A generalization of Roberts' counterexample to the fourteenth problem of Hilbert
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.
scientific article; zbMATH DE number 2157241