Note on a counterexample to Hilbert's fourteenth problem given by P. Roberts
Let \(L\) be a subfield of the function field \(F(x_1, \dots, x_n)\) in \(n\) variables over the ground field \(F \subset L\). Then Hilbert's 14-th problem (*) asks whether \(B : = L \cap F[x_1, \dots, x_n]\) is an \(F\)-algebra of finite type. Hilbert's original motivation grows out of his interest in the finiteness of the ring of invariants \(B = F[x_1, \dots, x_n]^G\), where \(G\) is a group acting linearly on the \(n\)-dimensional vector space over \(F\). In this situation \(L\) is the quotient field of \(F[x_1, \dots, x_n]^G\) and \(B = L \cap F [x_1, \dots, x_n]\). This was disproved by \textit{M. Nagata} by his famous counterexample [see Proc. Int. Congr. Math. 1958, 459-462 (1960; Zbl 0127.26302)]. More recently \textit{P. Roberts} [J. Algebra 132, No. 2, 461-478 (1990; Zbl 0716.13013)] constructed another counterexample to the general question (*). In the present paper the author shows first that P. Roberts' example is just the ring of invariants of a certain \(\mathbb{G}_a\)-action on \(\mathbb{A}^7\). Secondly she defines an explicit \((\mathbb{G}_a)^{12}\)-action whose ring of invariants is not finitely generated. This is based on a particular case of P. Roberts' example.
scientific article; zbMATH DE number 720650