Linear counterexamples to the fourteenth problem of Hilbert
In the early 1990s, \textit{A. A'Campo-Neuen} [Indag. Math., New Ser. 5, No. 3, 253--257 (1994; Zbl 0839.13004)] used Roberts' nonlinear counterexample to Hilbert's fourteenth problem to construct a linear counterexample, namely, a linear action of the vector group \(G={\mathbb{C}}^{12}\) on the vector space \(V= {\mathbb{C}}^{19}\). Smaller non-linear counterexamples due to \textit{D. Daigle} and the reviewer [J. Algebra 221, No.2, 528--535 (1999; Zbl 0963.13024)] appeared in the late 1990s. In the paper under review, the author uses these newer examples to contruct linear counterexamples, imitating the procedure used by A'Campo-Neuen. The key difference is that Tanimoto's construction uses non-abelian unipotent groups. Specifically, he gives a linear action of the group \(G_1=({\mathbb{C}}^{18}\rtimes {\mathbb{C}})\) on \({\mathbb{C}}^{25}\), and a linear action of \(G_2=({\mathbb{C}}^7\rtimes {\mathbb{C}})\) on \({\mathbb{C}}^{13}\), where in each case the ring of invariants is not finitely generated. It is an open question whether a counterexample to Hilbert's problem can be found for a group acting linearly on \({\mathbb{C}}^n\) when \(n\leq 12\).
scientific article; zbMATH DE number 2105798