On Freudenburg's counterexample to the fourteenth problem of Hilbert
type: Item
instance of: scholarly article
title: On Freudenburg's counterexample to the fourteenth problem of Hilbert
zbMATH Open document ID: 1113.13006
author: Ryuji Tanimoto
published in: Transformation Groups
publication date: 2007-01-24T00:00:00Z
review text: Let \(k\) be a field of characteristic zero and \(k[x_1,\dots,x_n]\) the polynomial algebra over \(k\). Recall that a derivation \(\partial \) of \(k[x_1,\dots,x_n]\) is called locally nilpotent if for any \(\alpha\in k[x_1,\dots,x_n]\) there exists \(m>0\) such that \(\partial^m(\alpha)=0\). It is known that the kernel \(R(\partial)\) of a locally nilpotent derivation \(\partial\) may be nonfinitely generated. The corresponding examples were proposed by P.~Roberts (\(n=7\)), G.~Freudenburg (\(n=6\)) and D.~Daigle-G.~Freudenburg (\(n=5\)). \textit{S.~Kuroda} [Tohoku Math. J. 56, 501--522 (2004; Zbl 1077.13003)] has shown that, in the case of Roberts' counterexample, the set of polynomials given by Roberts forms a generating set of \(R(\partial)\). Moreover, this generating set is a SAGBI basis of the kernel. In the present paper the author finds generating sets for Freudenburg's, Daigle-Freudenburg's and his own linear counterexamples. In these cases the generating sets also turn out to be SAGBI bases. Using these results, a negative answer to \textit{G. Freudenburg}'s question [~Transform. Groups 5, 61--71 (1999; Zbl 0993.13009)] is given. More precisely, it is proved that the kernel of Roberts' counterexample is nonfinitely generated over the image of the kernel of Freudenburg's counterexample under a homomorphism constructed by G.~Freudenburg. Another objectives of the paper are Hilbert series of the corresponding nonfinitely generated subalgebras. Considering some natural gradings and using the SAGBI basis, the author calculate explicitly the Hilbert series in all cases. All of them turn out to be represented by rational functions.
reviewed by: Ivan V. Arzhantsev
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 13N15
Mathematics Subject Classification ID: 14R10
zbMATH DE Number: 5118038
zbMATH Keywords: locally nilpotent derivations
zbMATH Keywords: kernels
zbMATH Keywords: generating sets
zbMATH Keywords: SAGBI basis
zbMATH Keywords: Hilbert series
MaRDI profile type: Publication
full work available at URL: s00031-004-1112-z
OpenAlex ID: W2027041318
Wikidata QID: Q124808911
DOI: 10.1007/S00031-004-1112-Z
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description: scientific article; zbMATH DE number 5118038