and relates this to algebraic and analytic compactification theories.

Findings

Counterexamples exist starting from dimension 2 in the general case.

Counterexamples for integrally closed subalgebras appear from dimension 3.

The study links algebraic intersection properties to compactification and moment problems.

Abstract

We study two variants of the following question: "Given two finitely generated subalgebras R_1, R_2 of C[x_1, \ldots, x_n], is their intersection also finitely generated?" We show that the smallest value of $n$ for which there is a counterexample is 2 in the general case, and 3 in the case that R_1 and R_2 are integrally closed. We also explain the relation of this question to the problem of constructing algebraic compactifications of C^n and to the moment problem on semialgebraic subsets of R^n. The counterexample for the general case is a simple modification of a construction of Neena Gupta, whereas the counterexample for the case of integrally closed subalgebras uses the theory of normal analytic compactifications of C^2 via "key forms" of valuations centered at infinity.