On Freudenburg's counterexample to the fourteenth problem of Hilbert
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.
scientific article; zbMATH DE number 5118038