On Freudenburg's counterexample to the Fourteenth Problem of Hilbert

In this paper we study Freudenburg's counterexample to the fourteenth problem of Hilbert and counterexamples derived from it. We shall construct a generating set of a nonfinitely generated Ga-invariant ring given in Freudenburg's counterexample by making use of an integral sequence which was constructed inductively by Freudenburg. This generating set shall be used in describing a generating set of a nonfinitely generated Ga-invariant ring given in Daigle and Freudenburg's counterexample. Using these generating sets, we shall determine the Hilbert series of the above Freudenburg's and Daigle and Freudenburg's nonfinitely generated Ga-invariant rings, and find that these Hilbert series are rational functions. Then we also show that the Hilbert series of nonfinitely generated invariant rings appearing in the author's linear counterexamples are rational functions.