When is the Intersection of Two Finitely Generated Subalgebras of a Polynomial Ring Also Finitely Generated?

We study two variants of the following question: “Given two finitely generated C -subalgebras R 1 , R 2 of C [ x 1 , … , 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...

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Bibliographic Details
Published in:Arnold mathematical journal 2017-09, Vol.3 (3), p.333-350
Main Author: Mondal, Pinaki
Format: Article
Language:English
Subjects:
Infinity
Mathematics
Mathematics and Statistics
Research Contribution
Set theory
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Summary:We study two variants of the following question: “Given two finitely generated C -subalgebras R 1 , R 2 of C [ x 1 , … , 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.
ISSN:2199-6792
2199-6806
DOI:10.1007/s40598-017-0068-8