On Roberts' counterexample to the fourteenth problem of Hilbert (Q1376308)

The aim of this paper is to understand better the counterexample to the fourteenth problem of Hilbert due to \textit{P. Roberts} [J. Algebra 132, No. 2, 461-473 (1990; Zbl 0716.13013)], who discovered a subfield \(K\) in the quotient field of the \(7\)-variable polynomial ring \(R\) over a field of characteristic zero such that \(K\cap R\) is not finitely generated as an algebra. The authors point out that \(K\cap R\) is the ring of invariants of the additive group scheme \(G_a\) acting algebraically on \(R.\) They develop a technique to construct \(G_a\)-invariant elements of any given degree (with respect to an appropriate grading) which are not contained in the subalgebra generated by lower degree invariants, providing a new proof of the fact that \(K\cap R\) is not finitely generated. Furthermore, the method is used also to generalize the counterexample of Roberts to analogous actions of \(G_a\) on polynomial rings in \(2m+1\) variables with \(m=3,4,\ldots\).