A generalization of Roberts' counterexample to the fourteenth problem of Hilbert
type: Item
instance of: scholarly article
title: A generalization of Roberts' counterexample to the fourteenth problem of Hilbert
zbMATH Open document ID: 1077.13003
author: Shigeru Kuroda
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2005-04-15T00:00:00Z
review text: Let \(K\) be a field, \(A\) a polynomial ring over \(K\) and \(L\) a subfield of the quotient field of \(A\) containing \(K\). The fourteenth problem of Hilbert asks whether the \(K\)-algebra \(L \cap A\) is finitely generated. After Nagata had found the first counterexample in 1958 as the invariant subring of a polynomial ring in \(32\) variables for a linear action of the \(13\)-dimensional additive group, it was Roberts who gave a simpler counterexample. It was later shown that this counterexample can be obtained as the invariant subring of \(A\) for a nonlinear action of the one-dimensional additive group \(G_a\). This was generalized to counterexamples in every dimension greater than or equal to five. The author further generalizes Roberts' counterexample and proves that there exists a lot of counterexamples of this kind. Sufficient conditions for a certain kind of \(G_a\)-invariant subrings of a polynomial ring not to be finitely generated are given. Also Roberts' counterexample is discussed in terms of the theory of SAGBI which is Subalgebra Analogue to Gröbner Bases for Ideals. As a consequence a generating set of it is determined. There are several more nice results included in the paper and the author finishes with a conjecture.
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 13E15
Mathematics Subject Classification ID: 13N15
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 14R20
zbMATH DE Number: 2157241
zbMATH Keywords: Gröbner bases
zbMATH Keywords: non-finitely generated algebra
zbMATH Keywords: Hilbert fourteenth problem
reviewed by: Lutz Strüngmann
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1113246747
OpenAlex ID: W1996573006
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Wikidata QID: Q124989325
DOI: 10.2748/TMJ/1113246747
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description: scientific article; zbMATH DE number 2157241