derivations
Andrzej Nowicki
N. Copernicus University, Faculty of Mathematics and Computer Science, ul. Chopina 12–18, 87–100 Toru´ n, Poland
(e-mail: anow@mat.uni.torun.pl)
The lecture at International Workshop on Affine Algebraic Geometry 8 – 14 December 1993, Technion, Haifa, Israel
1 Notations.
Let k be a commutative ring with 1, R a commutative k-algebra, and let d : R −→ R be a k-derivation of R. We denote by Rd the ring of constants of R with respect to d, that is, Rd = {x ∈ R; d(x) = 0}. Moreover, we denote by Nil(d) and Fin(d) the following subsets of R:
Nil(d) = {x ∈ R; ∃p∈N dp(x) = 0},
Fin(d) = {x ∈ R; ∃Mx⊂R x ∈ Mx, d(Mx) ⊆ Mx, Mx is a finite k-module}.
The subsets Rd, Nil(d) and Fin(d) are k-subalgebras of R and Rd ⊆ Nil(d) ⊆ Fin(d) ⊆ R.
The derivation d is called locally nilpotent (resp. locally finite) if Nil(d) = R (resp.
Fin(d) = R).
Let n be a natural number and let k[X] be the polynomial ring k[x1, . . . , xn]. In such a case every k-derivation d of k[X] has a unique presentation of the form
d = f1
∂
∂x1 + · · · + fn
∂
∂xn,
where f1, . . . , fn are polynomials from k[X], and it is easy to observe that
d is locally finite ⇐⇒ ∃s∈N∀i∈{1,...,n}∀p∈N deg dp(xi) 6 s. (1.1)
Assume now that Q ⊆ k. Let R[[t]] denote the formal power series ring in the variable t.
1
Then we have the k derivation ˜d : R[[t]] −→ R[[t]] defined by d(˜
∞
X
p=0
aptp) =
∞
X
p=0
d(ap)tp.
The derivation ˜d is the unique k-derivation of R[[t]] such that ˜d|R = d and ˜d(t) = 0. If ϕ is an element of R[[t]] then we denote by Ed(ϕ) the element of R[[t]] defined by
Ed(ϕ) =
∞
X
p=0
1 p!
d˜p(ϕ)tp.
Thus, we have a well defined mapping Ed: R[[t]] −→ R[[t]]. It is well known that Ed is a k[[t]]-automorphism of R[[t]] and ˜dEd = Edd. This automorphism is very popular in the˜ differential algebra. See, for example, papers of Seidenberg [23], [24] (1966-1967).
2 Systems of polynomial differential equations.
Let f1, . . . , fn be polynomials from k[Y ] = k[y1, . . . , yn], and let a1, . . . , an ∈ k. Then we may prove the following
Theorem 2.1. There exist unique series ϕ1, . . . , ϕn∈ k[[t]] such that
∂ϕi
∂t = fi(ϕ1, . . . , ϕn) (for i = 1, . . . , n)
and the constant terms of ϕ1, . . . , ϕn are equal to a1, . . . , an, respectively.
Put a = (a1, . . . , an). If polynomials f1, . . . , fn are fixed then the series ϕ1, . . . , ϕn from Theorem 2.1 are determined by a. Let us denote these series by ϕ1(a), . . . , ϕn(a).
Then we get
Theorem 2.2. There exist unique polynomials wij ∈ k[X] (for i ∈ {1, . . . , n} and j ∈ {0, 1, . . . }) such that
∀i∈{1,...,n}∀a∈kn ϕi(a) =
∞
X
j=0
wij(a)tj.
Put ψi =P∞
j=0wijtj, for i = 1, . . . , n. Then ψ1, . . . , ψn are series from k[X][[t]] and it is not difficult to see, by the above theorems, that they are unique series from k[X][[t]]
such that
∂ψi
∂t = fi(ψ1, . . . , ψn) (for i = 1, . . . , n)
and the constant terms of ψ1, . . . , ψn are equal to x1, . . . , xn, respectively.
In such a situation it is possible to prove the following
Theorem 2.3. If i ∈ {1, . . . , n} then ψi = Ed(xi), where d = f1∂x∂
1 + · · · + fn∂x∂
n.
3 Polynomial flows.
The problem presented in this section is rooted in the classical theory of ordinary differential equations. Let K denote either the field R of real numbers or the field C of complex numbers.
Consider a class C1 vector field f : Kn −→ Kn and the associated (autonomous) system of ordinary differential equations
∂x(t)
∂t = f (x(t)). (3.1)
For each initial condition (see e.g. [1])
x(0) = a ∈ Kn, (3.2)
there exists a unique maximal local solution (flow) x(t) = ϕ(t, a) = ϕa(t)
defined and satisfying (3.1) for all real t in a maximal open interval I(a) about t = 0, and also satisfying (3.2). The flow ϕ is called global if I(a) = R for each a ∈ Kn.
The flow ϕ(t, a) is called a polynomial flow if ϕ(t, a) is polynomial in a for each fixed t, that is, if ϕ(t, a) depends polynomially on the initial condition a.
Question 3.1. For which f , as above, is the flow ϕ(t, a) polynomial?
This question was raised by G. Meisters in [11] and discussed in many papers of Meis- ters, Olech, Bass, Coomes, Zurkowski and others (see [12], [3], see also [13] for information and a bibliography on this subject).
If a flow ϕ(t, a) is polynomial then, by (3.1), the vector field f , corresponding to this flow, must also be polynomial, that is, f = (f1, . . . , fn), where f1, . . . , fn are polynomials from K[x1, . . . , xn]. Moreover, if a flow ϕ(t, a) is polynomial then (see [2]) it is global and even (if K = C) entire ([3]).
In 1991 B. Coomes and V. Zurkowski in [4] proved (for K = C) that a polynomial vector field f = (f1, . . . , fn) has a polynomial flow if and only if the derivation f1 ∂
∂x1+· · ·+fn ∂
∂xn
is locally finite.
Let us return to Section 3. Let k be a ring containing Q and let k[X] = k[x1, . . . , xn] be a polynomial ring over k. Assume that f = (f1, . . . , fn) ∈ k[X]n and consider the autonomous system
∂X
∂t = f (X), X(0) = (x1, . . . , xn). (3.3) We already know, by Theorem 2.3, that the series Ed(x1), . . . , Ed(xn) form the flow of (3.3). These series are elements of the power series ring k[X][[t]]. Now one easily deduces that Question 3.1 reduces to the following
Question 3.2. When do series Ed(x1), . . . , Ed(xn) belong to k[[t]][X]?
Let us recall our notations:
k[X][[t]] = k[x1, . . . , xn][[t]], k[[t]][X] = k[[t]][x1, . . . , xn], k[[t, X]] = k[[t, x1, . . . , xn]].
There is a difference between ring k[X][[t]] and ring k[[t]][X]. They are both subrings of k[[t, X]] and k[[t]][X] $ k[X][[t]] $ k[[t, X]].
Proposition 3.3. Let g =P∞
p=0wptp be an element of k[X][[t]]. Then the following two conditions are equivalent:
(1) g ∈ k[[t]][X].
(2) There exists a natural number s such that deg wp 6 s, for any p ∈ N0. Now, using Proposition 3.3 and (1.1), one can easily deduce the following
Theorem 3.4 ([3]). Let k be a ring containing Q and let k[X] = k[x1, . . . , xn] be a polynomial ring over k. Assume that f = (f1, . . . , fn) ∈ k[X]n and consider the system (3.3) of differential equations. Then the following conditions are equivalent:
(1) The formal flow of (3.3) is polynomial.
(2) The derivation Pn i=1fi ∂
∂xi is locally finite.
4 The divergence.
If d : k[X] −→ k[X] is a k-derivation then we denote by d∗ the divergence of d, i. e., d∗ = ∂d(x1)
∂x1
+ · · · + ∂d(xn)
∂xn
.
H. Bass and G. Meisters in [2] showed that if the flow of a vector field f over C is polynomial, then the divergence of f is constant. This means, that the divergence of every locally finite C-derivation of C[x1, . . . , xn] is a complex number. Using this fact and a standard argument we get
Theorem 4.1. Let k be a reduced ring containing Q and let k[X] = k[x1, . . . , xn] be a polynomial ring over k. If d is a locally finite k-derivation of k[X] then d∗, the divergence of d, is an element of k.
Proof. Since d is locally finite, the mapping Edis a k[[t]]-automorphism of the polyno- mial ring k[[t]][X]. Hence J = Jac(Ed(x1), . . . , Ed(xn)), the jacobian of Ed, is an invertible element of k[[t]][X]. But k is reduced, so J ∈ k[[t]]. Put
J = a0+ a1t + a2t2+ · · · , (4.1) where a0, a1, . . . are elements from k. Then a0 = 1 and a1 = d∗. So, d∗ ∈ k.
Look at the proof of Theorem 4.1.
What is a2? What is a3? . . .
Lemma 4.2. Let d be a k-derivation of R, where R = k[x1, . . . , xn] or k[[x1, . . . , xx]]. Let h1, . . . , hn ∈ R. Then
d(Jac(h1, . . . , hn)) + Jac(h1, . . . , hn)d?+
n
X
p=1
Jac(h1, . . . , d(hp), . . . , hn).
Proposition 4.3. Let b = d∗. If b ∈ k then
Jac(Ed(x1), . . . , Ed(xn)) = ebt=
∞
X
p=0
1
p!bptp.
Theorem 4.4. Let k be a reduced ring containing Q and let k[X] = k[x1, . . . , xn] be a polynomial ring over k. If d is a locally nilpotent k-derivation of k[X] then d? = 0.
Proof. Look at the proof of Theorem 4.1. Since d is locally nilpotent, Ed is an automorphism of k[X][t]. Thus, J ∈ k[t] and, by Proposition 4.3, d∗ = 0.
The following example shows that, in general, the above property does not hold for non-reduced rings.
Example 4.5. Let k = Q[y]/(y2) and let d be the k derivation of k[x] (a polynomial ring in a one variable) defined by d(x) = ax2, where a = y + (y2). Since d2(x) = 2a2x3 = 0, d is locally nilpotent (and hence d is locally finite). But d? = 2ax 6∈ k.
Consider now Derk(k[X]), the k[X]-module of all k-derivations of k[X]. We know that Derk(k[X]) is a free k[X]-module on the basis {∂x∂
1, . . . , ∂x∂
n}. We say that a basis {d1, . . . , dn} of Derk(k[X]) is locally finite (resp. locally nilpotent) if each di is locally finite (resp. locally nilpotent). Note the following author’s result.
Theorem 4.6 ([15]). Let k be a ring containing Q and let k[X] = k[x1, . . . , xn] be the polynomial ring over k. The following conditions are equivalent.
(1) The Jacobian Conjecture is true in the n-variable case.
(2) Every commutative basis of the R-module Derk(k[X]) is locally finite.
(3) Every commutative basis of the R-module Derk(k[X]) is locally nilpotent. We may prove the following
Theorem 4.7. Let k be a ring containing Q. If {d1, . . . , dn} is a commutative basis of Derk(k[X]), then d∗i = 0 for all i = 1, . . . , n.
Note an application of the divergence for polynomial rings in two variables. Let d be a k-derivation of k[x, y]. Put d(x) = f , d(y) = g. Consider k[x, y]d, the ring of constants of k[x, y] with respect to d. It would be of considerable interest to find necessary and sufficient conditions on f and g for k[x, y]dto be nontrivial (that is, k[x, y]d 6= k). Observe that if 0 6= h ∈ k[x, y], then k[x, y]d= k[x, y]hd.
Proposition 4.8. Let k be a field of characteristic zero. Let d be a k-derivation of k[x, y].
Then k[x, y]d 6= k if and only if there exists a nonzero polynomial h ∈ k[x, y] such that (hd)∗ = 0.
5 Ring of constants.
Let k be a field and let d be a nonzero k-derivation of k[X] = k[x1, . . . , xn]. If char(k) = p > 0 then k[X]d is finitely generated over k [18].
Assume that char(k) = 0.
If n = 1 then k[X]d= k.
If n = 2 then k[X]d is finitely generated and k[X]d= k[f ] for some f ∈ k[X].
If n = 3 then k[X]d is finitely generated and the number of generators is unbounded [19]. See also [18] for details.
Recently, H. Derksen in [5], proved that if n = 32 then there exists a k-derivation d of k[X] such that k[X]d is not finitely generated.
In 1990, P. Roberts in [22], constructed a new counterexample to the fourteenth prob- lem of Hilbert. From this result and from the author’s result [16] one can deduce that if n = 7 then there exists a k-derivation d of k[X] such that k[X]d is not finitely generated.
The last fact implies that if n > 7 then there always exists such a k-derivation d of k[X] that k[X]d is not finitely generated.
Note also one of the classical results. The well known result of Weitzenb¨ock [28] states that every linear action of the additive group (k, +) on An has a finitely generated ring of invariants. A modern proof of this result is due to C.S.Seshadri [25]; cf.([14] pp. 36 – 40). In the vocabulary of derivations this is equivalent to the following
Theorem 5.1 (Weitzenb¨ock [28] 1932, Seshadri [25] 1961). Let d be a k-derivation of k[x1, . . . , xn] such that
d(xi) =
n
X
j=1
aijxj for i = 1, . . . , n, with aij ∈ k. (5.1) If the matrix [aij] is nilpotent then the ring of constants k[x1, . . . , xn]d is finitely generated over k.
In other words, the above theorem says that the ring of constants with respect to a locally nilpotent linear k-derivation of k[x1, . . . , xn] is finitely generated. Recently, A. Tyc [27] generalized this fact:
Theorem 5.2 ([27]). Every k-derivation of k[x1, . . . , xn] of the form (5.1) has a finitely generated (over k) ring of constants.
6 Jordan-Chevalley decomposition.
Let k be a field, V a finite dimensional linear k-space, and let ϕ : V −→ V be a k-endomorphism of V .
It is well known that if k is algebraically closed then there exists a basis B of V such that the matrix of ϕ with respect to B has the Jordan canonical form.
Such a situation is not only for algebraically closed fields. It is true also if the charac- teristic polynomial of ϕ is a product of linear factors. We are interested only in the case when k is a field of characteristic zero. In such a case the following theorem gives a more precise information (see [9] (p. 17) or [10] (XV exercise 14)).
Theorem 6.1. Let V be a finite dimensional linear space over a field k of characteristic zero, and let ϕ : V −→ V be a k-endomorphism. Then:
(1) There exist unique k-endomorphisms ϕs and ϕn of V such that (a) ϕ = ϕs+ ϕn,
(b) ϕs is semisimple, (c) ϕn is nilpotent, (d) ϕsϕn= ϕnϕs.
Moreover, the endomorphisms ϕs and ϕn have the following property:
(2) ϕs = u(ϕ) and ϕn= v(ϕ), for some polynomials u, v ∈ k[t] without constant terms.
In particular,
(i) ϕs and ϕn commute with any k-endomorphism of V commuting with ϕ.
(ii) If V0 ⊆ V1 ⊆ V are k-subspaces and ϕ maps V1 into V0 then ϕs and ϕn also map V1 into V0.
The decomposition ϕ = ϕs+ ϕn is called the Jordan-Chevalley decomposition of ϕ.
If V is not finitely dimensional then we have the following version of the Jordan- Chevalley decomposition
Theorem 6.2. Let V be a linear space over a field k of characteristic zero and let ϕ : V −→ V be a locally finite k-endomorphism. Then:
(1) There exist unique k-endomorphisms ϕs and ϕn of V such that (a) ϕ = ϕs+ ϕn,
(b) ϕs is semisimple, (c) ϕn is locally nilpotent, (d) ϕsϕn= ϕnϕs.
(2) The endomorphisms ϕsand ϕncommute with any k-endomorphism (not necessarily locally finite) of V commuting with ϕ.
(3) If V0 ⊆ V1 ⊆ V are k-subspaces and ϕ maps V1 into V0 then ϕs and ϕn also map V1 into V0.
The Jordan-Chevalley decomposition for derivations has the following form
Theorem 6.3 ([27]). Let d be a locally finite k-derivation of a k-algebra R, where k is a field of characteristic zero. Let d = ds+ dn be the Jordan-Chevalley decomposition of the endomorphism d. Then ds and dn are k-derivations of R. The derivation ds is semisimple, and the derivation dn is locally nilpotent.
From the above theorem one can easily deduce that, if R and d are as in the theorem, then
d(Rdn) ⊆ Rdn and d(Rds) ⊆ Rds. Moreover,
Rd = (Rdn)d0s, where d0s = ds|Rdn.
Now the mentioned result of Tyc (Theorem 5.2) follows from Weitzenb¨ock’s theorem (Theorem 5.1) and from the following
Theorem 6.4 ([27]). Let R be a noetherian k-algebra and let d be a semisimple k- derivation of R. Then Rd is a noetherian ring.
7 Semisimple and diagonalizable derivations.
In this section we assume that k is a field of characteristic zero, R is a k-algebra, and d : R −→ R is a k-derivation. An element x ∈ R is said to be semisimple with respect to d if there exists a finite d-invariant k-subspace of R containing x such that the k-endomorphism d|W is semisimple. We denote by Sem(d) the set of all semisimple (with respect to d) elements of R.
Proposition 7.1. If d is a k-derivation of a k-algebra R, where k is a field of character- istic zero, then Sem(d) is a k-subalgebra of R.
Proposition 7.2. If k is a field and d is a k-derivation of a k-algebra then Sem(d) ∩ Nil(d) = Ad.
We say that a k-derivation d is semisimple if d as the k-endomorphism is semisimple.
So, d : R −→ R is semisimple if and only if Sem(d) = R. Since Sem(d) ⊆ Nil(d), every semisimple derivation is locally finite.
Proposition 7.3. Let R = k[r1, . . . , rn] be a finitely generated k-algebra, where k is a field of characteristic zero. Let d be a k-derivation of R. The following conditions are equivalent:
(1) d is semisimple.
(2) There exist finite d-invariant k-subspaces W1, . . . , Wn of R such that ri ∈ Wi and d|Wi is semisimple, for all i = 1, . . . , n.
(3) There exists a finite d-invariant k-subspace W of R such that r1, . . . , rn ∈ W and d|W is semisimple.
The above proposition is very useful for a verification if a given derivation is semisim- ple. Look at the following examples in polynomial rings over a field k of characteristic zero.
Example 7.4. R = k[x1, . . . , xn], d(xi) = aixi for i = 1, . . . , n, where a1, . . . , an ∈ k.
The k-derivation d is semisimple.
Proof. Put Wi = kxi and use Proposition 7.3 (2).
Example 7.5. R = k[x, y], d(x) = y, d(y) = x. Then d is semisimple.
Proof. Since d(x + y) = x + y and d(x − y) = −(x − y), the polynomials x + y and x − y belong to Sem(d). Thus, by Proposition 7.1, we have: k[x, y] = k[x + y, x − y] ⊆ Sem(d) ⊆ k[x, y].
Example 7.6. R = k[x, y], d(x) = x, d(y) = y + f (x) − x∂f (x)∂x , where f (x) ∈ k[x] (for example: d(x) = x, d(y) = y + x2). Then d is semisimple.
Proof. Since d(x) = x and d(y + f (x)) = y + f (x), the polynomials x and y + f (x) belong to Sem(d). Thus, by Proposition 7.1, we have: k[x, y] = k[x, y + f (x)] ⊆ Sem(d) ⊆ k[x, y].
If d is a semisimple k-derivation of R and σ is a k-automorphism of R then the k- derivation σdσ−1 is also semisimple.
Example 7.7. R = k[x, y, z], d(x) = x, d(y) = y − x2, d(z) = z − y2 + 2x2y. Then d is semisimple.
Proof. Let δ = x∂x∂ + y∂y∂ + z∂z∂. Then δ is semisimple and d = σδσ−1, where σ is the k-automorphism of k[x, y, z] such that σ(x) = x, σ(y) = y + x2 and σ(z) = z + y2.
Let us introduce some new notion. Let d be a k-derivation of a polynomial ring R = k[x1, . . . , xn]. We say that d is diagonal if d(x1) = a1x1, . . . , d(xn) = anxn, for some a1, . . . , an ∈ k. We say that d is diagonalizable if d is equivalent to a diagonal k- derivation, that is, d is diagonalizable iff there exists a k-automorphism σ of R such that the derivation σdσ−1 is diagonal. It follows from Example 7.4 that every diagonalizable derivation is semisimple.
If R = k[x] is a polynomial ring in a one variable, then it is clear that every semisimple k-derivation of k[x] is diagonalizable.
Assume now that R = k[x, y]. In 1985 Bass and Meisters [2] gave a description, up to k-automorphisms of k[x, y], of all the locally finite k-derivations of k[x, y] (for k = R or C). An elegant proof of this fact based on the results of Rentchler [21], is given by A. van den Essen [6]. His proof is valid for an arbitrary field of characteristic zero. Using this fact and the Jordan-Chevalley decomposition we get
Theorem 7.8. If k be an algebraically closed field of characteristic zero then every semisim- ple k-derivation of k[x, y] is diagonalizable.
Question 7.9. Let d be a semisimple k-derivation of k[x1, . . . , xn], where k is an alge- braically closed field of characteristic zero. Is d diagonalizable?
Let n = 2. One of the eqivalent versions of the Jacobian Conjecture is as follows Conjecture 7.10. Let f ∈ k[x, y] and let d be the k-derivation of k[x, y] defined by
d(x) = −∂f
∂y, d(y) = ∂f
∂x.
Assume that there exists g ∈ k[x, y] such that d(g) = 1. Then d is locally finite.
Let d be as in the above conjecture. It is not difficult to see that id d is locally finite then d is locally nilpotent. So, the Jacobian Conjecture says that the derivation d is locally nilpotent.
Assume now that f, g ∈ k[x, y] and Jac(f, g) = 1. Consider the k-derivation δ of k[x, y]
defined by
δ(x) = −g2 ∂
∂x(f /g), δ(y) = g2 ∂
∂y(f /g)
(observe that δ(x), δ(y) ∈ k[x, y]). If the Jacobian Conjecture is true then the derivation δ is semisimple. It is difficult to prove that δ is semisimple and the author does not know if it is equivalent to the Jacobian Conjecture. We fixed here n = 2. For n > 2 there are similar problems.
8 Weitzenb¨ ock derivations.
This section contains the results of the author’s paper [17].
Let k be a field of characteristic zero and let d be a linear locally nilpotent k-derivation of the polynomial ring k[X] = k[x1, . . . , xn].
Let us recall that Weitzenb¨ock’s theorem (Theorem 5.1) states that the ring of constant of d is finitely generated over k.
All the known proofs of Weitzenb¨ock’s theorem are not constructive. Given a linear k-derivation of k[x1, . . . , xn] it is not easy to describe its ring of constants even if we assume that the corresponding matrix is nilpotent.
A k-derivation d of k[x1, . . . , xn] is called a Weitzenb¨ock derivation if d is linear and its matrix is a Jordan’s matrix with the zeros on the main diagonal. It is obvious that when studying the ring of constants of a linear locally nilpotent k-derivation d, we may always suppose that d is a Weitzenb¨ock derivation.
Let us make precise some notations. Assume that n > 2 and let Rn= k[x0, x1, . . . , xn−1]
be the polynomial ring in n variables over k.
Let us consider only one Jordan’s cell:
J =
0 1 0 · · · 0 0 0 0 1 · · · 0 0
· · · · 0 0 0 · · · 0 1 0 0 0 · · · 0 0
.
Let dn : Rn −→ Rn be the Weitzenb¨ock derivation of Rn corresponding to the matrix J , i. e.,
dn(x0) = 0 dn(x1) = x0 dn(x2) = x1
· · ·
dn(xn−1) = xn−2
(8.1)
Such a Weitzenb¨ock derivation will be called basic.
We would like to find (for any n) a finite set of generators over k of the ring Rdnn. If n = 2 then it is easy. Since d2 = x0∂x∂
1, Rd22 = k[x0]. For arbitrary n the problem seems to be difficult. In [8] we may find a finite set of polynomials which is proposed as
a generating set of Rdnn. But, as shows L. Tan in [26], it is not a generating set. Full lists of generators, for n = 3 or 4, are described in [26].
The case n = 5, as it is mentioned in [26] (see also [20]), is complicated and a finite set of generators was unknown. We present here such a set.
We present also finite sets of generators for n 6 4 and we give some observations for n = 6. Several cases of nonbasic Weitzenb¨ock derivations are also described.
Our calculations are based on methods from the theory of Gr¨obner bases and we use a modification of A. van den Essen’s algorithm [7].
Denote by t the variable x0. If i > 2 then we put
zi = xi1+ i!
i − 1
i−2
X
p=0
(−1)i−1−p1
i!ti−1−pxi1xi−p. (8.2) In particular,
z2 = x21− 2tx2,
z3 = x31− 3tx1x2+ 3t2x3,
z4 = x41− 4tx21x2+ 8t2x1x3− 8t3x4,
z5 = x51− 5tx31x2+ 15t2x12x3− 30t3x1x4+ 30t4x5.
(8.3)
It is not difficult to see that d(zi) = 0. Moreover, we have:
k[t, z2, . . . , zn−1] ⊆ Rdnn ⊆ k[t, z2, . . . , zn−1][1/t] (8.4) and
k[t, z2, . . . , zn−1][1/t] ∩ Rn= Rndn. Put:
u = x21x22− 2x31x3+ 6tx1x2x3− 83tx32− 3t2x23, p2 = 2x1x3− x22− 2tx4,
p3 = 6x21x4− 6x1x2x3+ 2x32− 12tx2x4+ 9tx23. Proposition 8.1.
(1) Rd33 = k[t, z2], (2) Rd44 = k[t, z2, z3, u],
(3) Rd55 = B = k[t, z2, z3, p2, p3].
Let Sn= k[x1, y1, x2, y2, . . . , xn, yn] be the polynomial ring over k. Let us consider the k-derivation δn of Sn defined as
δn= y1
∂
∂x1 + · · · + yn
∂
∂xn, (8.5)
that is, δn(y1) = 0, δn(x1) = y1, . . . , δn(yn) = 0, δn(xn) = yn. It is a nonbasic Weitzenb¨ock derivation and all cells of its Jordan matrix have the dimension equal 2. We will present finite generating sets of the ring Snδn for n 6 4.
The derivation δnis an A-derivation of Snsuch that δn(xi) ∈ A, for i = 1, . . . , n, where A = k[y1, . . . , yn].
Consider a more general case. Let A be a k-domain containing k and let d be an A-derivation of A[x1, . . . , xn] such that d(xi) ∈ A. Of course such a derivation is locally nilpotent. If A is a field, then following describes the ring of constants of d.
Proposition 8.2. Let A be a field containing k and let d be a nonzero A-derivation of A[x1, . . . , xn] such that d(x1) = a1, . . . , d(xn) = an, where a1, . . . , an ∈ A and a1 6= 0.
Then
A[x1, . . . , xn]d= A[u2, . . . , un] where up = apx1− a1xp, for p = 2, . . . , n.
Consider the following
Question 8.3. Is the thesis of the above proposition true in the case when A is a k- domain?
If n = 2 and A is a UFD, then we have a positive answer to this question.
Proposition 8.4. Let A[x, y] be the polynomial ring in two variables over a unique fac- torization k-domain A. Assume that d : A[x, y] −→ A[x, y] is an A-derivation such that d(x) = a, d(y) = b, where a, b are coprime elements from A. Then A[x, y]d= A[bx − ay].
The following example shows that Question 8.3 has a negative answer in general.
Example 8.5. Let A = Q[t], R = A[x, y, z]. Consider the A-derivation d of R such that d(x) = 2t, d(y) = 1 + t, d(z) = 1 − t and let F = x − y + z. Then d(F ) = 0 and F 6∈ A[u2, u3], where u2 = (1 + t)x − 2ty and u2 = (1 − t)x − 2tz.
Let us return to the derivation δn defined by (8.5). It is clear that S1δ1 = k[y1] and, as a consequence of Proposition 8.4, we get
Proposition 8.6. S2δ2 = k[y1, y2, x1y2− x2y1]. If i < j then we denote
uij = yixj − yjxi. In particular ˜x2 = u12, . . . , ˜xn= u1n.
Proposition 8.7. B ⊆ Snδn ⊆ B[1/y1], where B = k[y1, . . . , yn, u12, . . . , u1n].
Let B be the algebra as in Proposition 8.7. In this case Question 8.3 reduces to the question: ”Is Snδn equal to B?”. It is not true in general.
Example 8.8. u236∈ k[y1, y2, y3, u12, u13].
Proposition 8.9. S3δ3 = k[y1, y2, y3, u12, u13, u23].
Proposition 8.10. S4δ4 = k[y1, y2, y3, y4, u12, u13, u14, u23, u24, u34]. Conjecture 8.11. Snδn = k[y1, . . . , yn, u12, u13, . . . , un−1,n].
Proposition 8.12. Let R = k[x1, x2, y1, y2, y3] and
d(x1) = 0 d(x2) = x1
d(y1) = 0 d(y2) = y1 d(y3) = y2.
Then Rd= k[x1, y1, x1y2− x2y1, y22− 2y1y3, 2x21y3− 2x1x2y2+ x22y1]. Proposition 8.13. Let R = k[x1, x2, x3, y1, y2, y3] and
d(x1) = 0 d(x2) = x1 d(x3) = x2
d(y1) = 0 d(y2) = y1 d(y3) = y2. Then Rd= k[x1, y1, f1, f2, f3, f4, f5, f6], where
f1 = 2x1x3− x22 f2 = x1y2− x2y1,
f3 = x1y3− x2y2+ x3y1, f4 = 2y1y3− y22,
f5 = x1y22− 2x2y1y2+ 2x3y12, f6 = 2x21y3− 2x1x2y2 + x22y1.
N. Onoda in [20] announces the following theorem concerning to the rings of constants of Weitzenb¨ock derivations.
Theorem 8.14 ([20]). Let dn be the Weitzenb¨ock derivation defined by (8.1) over an algebraically closed field k (of characteristic zero) and let C = Rndn. Then
(1) C is a Gorenstein ring.
(2) (a) If n = 1 then C ∼= k[x].
(b) If n = 2 then C ∼= k[x, y].
(c) If n = 3 then C ∼= k[x, y, z, u]/(x2u + y3+ z2).
(d) If n = 4 then C ∼= k[x, y, z, u, v]/(x3v + y3+ z2+ x2yu).
(3) If n 6 4 then C is a complete intersection.
(4) If n = 5 then C is not a complete intersection.
References
[1] V.I. Arnold, Ordinary Differential Equations, MIT Press, 1973 – 1981.
[2] H. Bass, G.H. Meisters, Polynomial flows in the plane, J. Algebra, 55(1985), 173 – 208.
[3] B.A. Coomes, Polynomial flows on Cn, Trans. Amer. Math. Soc., 320(1990), 493 – 506.
[4] B.A. Coomes, V. Zurkowski, Linearization of polynomial flows and spectra of deriva- tions, J. Dynamics and Diff. Equations, 3(1991), 29 – 66.
[5] H.G.J. Derksen, The kernel of a derivation, J. Pure Appl. Algebra 84(1993), 13 – 16.
[6] A. van den Essen, Locally finite and locally nilpotent derivations with applications to polynomial flows and polynomial morphisms, Proc. Amer. Math. Soc., 116(1992), 861 – 871.
[7] A. van den Essen, An algorithm to compute the invariant ring of a Ga-action on an affine variety, Catholic University, Nijmegen, Report 9202(1992), 1 – 8.
[8] A. Fauntleroy, Actions on affine spaces and associated rings of invariants, J.Pure Appl.Algebra 9(1977), 195 – 206.
[9] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer- Verlag, New York Heidelberg Berlin, 1972.
[10] S. Lang, Algebra, Addison–Wesley Publ. Comp. 1965.
[11] G.H. Meisters, Polynomial dependence on initial conditions, 10th Midwest Confer- ence on Differential Equations, Fargo, N. D., 1981.0
[12] G.H. Meisters, Polynomial flows on Rn, Proceedings of the Semester on Dynamical Systems, Banach Center Publ., PWN, Warsaw, 23(1989), 9 – 24.
[13] G.H. Meisters, Bibliography on Polynomial Maps, 1992.
[14] M. Nagata, Lectures on the Fourteenth Problem of Hilbert, Lect.Notes 31, Tata Institute, Bombay, 1965.
[15] A. Nowicki, Commutative basis of derivations in polynomial and power series rings, J. Pure Appl. Algebra,40(1986), 279–283.
[16] A. Nowicki, Rings and fields of constants for derivations in characteristic zero, J. Pure Appl. Algebra, 96(1994), 47–55.
[17] A. Nowicki, An application of Gr¨obner bases to some Weitzenb¨ock derivations, Toru´n, 1993.
[18] A. Nowicki, M. Nagata, Rings of constants for k–derivations in k[x1, . . . , xn], J. Math. Kyoto Univ., 28(1988), 111 – 118.
[19] A. Nowicki, J.-M.Strelcyn, Generators of rings of constants for some diagonal deriva- tions in polynomial rings, J. Pure Appl. Algebra, 101(1995), 207–212.
[20] N. Onoda, Linear actions of Ga on polynomial rings, Proc. 25th Symp.Ring Theory, Matsumoto 1992, 11 – 16.
[21] R. Rentschler, Op´erations du groupe additif sur le plan affine, C.R.Acad.Sc.Paris 267(1968), 384 – 387.
[22] P. Roberts, An infinitely generated symbolic blow-up in a power series ring and a new counterexample to Hilbert’s fourteenth problem, J.Algebra 132(1990), 461 – 473.
[23] A. Seidenberg, Derivations and integral closure, Pac. J. Math., 16(1966), 167 – 173.
[24] A. Seidenberg, Differential ideals and rings of finitely generated type, Amer. J. Math., 89(1967), 22 – 42.
[25] C.S. Seshadri, On a theorem of Weitzenb¨ock in invariant theory, J.Math.Kyoto Univ.
1(1961), 403 – 409.
[26] L. Tan, An algorithm for explicit generators of the invariants of the basic Ga-actions, Comm. in Algebra 17(1989), 565 – 572.
[27] A. Tyc, Jordan decomposition and the ring of constants of locally finite derivations, Preprint 1993.
[28] R. Weitzenb¨ock, ¨Uber die invaranten Gruppen, Acta.Math. 58(1932), 231 – 293.
