Counterexamples to the Fourteenth Problem ofHilbert
Shigeru Kuroda
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 subfieldof 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 theanswer 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 wheren = 32 and trans.degK L = 4. In 1990, Roberts [16] constructed a counterexample ofdifferent 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 thistalk, we give various kinds of new counterexamples obtained by applying our theory.
After giving the first counterexample, Nagata presented a question whether thereexists 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 toZariski. This was a longstanding open question, but we finally answer it by giving
counterexamples (cf. [9]). Concerning the dimension n of K[X], counterexamples havebeen 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 Land 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., aK-linear map K[X] → K[X] satisfying D(f g) = D(f )g + f D(g) for f, g ∈ K[X]. Then,
D extends uniquely to a derivation of K(X). The kernels K[X]D and K(X)D of D andits extension to K(X) are a K-subalgebra of K[X] and a subfield of K(X) containing K,respectively. Since K[X]D = K(X)D ∩ K[X], the problem of finite generation of K[X]Dis a kind of the Fourteenth Problem of Hilbert.
The result of Zariski implies that K[X]D is always finitely generated if n ≤ 3. On theother hand, Derksen [3] first showed the existence of D for which K[X]D is not finitelygenerated 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] forPartly supported by the Grant-in-Aid for JSPS Fellows, The Ministry of Education, Science, Sportsand Culture, Japan.

1n = 7. For n = 6 and n = 5, Freudenburg [6] and Daigle-Freudenburg [1] gave Dfor which K[X]D is not finitely generated, respectively. Thus, the problem of finite
generation of K[X]D was open only for n = 4. We give a negative answer to this open
case by showing that our counterexamples for n = 4 can be realized as the kernel of aderivation (cf. [11]).
To explain this, we give an outline of the construction of our counterexamples above.
First, we recall some notions on derivations. Let D be a derivation of K[X]. Then,
D is said to be triangular if D(Xi ) is in K[X1 , . . . , Xi−1 ] for each i. We say that D islocally nilpotent if, for each f ∈ K[X], there exists l such that Dl (f ) = 0. Note that atriangular derivation is locally nilpotent. An element g ∈ K[X] is said to be a slice of D
if D(g) = 1. Assume that D is locally nilpotent and having a slice g. We define
σgD (f ) =l−1XDi (f )i=0i!

(−g)ifor each f ∈ K[X], where l ∈ N such that Dl (f ) = 0. Then, σgD (f ) is in K[X]D , sinceD(σgD (f )) =l−1 µ i+1
X
D (f )i=0i!

Di (f )(−g) − i(−g)i−1i!
i¶=Dl (f )(−g)l−1 = 0.
(l − 1)!

It is well-known that K[X]D = K[σgD (X1 ), . . . , σgD (Xn )]. Now, let K[Y ] = K[Y1 , . . . , Ym ]be the polynomial ring in m variables for some m ∈ N, and τ : Zm → Zn a Z-linear
map. We define a homomorphism hτ : K[Y ] → K(X) of K-algebras by hτ (Y b ) = X τ (b)for each b. Here, we denote X a = X1a1 · · · Xnan and Y b = X1b1 · · · Ynbm for a = (a1 , . . . , an )and b = (b1 , . . . , bm ). For a locally nilpotent derivation D of K[Y ], we define L(D, τ )to be the field of fractions of hτ (K[Y ]D ). Then, we may obtain a great variety ofcounterexamples to the Fourteenth Problem of Hilbert as L(D, τ ) for suitable D and τ .
Our counterexample for n = 4 can be obtained as L(D, τ ) for a triangular derivation Dof K[Y ] for m = 4 with D(Yi ) 6= 0 for each i and an suitable injection τ .
Although L(D, τ ) is defined by using the kernel of a derivation, L(D, τ ) ∩ K[X] is notnecessarily equal to the kernel of a derivation of K[X]. We give the following sufficientcondition on D and τ for L(D, τ )∩K[X] to be the kernel of a derivation of K[X] (cf. [11]).
Assume that τ is injective. Then, L(D, τ )∩K[X] is equal to the kernel of some derivationPPof K[X] if and only if Zn ∩ i∈ID Rτ (ei ) = i∈ID Zτ (ei ), where ID = {i | D(Yi ) = 0}and e1 , . . . , en are the coordinate unit vectors of Rn . By this criterion, we know that ourcounterexample for n = 4 can be realized as the kernel of a derivation.
When D is locally nilpotent, the problem of finite generation of K[X]D for n = 4
has not been settled yet. The derivations given by Deveney-Finston, Freudenburg, andDaigle-Freudenburg are in fact triangular derivations. Hence, the problem has negative
answers for n ≥ 5. However, our derivation D of K[X] for n = 4 is not locally nilpotent.
0Furthermore, K[X]D is not contained in K[X]D for any locally nilpotent derivation D0 ofK[X]. In 2001, Daigle-Freudenburg [2] showed that the kernel of a triangular derivationof K[X] is always finitely generated when n = 4. So, we must consider locally nilpotent
derivations whose kernels are not equal to the kernels of any triangular derivations.
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Towards the solution of this open problem, the study of locally nilpotent derivationsof K[X] of rank n is of great important. Here, the rank of D is by definition the least
integer r for which there exist f1 , . . . , fn ∈ K[X] with K[f1 , . . . , fn ] = K[X] such that
D(fi ) = 0 for i = 1, . . . , n − r. The rank of a triangular derivation of K[X] is known tobe less than n if n ≥ 2. Hence, the kernel of a locally nilpotent derivation D of K[X]of rank n is never isomorphic to the kernel of any triangular derivation of K[X] via anautomorphism of K[X] when n ≥ 2. Although Freudenburg [5] has already given an
example of a locally nilpotent derivation of K[X] of rank n, the construction of suchderivations is quite difficult in general. So, finite generation of their kernels has not beenstudied. By applying our theory, we construct the first example of a locally nilpotent
derivation of K[X] of rank n whose kernel is not finitely generated for each n ≥ 5(cf. [14]).
As mentioned, a locally nilpotent derivation of K[X] with a slice is generated by atmost n elements. So, van den Essen posed a question whether the kernel of a derivationof K[X] with a slice is always finitely generated, and also conjectured that the answer
is no. We show that the conjecture is true by giving an explicit example.
We also have a condition for finite generation of the kernel of a derivation of K[X].
For a Laurent polynomial f in X1 , . . . , Xn , we define supp(f ) to be the set of a ∈ Zn
such that the monomial X a appears in f with nonzero coefficient. Let D be a derivation
D of K[X]. Then, define the support supp(D) of D bysupp(D) =n[supp(Xi−1 D(Xi )).

i=1

We call the convex hull New(D) of supp(D) in Rn the Newton polytope of D. The dimension of New(D) is defined as the dimension of the R-vector subspace of Rn generated
by a − b for a, b ∈ New(D). Then, our result is that K[X]D is finitely generated if thedimension of New(D) is at most two (cf. [7]). We note that there exists D for which
K[X]D is not finitely generated and the dimension of New(D) is three.
Finally, we discuss the case where K(X) is algebraic over L. In this case, there
are only few studies so far. It had even been unknown whether there existed a counterexample until when we gave one for n = 3 (cf. [10]). Here, due to Zariski, K(X) isnecessarily algebraic over L if n = 3 and L ∩ K[X] is not finitely generated. For ourcounterexamples in [10], the extension degrees [K(X) : L] of K(X) over L can only becertain even numbers not less than 22. However, by using a different construction, wegive a counterexample with [K(X) : L] = d for each d ≥ 3 when n = 3 (cf. [12]). We
remark that the case where [K(X) : L] = 2 is open for any n ≥ 3.
Assume that a finite group G acts on K(X). Then, K(X) is algebraic over thesubfield K(X)G of G-invariant elements of K(X). We define K[X]G = K(X)G ∩ K[X].
By Noether’s theorem, K[X]G is finitely generated if σ · f is in K[X] for each σ ∈ Gand f ∈ K[X]. However, finite generation of K[X]G was unknown in general. Using
our theory, we give examples of actions of G on K(X) for which K[X]G are not finitelygenerated (cf. [13]). We may construct such actions for each finite group containing atleast three elements.
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[12] S. Kuroda, Hilbert’s Fourteenth Problem and algebraic extensions, preprint.
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Research Institute for Mathematical Sciences
Kyoto University
Kyoto 606-8502
Japan
E-mail address: kuroda@kurims.kyoto-u.ac.jp
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