Derivations of polynomial rings over a field of characteristic zero

Andrzej Nowicki

Faculty of Mathematics and Computer Science, N. Copernicus University, 87-100 Toruń, Poland

The lecture on the 37th Symposium on Ring and Representation Theory, Hirosaki University, Japan, December 13, 2003

Let k be a commutative ring containing ℚ and let A be a commutative k-algebra containing k. A k-linear mapping d : A → A is called a k-derivation of A if it satisfies the Leibniz rule: d(ab) = ad(b) + bd(a).

Let d be a k-derivation of A. W denote by A^{d} the kernel of d, that is,

A^{d} = Ker d = {a ∈ A; d(a) = 0}.

This set is a k-subalgebra of A which we call the ring of constants of d. If A is a domain and k is a field, then we denote by A_{0} the field of quotients of A and we denote also by d the unique extension of d to A_{0}. In this case A_{0}^{d} is a subfield of A_{0} containing k.

Let D be a family of k-derivations of A. Then we have the ring of constants A^{D} = ⋂_{d∈D} A^{d} = {a ∈ A; d(a) = 0 for all d ∈ D}. We will mostly consider derivations of a polynomial ring in a finite set of variables. In such a case the rings of the form A^{D} are not interesting for us. It is known ([23], [22]) that in this case every ring A^{D} is of the form A^{d} for some k-derivation d of A.

If d is a derivation of A, then we denote by Nil(d) the following subset of A:

Nil(d) := {a ∈ A; ∃_{n⩾0} d^{n}(a) = 0} .

This subset is also a k-subalgebra of A and we have: k ⊆ A^{d} ⊆ Nil(d) ⊆ A. We say that a derivation d is locally nilpotent if A = Nil(d).

Assume now that A = k[X] := k[x_{1}, …, x_{n}] is the polynomial ring over k. In this case we know a description of all k-derivations of A. If d is a kderivation of k[X], then we have the polynomials f_{1} := d(x_{1}), …, f_{n} := d(x_{n}), belonging to k[X], and then

d = f_{1} \frac{∂}{∂x_{1}} + ⋯ + f_{n} \frac{∂}{∂x_{n}} .

Every k-derivation d of k[X] is uniquely determined by a sequence (f_{1}, …, f_{n}) of polynomials from k[X]. If d is a k-derivation of k[X], then d is locally nilpotent if and only if x_{1}, …, x_{n} ∈ Nil(d).

Locally nilpotent derivations of polynomial rings play an important role in algebra and algebraic geometry. It is well known that many open famous problems may be formulated using derivations or locally nilpotent derivations and their rings of constants. On the list of such problems are: the cancellation problem, the embedding problem, the linearization problem, the Jacobian conjecture, tame generator conjecture, the fourteenth problem of Hilbert and others (see, for example, [17], [12], [10]). On this lecture we present some old and new results concerning the fourteenth problem of Hilbert.

Let k be a field of characteristic zero, n ⩾ 1, k[X] := k[x_{1}, …, x_{n}] the polynomial ring over k, and k(X) := k(x_{1}, …, x_{n}) the field of rational functions over k. Assume that L is a subfield of k(X) containing k. The fourteenth problem of Hilbert is the following question ([21]).

Is the ring L ∩ k[X] finitely generated over k?

In 1954 Zariski ([29]) proved that the answer is affirmative if tr.deg_{k}L ⩽ 2. It is known, by a famous counterexample of Nagata ([21]), that if tr.deg_{k}L ⩾ 4, then it is possible to obtain a negative answer. If tr.deg_{k}L = 3, then the problem is still open.

Assume that d is a k-derivation of k[X]. Then we have the field L := (k[X]^{d})_{0}, the field of quotients of the constant ring k[X]^{d}, which is a subfield of k(X) containing k. The intersection L ∩ k[X] is equal to k[X]^{d}. So, in this case, the fourteenth problem of Hilbert is the following question.

Is the ring k[X]^{d} finitely generated over k?

In this question k is of characteristic zero. If char(k) > 0, then it is easy to show ([25]) that the answer is affirmative. So, let again char(k) = 0. Using the above mentioned result of Zariski it is not difficult to show (see [25]) that if n ⩽ 3, then the answer is affirmative. What does happen for n ⩾ 4?

In 1993 Derksen ([7]) showed that the ring from the Nagata counterexample is of the form k[X]^{d} for some derivation d of k[X] with n = 32. Thus, he proved:

Theorem 1 (Derksen). There exists a k-derivation d of k[X] := k[x_{1}, …, x_{32}] such that k[X]^{d} is not finitely generated over k.

If G is a subgroup of Aut_{k}(k[X]), the group of all k-automorphisms of k[X], then we denote by k[X]^{G} the subalgebra of invariants of G, that is,

k[X]^{G} = {f ∈ k[X]; σ(f) = f for all σ ∈ G}.

In 1994 the author, inspired by the above Derksen theorem, proved:

Theorem 2. If G ⊆ GL_{n}(k) is a connected algebraic group, then there exists a k derivation d of k[X] such that k[X]^{G} = k[X]^{d}.

In 1990 Roberts ([27]) gave a new counterexample to the fourteenth problem of Hilbert with n = 7. In 1994 Deveney and Finston ([9]) realized the Roberts counterexample in the following form.

Theorem 3. Let d be the k-derivation of k[X] := k[x_{1}, x_{2}, x_{3}, y_{1}, y_{2}, y_{3}, y_{4}] defined by d(x_{1}) = d(x_{2}) = d(x_{3}) = 0 and

d(y_{1}) = x_{1}^{3}, d(y_{2}) = x_{2}^{3}, d(y_{3}) = x_{3}^{3}, d(y_{4}) = (x_{1}x_{2}x_{3})^{2}.

Then the ring k[X]^{d} is not finitely generated over k.

Using this theorem one can easily deduce that if n ⩾ 7, then there always exists a k-derivation d of k[X] such that the ring of constants k[X]^{d} is not finitely generated over k.

Observe that the derivation d from the above theorem is locally nilpotent. This derivation has no slice. We say that a locally nilpotent k-derivation d of a k-algebra A has a slice if there exists an element s ∈ A such that d(s) = 1. It is well known (see for example [12] or [22]) that if A is a finitely generated k-algebra and d is a locally nilpotent k-derivation of A having a slice, then the ring of constants A^{d} is finitely generated over k.

Similar examples of derivations for n ⩾ 7 one can find, for instance, in [16] and [13]. Later, in 1998, Freudenburg ([14]) constructed a locally nilpotent derivation with the same property for n = 6.

Theorem 4 (Freudenburg). Let d be the k-derivation of k[X] := k[x_{1}, x_{2}, y_{1}, y_{2}, y_{3}, y_{4}] defined by d(x_{1}) = d(x_{2}) = 0 and

d(y_{1}) = x_{1}^{3}, d(y_{2}) = x_{2}^{3}y_{1}, d(y_{3}) = x_{2}^{3}y_{2}, d(y_{4}) = x_{1}^{2}x_{2}^{2}.

Then the ring k[X]^{d} is not finitely generated over k.

In 1999, Daigle and Freudenburg ([4]) gave a similar example with n = 5.

Theorem 5 (Daigle and Freudenburg). Let d be the k-derivation of the polynomial ring k[X] := k[a, b, x, y, z] defined by d(a) = d(b) = 0 and

d(x) = a^{2}, d(y) = ax + b, d(z) = y.

Then the ring k[X]^{d} is not finitely generated over k.

Observe that also in this case the derivation d is locally nilpotent. So, if n ⩽ 3, then k[X]^{d} is always finitely generated over k, and if n ⩾ 5, then there exists a k-derivation (even locally nilpotent) of k[X] with non-finitely generated ring of constants.

For n = 4 the problem is open. In this case there is no counterexample for arbitrary derivations and we do not know if k[X]^{d} is finitely generated for locally nilpotent derivations. In the last case we know, by the result of Maubach ([19]), that k[X]^{d} is finitely generated if d is locally nilpotent and the polynomials d(x_{1}), d(x_{2}), d(x_{3}) and d(x_{4}) are monomials. Recently Daigle and Freudenburg ([6]) proved that the ring of constants of any triangular derivation of k[x_{1}, x_{2}, x_{3}, x_{4}] is finitely generated over k. We say that a derivation d of k[X] is triangular if d(x_{i}) ∈ k[x_{1}, …, x_{i−1}] for all i = 1, …, n. Every triangular derivation of k[X] is of course locally nilpotent.

Let d be a k-derivation of k[X] = k[x_{1}, …, x_{n}], where k is a ring containing ℚ. If k[X]^{d} ≠ k and k[X]^{d} is finitely generated over k then we denote by γ(d) the minimal number of polynomials in k[X] ∖ k which generate k[X]^{d} over k. Moreover, we assume that γ(d) = 0 iff k[X]^{d} = k, and γ(d) = ∞ iff k[X]^{d} is not finitely generated over k. We already know from the previous section that there exist a natural number n and a k-derivation d of k[X] such that γ(d) = ∞.

If k is not a domain, then there always exist k-derivations of k[X] (even for n = 1) with non-finitely generated ring of constants. It follows from the following proposition which we may find in [12].

Proposition 6. Assume that k contains two nonzero elements a and b such that ab = 0. Let d be the k-derivation of k[X] defined by: d(x_{1}) = ax_{1}, d(x_{2}) = 0, …, d(x_{n}) = 0. Then γ(d) = ∞. More exactly, k[X]^{d} = k[x_{2}, x_{3}, …, x_{n}, bx_{1}, bx_{1}^{2}, bx_{1}^{3}, …].

If d = 0, then it is clear that γ(d) = n. If n = 1, k is a domain and d ≠ 0, then of course γ(d) = 0. Now we assume that k is a domain, d ≠ 0 and n ⩾ 2.

It is known ([28], [11]) that if k is a field then every Dedekind k-subalgebra of k[X] is a polynomial ring in one variable over k. As a consequence of this fact we obtain Theorem 7. Let d be a nonzero k-derivation of k[X], where k is a field of characteristic zero. If tr.deg_{k}(k[X]^{d}) ⩽ 1, then γ(d) ⩽ 1.

It follows from the above theorem that if k is a field of characteristic zero and d is a nonzero k-derivation of k[x, y], then γ(d) ⩽ 1. The same is true when k is a UFD.

Theorem 8 ([1]). Let k be a UFD containing ℤ. If d is a nonzero kderivation of k[x, y], then γ(d) ⩽ 1.

If k is not a UFD (even if k is noetherian), then the following proposition shows that Theorem 8 does not always hold.

Proposition 9 (Berson). Let k := ℂ[t^{2}, t^{3}]. Consider the k-derivation d of k[x, y] defined by

d(x) = t^{3}, d(y) = −t^{2}.

Then k is not a UFD, and k[x, y]^{d} is not finitely generated over k.

Observe that the derivation d from the above example is locally nilpotent. Similar examples of locally nilpotent derivations of k[x, y] with non-finitely generated rings of constants one can find in [2].

Now let n = 3. We already know from if k is a field, then γ(d) < ∞. However in this case the number γ(d) is unbounded. Strelcyn and the author ([26]) proved that if n ⩾ 3 and r ⩾ 0, then there exists a k-derivation d of k[X] such that γ(d) = r.

In some cases the number γ(d) is bounded. As a consequence of Theorem 8 we obtain that if k is a UFD containing ℤ and d is a nonzero k-derivation of k[x, y, z] such that d(x) = 0, then γ(d) ⩽ 2. For locally nilpotent derivations over a field the number γ(d) is also bounded.

Theorem 10 (Miyanishi [20]). If d is a nonzero locally nilpotent derivation of k[x, y, z], where k is a field of characteristic zero, then k[x, y, z]^{d} = k[f, g] for some algebraically independent polynomials f, g ∈ k[x, y, z].

There exists also the following homogeneous version of this theorem.

Theorem 11 ([30], [3]). Let d be a nonzero homogeneous locally nilpotent derivation of k[x, y, z]. Assume that the degrees of x, y, z are positive. Then k[x, y, z]^{d} = k[f, g], for some homogeneous polynomials f, g ∈ k[x, y, z].

Note also the following recent two results for n = 4.

Theorem 12 ([8]). If k is a field of characteristic zero and d is a nonzero triangular derivation of k[x_{1}, x_{2}, x_{3}, x_{4}] with a slice, then γ(d) = 3.

Theorem 13 ([5]). For any integer n ⩾ 3 there exists a triangular derivation d of k[x_{1}, x_{2}, x_{3}, x_{4}] such that n ⩽ γ(d) ⩽ n + 1.

Various facts and results concerning this subject we may find also in [12], [18], [15], [24], [22].

References

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