1 Locally nilpotent derivations

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) December 29, 1993

Let k be a field of characteristic zero, let n > 1 be a natural number, and let k[x₁, . . . , x_n] be the polynomial ring over k.

The well known result of Weitzenb¨ock [22] states that every linear action of the additive group G_aon Aⁿhas a finitely generated ring of invariants. In the vocabulary of differential equations this means that every system of polynomial differential equations of the form









 dx₁

dt = a₁₁x₁+ · · · + a_1nx_n

· · · dx_n

dt = an1x1+ · · · + annxn

where [a_ij] is a nilpotent n×n matrix over k, has a finite subset of k[x₁, . . . , x_n] generating over k the ring of all polynomial first integrals.

In the vocabulary of derivations this is equivalent to the following assertion. Let d be a locally nilpotent k-derivation of the polynomial ring k[x₁, . . . , x_n] which is linear (i. e., the polynomials d(x₁), . . . , d(x_n) are linear forms). Then the ring of constant of d is finitely generated over k.

Recently, A. Tyc [21] generalized this fact. He proved, using Weitzenb¨ock theorem, that every (that is, not only locally nilpotent) linear k-derivation has a finitely generated ring of constants.

A modern proof of Weitzenb¨ock theorem is due to C.S.Seshadri [17] (see also [9]). All the known proofs are not constructive. Given a linear k-derivation of k[x₁, . . . , x_n] it is not easy to describe its ring of constants even if we assume that the corresponding matrix is nilpotent.

We say that a k-derivation d of k[x₁, . . . , x_n] is Weitzenb¨ock 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, without loss of generality, we may suppose that d is a Weitzenb¨ock derivation.

1

Let us make precise some notations. Assume that n > 2 and let Rn = k[x₀, x₁, . . . , x_n−1] be the polynomial ring in n variables over k. 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 d_n : R_n−→ R_n be the Weitzenb¨ock derivation of R_n corresponding to J , i. e.,















d_n(x₀) = 0 dn(x1) = x0

d_n(x₂) = x₁

· · ·

dn(xn−1) = xn−2.

(0.1)

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 R^d_nⁿ, the ring of constants of d_n. If n = 2 then it is easy. Since d₂ = x_0∂x^∂

1, R^d₂² = 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 R^d_nⁿ. But, as shows L. Tan in [20], it is not a generating set. Full lists of generators, for n = 3 or 4, are described in [20].

The case n = 5, as it is mentioned in [20] (see also [13]), is complicated and a finite set of generators was unknown. We present here (see Theorem 6.2) 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. van den Essen’s algorithm [5], [6].

We assume that the reader is familiar with the basic notions of Gr¨obner bases (see for example [2], [14]). If I is an ideal then we denote by GB(I) its reduced Gr¨obner basis.

We say that an k-algebra A is a k-domain if A is without zero divisors.

Acknowledgments. The author wishes to thank Jean Moulin Ollagnier for his sug- gesting introduction to Gr¨obner bases. Also the author wishes to thank Kazuo Kishimoto and Jean-Marie Strelcyn for many inspiring discussions.

1 Locally nilpotent derivations

In this section we recall some facts concerning to locally nilpotent derivations.

Let A be a finitely generated (commutative) k-domain and let d : A −→ A be a nonzero k-derivation. We denote by A^d the ring of constants of d, i. e., A^d = {a ∈ A; d(a) = 0}.

We say that d is locally nilpotent if for any a ∈ A there exists m > 0 such that d^m(a) = 0.

In such a case, if a 6= 0, then we denote by deg_d(a) the number s such that d^s(a) 6= 0 and d^s+1= 0.

Assume that d is locally nilpotent.

If there exists an element b ∈ A such that d(b) = 1 then it is well known (see for example [15], [4], [7]) that A^dis a finitely generated k-algebra and there exists an explicit formula for a finite generating set of A^d.

Now we do not assume that there exists an element b in A with d(b) = 1. Since d is a nonzero locally nilpotent k-derivation of A, there exists an element c ∈ A such that deg_d(c) = 1. Put h = d(c). Then, from [20], [5] and [7], one can deduce the following Proposition 1.1. Let A = k[a₁, . . . , a_n] be a finitely generated k-domain, and let d be a nonzero locally nilpotent k-derivation of A. Then

k[h, ˜a₁, . . . , ˜a_n] ⊆ A^d ⊆ k[h, ˜a₁, . . . , ˜a_n][1/h], k[h, ˜a₁, . . . , ˜a_n][1/h] ∩ A = A^d,

where the elements ˜a₁, . . . , ˜a_n are defined by the formula:

˜ a_i =

si

X

p=0

1

p!d^p(a_i)(−c)^ph^si−p (1.1) for i = 1, . . . , n, with s_i = deg_d(a_i). 

Let M be the family of all k-subalgebras of A containing h such that B ⊆ A^d⊆ B[1/h]

and, if B ∈ M, then denote by B the k-subalgebra of A generated by the set {a ∈ A; ah ∈ B}. Proposition 1.1 implies that there exists a finitely generated k-algebra B belonging to M. Moreover, if B ∈ M then it is easy to see that B ⊆ B and B ∈ M.

Assume that A = k[a₁, . . . , a_n] and let ˜a₁, . . . , ˜a_n be the elements of A defined by (1.1).

For every integer m > 0 we define (as in [5]) the following sequence (Bm) of k- subalgebras of A:

( B₀ = k[h, ˜a₁, . . . , ˜a_n],

Bm = Bm−1, for m > 1. (1.2)

Each B_m is a finitely generated algebra over k belonging to M ([5] Lemma 2.5).

Moreover, B₀ ⊆ B₁ ⊆ · · · ⊆ A^d, S∞

m=0B_m = A^d, and we have

Lemma 1.2 ([5]). If A^d is finitely generated over k, then A^d = B_m for some m > 0.  As a consequence of the above facts we get

Proposition 1.3. Let k be a field of characteristic zero, A = k[a1, . . . , an] a finitely generated k-algebra without zero divisors, and d a nonzero locally nilpotent k-derivation of A. Let (B_m) be the sequence of k-algebras defined by (1.2). The following conditions are equivalent:

(1) A^d is finitely generated over k.

(2) A^d= B_m, for some m > 0.

(3) The sequence B0 ⊆ B1 ⊆ . . . is stationary.

(4) There exists m > 0 such that Bm = B_m+1.

(5) There exists a finitely generated k-algebra B ∈ M such that B = B.

Proof. The implication (1) ⇒ (2) follows from Lemma 1.2. The implications (2)

⇒ (3) ⇒ (4) ⇒ (5) are evident. It remains to show (5) ⇒ (1). Let x ∈ A^d. Since A^d ⊆ B[1/h], there exists s > 0 such that h^sx ∈ B. Hence h^s−1x ∈ B = B, and consequently h^s−2x, . . . , h¹x, x ∈ B. So, A^d= B is finitely generated over k. 

From the proof of the implication (5) ⇒ (1) we obtain the following

Corollary 1.4. Let k be a field of characteristic zero, A = k[a1, . . . , an] a finitely generated k-domain, and d a nonzero locally nilpotent k-derivation of A. Assume that there exists a finitely generated k-algebra B ∈ M such that B = B. Then A^d is finitely generated over k and A^d= B. 

2 An open problem and A. van den Essen’s algorithm

Let A denote the polynomial ring k[x₁, . . . , x_n].

We know that if d is an arbitrary k-derivation of A and n 6 3 then A^d is a finitely generated k-algebra. This result is due to Zariski [23] (see [12] for details). H. Derksen in [3] showed that if n = 32, then it is possible to construct such a k-derivation d of A that A^d is not finitely generated. His example is based on the well known Nagata counterexample [10] to the fourteenth problem of Hilbert. The existence of derivations of A with the non-finitely generated algebra A^d follows also from the author paper [11].

By [16] and [11], it is possible to show that if n = 7 then also exists such a derivation.

This fact implies that if n > 7 then there always exists a k-derivation d of A such that A^d is not finitely generated. There is a still open question for remaining n, for instance, if n = 4. This question is open even if we assume that d is locally nilpotent.

In 1989 L. Tan in [20] gave a method of computing generators of the ring of constants for basic Weitzenb¨ock derivations. In this case it is known that the ring of constants is finitely generated.

Later, in 1992, A. van den Essen ([5], [6]) showed that this method works in a more general case. He gave an algorithm computing a finite set of generators of the ring of constants for any locally nilpotent derivation (of a finitely generated k-domain), in case when the finiteness of the ring of constants is known.

Assume now that A = k[a₁, . . . , a_n] is a finitely generated k-domain, and d is a locally nilpotent k-derivation of A. By A. van den Essen’s algorithm we may compute a finite set of generators of the algebras B_m defined in (1.2). If we know that A^d is finitely generated than we know, by Lemma 1.2, that A^d= B_m, for some m. The ascending chain B0 ⊆ B1 ⊆ . . . becomes stationary after a finite number of steps where they are equal to A^d. Using the theory of Gr¨obner basis we may compute B_m from B_m−1 for all m > 1 (see [5] Section 3 or [6] Section 4.2) and also, using the algebra membership algorithm given in [18], we may decide if Bm = Bm−1.

It is remarkable that the assumption of finiteness of A^d, which is in A. van den Essen’s algorithm, is unessential. By this assumption we know only that our calculations must be stopped in a finite number of steps, but we do not know when. We may use the same procedure for arbitrary locally nilpotent derivation of A. If we obtain the success, i. e.,

the calculations are stopped, then, by Theorem 1.3, we have a proof that A^d is finitely generated, and we have a finite set of its generators.

The author was trying to use the above algorithm to several derivations of polynomial rings. There exist some difficulties. If we start from the algebra B₀ than, in many cases, we must (for the next steps) introduce new polynomial rings in much bigger number of variables. This number is often too big for a computer.

However there exists an another way. We may start not from B₀ but from an algebra B which is in family M described in Section 1 and, in the next steps, we construct the algebras of the form B. By this way the author described generators for some locally nilpotent derivations studied in the next sections.

3 Properties of the basic Weitzenb¨ ock derivations

Let d = d_n be a basic Weitzenb¨ock derivation. Then d^p(x_i) = x_i−p, for p = 0, 1, . . . , i, and deg_d(xi) = i, for i = 0, 1, . . . , n − 1. The variables x0 and x1 play the same role as elements h and c from Section 1, that is d(x₁) = x₀ 6= 0, d²(x₁) = 0. Thus, by (1.1), putting a_i = x_i, s_i = i, c = x₁, h = x₀ , we get

˜

x0 = x0, ˜x1 = 0 and, for i > 2,

˜ xi =

i

X

p=0

(−1)^p1

p!x^i−p₀ x^p₁xi−p

= (−1)ⁱ1

i!x⁰₀xⁱ₁x₀+ (−1)ⁱ⁻¹x¹₀xⁱ⁻¹₁ x₁+

i−2

X

p=0

(−1)^p1

p!x^i−p₀ x^p₁x_i−p

= (−1)ⁱ⁻¹i − 1

i! x₀· ( xⁱ₁+ i!

i − 1

i−2

X

p=0

(−1)^i−1−p1

i!x^i−1−p₀ xⁱ₁x_i−p ).

We know, from Section 1, that d(˜x_i) = 0, for i = 0, 1, . . . , n − 1.

Let us denote by t the variable x₀.

Multiplying every element ˜x_i (for i > 2) by (−1)^{i−1 i!}_i−1 and dividing by t (recall that d(t) = 0) we obtain the polynomial z_i such that

z_i = xⁱ₁+ i!

i − 1

i−2

X

p=0

(−1)^i−1−p1

i!t^i−1−pxⁱ₁x_i−p (3.1)

and d(zi) = 0.

In particular,















z₂ = x²₁− 2tx₂,

z₃ = x³₁− 3tx₁x₂+ 3t²x₃,

z₄ = x⁴₁− 4tx²₁x₂+ 8t²x₁x₃− 8t³x₄,

z₅ = x⁵₁− 5tx³₁x₂+ 15t²x₁²x₃− 30t³x₁x₄+ 30t⁴x₅.

(3.2)

The above polynomials we may find also in the mentioned Weitzenb¨ock’s paper [22] p.242.

It is not difficult tu see that they are algebraically independent over k(t, x1) and hence, they are algebraically independent over k. It follows from Proposition 1.1 that

k[t, z2, . . . , zn−1] ⊆ R^d_nⁿ ⊆ k[t, z2, . . . , zn−1][1/t] (3.3) and

k[t, z2, . . . , zn−1][1/t] ∩ Rn= R_n^dn.

Therefore the k-algebra k[t, z₂, . . . , z_n−1] is an element of the family M defined in Section 1.

The derivation d_nis homogeneous in the usual sense. If F is a homogeneous polynomial from R_n of degree m then d_n(F ) is homogeneous of the same degree m.

There exists also an another homogeneity of dn. Consider the finite sequence α = (0, 1, 2, . . . , n − 1). We say that a polynomial F ∈ R_n is an α-form of degree s if every monomial of F is of the form axⁱ₀⁰xⁱ₁¹. . . , xⁱ_n−1ⁿ⁻¹, with a ∈ k, where

0i0+ 1i1+ 2i2+ · · · + (n − 1)in−1 = s.

It is easy to see that if F ∈ R_nis an α-form of degree s then d_n(F ) is an α-form of degree s − 1. Thus we get

Proposition 3.1. The ring R^d_nⁿ is generated over k by homogeneous polynomials which are α-forms. 

Note that every polynomial z_i is homogeneous of degree i and it is also an α-form of degree i.

4 Three variables

Proposition 4.1. R^d₃³ = k[t, z₂].

Proof. Let B = k[t, z₂]. By (3.3) we know that B ∈ M. Consider the ideal I = (t − y₁, z₂− y₂, t) = (t, y₁, x²₁− y₂)

of the polynomial ring k[t, x₁, x₂, y₁, y₂]. Then

GB(I) = {t, y1, x²₁− y2} and GB(I) ∩ k[y1, y2] = {y1}.

Hence B = B and R^d₃³ = B, by Theorem 1.3. 

5 Four variables

Proposition 5.1. R^d₄⁴ = k[t, z₂, z₃, u],

where u = x²₁x²₂− 2x³₁x₃+ 6tx₁x₂x₃−⁸₃tx³₂− 3t²x²₃.

Proof. Let B = k[t, z₂, z₃, u]. Observe that 3t²u = z₂³ − z₃². Thus d₄(u) = 0 and we see, by (3.3), that B is a finitely generated algebra from the family M.

Now consider the ideal

I = (t − y₁, z₂− y₂, z₃− y₃, u − y₄, t)

= (t, y₁, x²₁− y₂, x³₁− y₃, −2x³₁x₃+ x²₁x²₂− y₄) of the polynomial ring A = k[t, x₁, x₂, x₃, y₁, y₂, y₃, y₄].

Let us fix in A the lexicographic ordering x₁ > x₂ > x₃ > t > y₂ > y₃ > y₄ > y₁ and let GB(I) be the reduced Gr¨obner basis of I with respect to this ordering. Then we have

GB(I) = {t, y₁, x²₁− y₂, x₁y₂− y₃, x₁y₃− y₂², y³₂− y₃², x₂²y₂− 2x₃y₃− y₄, x₁y₄− x²₂y₃+ 2x₃y²₂, x₂²y₃²− 2x₃y₂²y₃− y₂²y₄ }

and thus

GB(I) ∩ k[y₁, y₂, y₃, y₄] = {y₁, y₂³− y₃²}.

Now it is clear that B = B. Therefore, by Theorem 1.3, R^d₄⁴ = B. 

6 Five variables

In view of Proposition 3.1 we know that all generators of the ring R^d₅⁵ are homogeneous in the usual sense and also they are homogeneous with respect to the direction α = (0, 1, 2, 3, 4). It is not difficult to find such homogeneous polynomials which are of small degrees. Here are four of them:

Table 6.1.

z₂ x²₁− 2tx₂ 2 2

p₂ 2x₁x₃− x²₂− 2tx₄ 2 4 z₃ x³₁− 3tx₁x₂ + 3t²x₃ 3 3 p₃ 6x²₁x₄− 6x₁x₂x₃+ 2x³₂− 12tx₂x₄+ 9tx²₃ 3 6

In the first column we write the name of a polynomial and in the last two columns its degree and α-degree respectively.

We will show that {t, z₂, z₃, p₂, p₃} is a set of generators of the k-algebra R^d₅⁵. For this aim let us denote:

B = k[t, z₂, z₃, p₂, p₃].

Observe at first that the polynomial z₄ (see (3.2)) belongs to B. In fact, z₄ = z₂²+ 4t²p₂.

So we have:

B ⊆ R^d₅⁵ ⊆ B[1/t], that is, B is an element of M.

Note that the polynomial u from Proposition 5.1 also belongs to B (since u = −z2p2−

1 3tp₃).

Theorem 6.2. R^d₅⁵ = B = k[t, z2, z3, p2, p3].

Proof. Consider the polynomial ring

A = k[t, x₁, x₂, x₃, x₄, y₁, y₂, y₃, y₄, y₅] and its ideal

I = (t − y₁, z₂− y₂, p₂− y₃, z₃ − y₄, p₃− y₅, t)

= (t, y₁, x²₁− y₂, 2x₁x₃− x²₂− y₃, x³₁− y₄, 6x²₁x₄− 6x₁x₂x₃+ 2x³₂− y₅).

Using the computer program CoCoA [1] we compute that the elements

b₁, . . . , b₂₇ below form the set GB(I), the reduced Gr¨obner basis of I (with respect to the lexicographic ordering x₁ > · · · > x₄ > t > y₂ > · · · > y₅ > y₁).

b₁ = y₁, b₂ = x²₁− y₂,

b₃ = x₁x₃− 1/2x²₂− 1/2y₃, b₄ = x₁y₂− y₄,

b₅ = x³₂+ 3x₂y₃− 6x₄y₂+ 2y₅, b₆ = t,

b₇ = x²₂y₂− 2x₃y₄+ y₂y₃, b₈ = x₁y₄− y₂²,

b₉ = x₁x²₂+ x₁y₃− 2x₃y₂, b₁₀ = y³₂− y₄²,

b₁₁ = x²₂y₄− 2x₃y₂²+ y₃y₄,

b₁₂ = x₂x₃y₄+ x₂y₂y₃− 3x₄y₂²+ y₂y₅, b₁₃ = x²₂y₃− 6x₂x₄y₂+ 2x₂y₅+ 4x²₃y₂− y₃², b₁₄ = x₁x₂y₃+ x₁y₅+ x₂x₃y₂− 3x₄y₄,

b₁₅ = x₂x₄y₂y₄ − 1/3x₂y₄y₅− 2/3x₃²y₂y₄− 1/3x₃y₂²y₃+ 1/3y²₃y₄, b₁₆ = x₂x₄y²₂ − 1/3x₂y₂y₅− 2/3x²₃y₂²− 1/3x₃y₃y₄ + 1/3y₂y²₃, b₁₇ = x₂x₃y²₂ + x₂y₃y₄− 3x₄y₂y₄+ y₄y₅,

b₁₈ = x²₂y₅+ 2x₂x²₃y₂− 2x₂y₃²− 6x₃x₄y₄+ 6x₄y₂y₃− y₃y₅, b₁₉ = x₁x₂y₅− x₁y₃²− 3x₂x₄y₄+ 2x²₃y₄+ x₃y₂y₃,

b₂₀ = x₂x₄y₄² − 1/3x₂y₂²y₅− 2/3x₃²y²₄− 1/3x₃y₂y₃y₄+ 1/3y₂²y²₃,

b₂₁ = x₂x²₃y₂y₃ + 3x₂x₄y₂y₅− x₂y₃³− x₂y₅²− 2x²₃y₂y₅− 3x₃x₄y₃y₄+ 3x₄y₂y₃², b₂₂ = x₁y³₃+ x₁y²₅− 2x³₃y²₂ − 3x²₃y₃y₄+ 9x²₄y₂y₄− 6x₄y₄y₅,

b₂₃ = x³₃y₄+ 3/2x²₃y₂y₃− 9/2x₄²y₂²+ 3x₄y₂y₅− 1/2y₃³− 1/2y₅²,

b₂₄ = x₂x₃y₂y₅ + 3x₂x₄y₃y₄+ 2x³₃y²₂+ x₃²y₃y₄− x₃y₂y₃²− 9x²₄y₂y₄+ 3x₄y₄y₅,

b₂₅ = x₂x₄y₂y₃³+ x₂x₄y₂y₅²− 1/3x₂y₃³y₅− 1/3x₂y₅³− 2/3x⁴₃y²₂y₃+ 5/6x²₃y₂y₃³

−2/3x²₃y₂y₅²+ 3x₃x²₄y₂y₃y₄− 2x₃x₄y₃y₄y₅ − 9/2x²₄y₂²y²₃+ 3x₄y₂y₃²y₅

−1/6y⁵₃− 1/6y₃²y²₅,

b₂₆ = x₂x₃y³₃y₅²+ x₂x₃y₅⁴+ 9x₂x²₄y⁴₃y₄+ 9x₂x²₄y₃y₄y²₅+ 2x⁵₃y₂²y₃y₅

+6x³₃x₄y₂²y₃³+ 6x₃³x₄y₂²y₅²− 5/2x³₃y₂y³₃y₅+ 2x³₃y₂y₅³− 9x²₃x²₄y₂y₃y₄y₅

+3x²₃x₄y₃⁴y₄+ 9x²₃x₄y₃y₄y²₅ + 27/2x₃x²₄y₂²y₃²y₅− 3x₃x₄y₂y₃⁵ − 12x₃x₄y₂y₃²y²₅ +1/2x₃y₃⁵y₅+ 1/2x₃y₃²y³₅ − 27x³₄y₂y₃³y₄− 27x₄³y₂y₄y₅²+ 9x²₄y₃³y₄y₅+ 9x²₄y₄y³₅, b₂₇ = x₂x²₃y³₃y₅+ x₂x²₃y₅³− 3x₂x₄y₃⁵− 3x₂x₄y²₃y₅²+ 2x⁶₃y²₂y₃+ 6x₃⁴x₄y₂²y₅− 5/2x⁴₃y₂y₃³

+2x⁴₃y₂y²₅+ 27x²₃x²₄y₂²y²₃− 63/2x²₃x₄y₂y²₃y₅+ 1/2x²₃y⁵₃+ 1/2x²₃y₃²y₅²

−27x₃x³₄y₂y₄y₅− 9x₃x²₄y₃³y₄+ 9x₃x²₄y₄y₅²− 81/2x⁴₄y₃y₄²+ 189/2x³₄y₂²y₃y₅ +9/2x²₄y₂y₃⁴− 81/2x²₄y₂y₃y₅²+ 9/2x₄y₃⁴y₅+ 9/2x₄y₃y₅³.

Looking at the above elements we see that only two from them, namely b₁ and b₁₀, belong to k[y₁, . . . , y₅], i. e.

GB(I) ∩ k[y₁, . . . , y₅] = {b₁, b₁₀}.

For us only b₁₀(y₂, y₄) = y₂³− y²₄ is an interesting element. Set v = b₁₀(z₂, z₃). Then v = z₂³− z₃² = −6t²w,

where

w = x³₁x₃− 1

2x²₁x²₂− 3tx₁x₂x₃+4

3tx³₂+ 3 2t²x²₃. Observe that

w = z₂p₂+1 3tp₃.

This means that w ∈ B. So B = B and therefore, by Theorem 1.3, R^d₅⁵ = B = k[t, z₂, z₃, p₂, p₃]. 

7 Six variables

The case six variables seems to be more complicated. The ring R^d₆⁶ contains all the polynomials from the following table.

Table 7.1.

z₂ x²₁− 2tx₂ 2 2

p₂ −x²₂+ 2x₁x₃− 2tx₄ 2 4

z₃ x³₁− 3tx₁x₂ + 3t²x₃ 3 3

q₃ x₁x²₂− 2x²₁x₃− tx₂x₃+ 5tx₁x₄− 5t²x₅ 3 5 p₃ 2x³₂− 6x₁x₂x₃+ 6x²₁x₄+ 9tx²₃− 12tx₂x₄ 3 6 f₆ x₁x³₂− 3x²₁x₂x₃+ 3x³₁x₄− tx²₂x₃+ 6tx₁x²₃− 4tx₁x₂x₄− 5tx²₁x₅− 6t²x₃x₄

+10t²x₂x₅ 4 7

f₇ −4x⁴₂+ 16x₁x₂²x₃− 21x²₁x²₃+ 10x₁²x₂x₄ − 10x³₁x₅+ 12tx₂x²₃− 40tx²₂x₄

+30tx₁x₃x₄+ 30tx₁x₂x₅− 30t²x₃x₅ 4 8 f₈ 8x²₂x²₃− 18x₁x³₃− 16x³₂x₄+ 38x₁x₂x₃x₄− 9x²₁x²₄+ 10x₁x²₂x₅− 20x²₁x₃x₅

+18tx²₃x₄− 32tx₂x²₄− 10tx₂x₃x₅+ 50tx₁x₄x₅− 25t²x²₅ 4 10 f₉ x₁x⁴₂− 4x²₁x²₂x₃+ 3/2x³₁x²₃+ 5x³₁x₂x₄− 5x⁴₁x₅− tx³₂x₃+ 21/2tx₁x₂x²₃

−10tx₁x²₂x₄− 15tx²₁x₃x₄+ 20tx²₁x₂x₅− 27/2t²x³₃+ 30t²x₂x₃x₄ − 20t²x²₂x₅ 5 9

f₁₀ −1/2x₁x²₂x²₃+ 3x²₁x³₃+ x₁x³₂x₄− 8x²₁x₂x₃x₄+ 9x³₁x²₄+ 5x²₁x²₂x₅

−10x³₁x₃x₅− 9/2tx₂x³₃+ 14tx²₂x₃x₄− 3/2tx₁x²₃x₄− 22tx₁x₂x²₄− 15tx³₂x₅

+35tx₁x₂x₃x₅− 5tx²₁x₄x₅+ 12t²x₃x²₄− 45/2t²x²₃x₅+ 10t²x₂x₄x₅ 5 11 f₁₁ 5x³₂x²₃− 15x₁x₂x³₃− 10x⁴₂x₄+ 30x₁x²₂x₃x₄+ 15x²₁x₃²x₄− 30x²₁x₂x²₄

+10x₁x³₂x₅ − 30x₁²x₂x₃x₅+ 30x³₁x₄x₅+ 81/4tx⁴₃− 66tx₂x²₃x₄+ 56tx²₂x²₄

−12tx₁x₃x²₄− 10tx²₂x₃x₅+ 60tx₁x²₃x₅− 40tx₁x₂x₄x₅− 25tx²₁x²₅+ 24t²x³₄

−60t²x₃x₄x₅+ 50t²x₂x²₅ 5 12

f₁₂ −5/2x₁x³₂x²₃+ 15/2x²₁x₂x³₃+ 5x₁x⁴₂x₄− 15x²₁x²₂x₃x₄− 15/2x³₁x²₃x₄ +15x³₁x₂x²₄ − 5x²₁x³₂x₅+ 15x³₁x₂x₃x₅− 15x⁴₁x₄x₅+ 15/2tx²₂x³₃

−27tx₁x⁴₃− 20tx³₂x₃x₄+ 78tx₁x₂x₃²x₄− 8tx₁x²₂x²₄− 39tx²₁x₃x²₄ +15tx⁴₂x₅− 50tx₁x₂²x₃x₅+ 45/2tx²₁x²₃x₅+ 10tx²₁x₂x₄x₅+ 25tx³₁x²₅ +27t²x³₃x₄− 72t²x₂x₃x²₄+ 60t²x₁x³₄− 15t²x₂x²₃x₅+ 80t²x²₂x₄x₅

−30t²x₁x₃x₄x₅− 75t²x₁x₂x²₅ − 60t³x²₄x₅ + 75t³x₃x²₅ 6 13 f₁₃ 2x⁷₂− 14x₁x⁵₂x₃+ 27x²₁x³₂x²₃− 9x₁³x₂x³₃+ 16x²₁x⁴₂x₄− 54x³₁x²₂x₃x₄

+9x⁴₁x²₃x₄+ 30x₁⁴x₂x²₄− 10x³₁x³₂x₅+ 30x⁴₁x₂x₃x₅− 30x⁵₁x₄x₅ +19tx⁴₂x²₃− 66tx₁x²₂x³₃+ 63/4tx²₁x⁴₃− 24tx⁵₂x₄+ 68tx₁x³₂x₃x₄

+96tx²₁x₂x²₃x₄− 92tx²₁x²₂x²₄− 36tx³₁x₃x²₄+ 20tx₁x⁴₂x₅− 50tx²₁x²₂x₃x₅

−60tx³₁x²₃x₅+ 100tx₁³x₂x₄x₅+ 25tx⁴₁x²₅+ 81/2t²x₂x⁴₃− 96t²x²₂x²₃x₄

−72t²x₁x³₃x₄+ 72t²x³₂x²₄+ 48t²x₁x₂x₃x²₄− 20t²x³₂x₃x₅+ 120t²x₁x₂x²₃x₅

−80t²x₁x²₂x₄x₅+ 60t²x²₁x₃x₄x₅− 100t²x²₁x₂x²₅+ 36t³x²₃x²₄− 120t³x₂x₃x₄x₅

+100t³x²₂x²₅ 6 14

In the first column we write the name of a polynomial and in the last two columns its degree and α-degree respectively, where α = (0, 1, 2, 3, 4, 5).

Let B be the k-subalgebra of R₆ generated by all the polynomials from Table 7.1. The polynomial z₅ (see (3.2)) belongs to B. In fact,

z₅ = z₂z₃− 6t²q₃. So we have:

B ⊆ R^d₆⁶ ⊆ B[1/t], that is B is an element of M. Is R^d₆⁶ equal to B ?

8 Weitzenb¨ ock derivations with 2 × 2 cells

Throughout this section k is a field of characteristic zero.

Let S_n= k[x₁, y₁, x₂, y₂, . . . , x_n, y_n] be the polynomial ring over k. Let us consider the k-derivation δ_n of S_n defined as

δ_n= y₁ ∂

∂x₁ + · · · + y_n ∂

∂x_n, (8.1)

that is, δ_n(y₁) = 0, δ_n(x₁) = y₁, . . . , δ_n(y_n) = 0, δ_n(x_n) = y_n. 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 S_n^δn for n 6 4.

The derivation δ_nis an A-derivation of S_nsuch that δ_n(x_i) ∈ 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, . . . , x_n] such that d(x_i) ∈ A. Of course such a derivation is locally nilpotent. If A is a field the following proposition describes the ring of constants of d.

Proposition 8.1. Let A be a field containing k and let d be a nonzero A-derivation of A[x₁, . . . , x_n] such that d(x₁) = a₁, . . . , d(x_n) = a_n, where a₁, . . . , a_n ∈ A and a₁ 6= 0.

Then

A[x₁, . . . , x_n]^d= A[u₂, . . . , u_n] where u_p = a_px₁− a₁x_p, for p = 2, . . . , n.

Proof. Put b = a⁻¹₁ x₁. Then d(b) = 1 and our proposition follows from the facts mentioned in Section 1. 

Consider the following

Question 8.2. 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 8.3. 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].

Proof. Put z = bx−ay. It is clear that A[z] ⊆ A[x, y]^d(since d(z)=0). Assume that F is an element of A[x, y] with d(F ) = 0. We must show that F ∈ A[z]. As d is homogeneous, one may assume that F is homogeneous of degree s > 1. By Proposition 8.1 we know that F ∈ A_o[z], where A_o is the field of fractions of A. So, by homogeneity, F = ^p_qz^s for some coprime p, q ∈ A. Then qF = pz^s and hence q | z = bx − ay. But a and b are coprime, so q is invertible in A and we have F = pq⁻¹z^s ∈ A[z]. 

The following example shows that Question 8.2 has a negative answer in general.

Example 8.4. 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 δ_n defined by (8.1). It is clear that S₁^δ1 = k[y₁] and, as a consequence of Proposition 8.3, we get

Proposition 8.5. S₂^δ2 = k[y1, y2, x1y2− x2y1]. 

The variables y₁ and x₁ play the same role as elements h and c from Section 1, i. e., δ_n(x₁) = y₁ 6= 0, δ_n²(x₁) = 0. Thus, by (1.1), ˜y_i = y_i for i = 1, . . . , n and moreover,

˜

x₁ = 0, and ˜x_i = y₁x_i− y_ix₁ for i = 2, . . . , n. If i < j, then we denote u_ij = y_ix_j − y_jx_i.

In particular, ˜x₂ = u₁₂, . . . , ˜x_n= u_1n. So, by Proposition 1.1, we get