Counterexamples to the Fourteenth Problem of Hilbert

Let K be a field, K[X] = K[X1, . . . , Xn] the polynomial ring in n variables over K for some n ∈ N, and K(X) the field of fractions of K[X]. Assume that L is a subfield of K(X) containing K. Then, the Fourteenth Problem of Hilbert asks whether the Ksubalgebra L ∩K[X] of K[X] is finitely generated. Zariski [17] showed in 1954 that the answer to this problem is affirmative if the transcendence degree trans.degK L of L over K is at most two, while Nagata [15] gave the first counterexample in 1958 in the case where n = 32 and trans.degK L = 4. In 1990, Roberts [16] constructed a counterexample of different type when n = 7 and trans.degK L = 6. Following Nagata and Roberts, several new counterexamples have been constructed. We also developed a powerful theory on the Fourteenth Problem of Hilbert by generalizing the construction of Roberts. In this talk, we give various kinds of new counterexamples obtained by applying our theory. After giving the first counterexample, Nagata presented a question whether there exists a counterexample to the Fourteenth Problem of Hilbert when trans.degK L = 3, because the answer had been known to be affirmative if trans.degK L ≤ 2 due to Zariski. This was a longstanding open question, but we finally answer it by giving counterexamples (cf. [9]). Concerning the dimension n of K[X], counterexamples have been found for n ≥ 5, while there exists no counterexamples when n ≤ 2 due to Zariski. The cases where n = 3, 4 were open, but we also give counterexamples in these cases (cf. [9], [10]). Thereby, the Fourteenth Problem of Hilbert is settled for all trans.degK L and n at last. The problem of finite generation of the kernel of a derivation is an important special case of the Fourteenth Problem of Hilbert. Let D be a derivation of K[X] over K, i.e., a K-linear map K[X] → K[X] satisfying D(fg) = D(f)g + fD(g) for f, g ∈ K[X]. Then, D extends uniquely to a derivation of K(X). The kernels K[X] and K(X) of D and its extension to K(X) are a K-subalgebra of K[X] and a subfield of K(X) containing K, respectively. Since K[X] = K(X) ∩K[X], the problem of finite generation of K[X] is a kind of the Fourteenth Problem of Hilbert. The result of Zariski implies that K[X] is always finitely generated if n ≤ 3. On the other hand, Derksen [3] first showed the existence of D for which K[X] is not finitely generated when n = 32 by using the counterexample of Nagata. In 1994, DeveneyFinston [4] described the counterexample of Roberts by using a derivation of K[X] for Partly supported by the Grant-in-Aid for JSPS Fellows, The Ministry of Education, Science, Sports and Culture, Japan.