Locally nilpotent derivation

In mathematics, a derivation [math]\displaystyle{ \partial }[/math] of a commutative ring [math]\displaystyle{ A }[/math] is called a locally nilpotent derivation (LND) if every element of [math]\displaystyle{ A }[/math] is annihilated by some power of [math]\displaystyle{ \partial }[/math].

One motivation for the study of locally nilpotent derivations comes from the fact that some of the counterexamples to Hilbert's 14th problem are obtained as the kernels of a derivation on a polynomial ring.

Over a field [math]\displaystyle{ k }[/math] of characteristic zero, to give a locally nilpotent derivation on the integral domain [math]\displaystyle{ A }[/math], finitely generated over the field, is equivalent to giving an action of the additive group [math]\displaystyle{ (k,+) }[/math] to the affine variety [math]\displaystyle{ X = \operatorname{Spec}(A) }[/math]. Roughly speaking, an affine variety admitting "plenty" of actions of the additive group is considered similar to an affine space.

Definition

Let [math]\displaystyle{ A }[/math] be a ring. Recall that a derivation of [math]\displaystyle{ A }[/math] is a map [math]\displaystyle{ \partial\colon\, A\to A }[/math] satisfying the Leibniz rule [math]\displaystyle{ \partial (ab)=(\partial a)b+a(\partial b) }[/math] for any [math]\displaystyle{ a,b\in A }[/math]. If [math]\displaystyle{ A }[/math] is an algebra over a field [math]\displaystyle{ k }[/math], we additionally require [math]\displaystyle{ \partial }[/math] to be [math]\displaystyle{ k }[/math]-linear, so [math]\displaystyle{ k\subseteq \ker \partial }[/math].

A derivation [math]\displaystyle{ \partial }[/math] is called a locally nilpotent derivation (LND) if for every [math]\displaystyle{ a \in A }[/math], there exists a positive integer [math]\displaystyle{ n }[/math] such that [math]\displaystyle{ \partial^{n}(a)=0 }[/math].

If [math]\displaystyle{ A }[/math] is graded, we say that a locally nilpotent derivation [math]\displaystyle{ \partial }[/math] is homogeneous (of degree [math]\displaystyle{ d }[/math]) if [math]\displaystyle{ \deg \partial a=\deg a +d }[/math] for every [math]\displaystyle{ a\in A }[/math].

The set of locally nilpotent derivations of a ring [math]\displaystyle{ A }[/math] is denoted by [math]\displaystyle{ \operatorname{LND}(A) }[/math]. Note that this set has no obvious structure: it is neither closed under addition (e.g. if [math]\displaystyle{ \partial_{1}=y\tfrac{\partial}{\partial x} }[/math], [math]\displaystyle{ \partial_{2}=x\tfrac{\partial}{\partial y} }[/math] then [math]\displaystyle{ \partial_{1},\partial_{2}\in \operatorname{LND}(k[x,y]) }[/math] but [math]\displaystyle{ (\partial_{1}+\partial_{2})^{2}(x)=x }[/math], so [math]\displaystyle{ \partial_{1}+\partial_{2}\not\in \operatorname{LND}(k[x,y]) }[/math]) nor under multiplication by elements of [math]\displaystyle{ A }[/math] (e.g. [math]\displaystyle{ \tfrac{\partial}{\partial x}\in \operatorname{LND}(k[x]) }[/math], but [math]\displaystyle{ x\tfrac{\partial}{\partial x}\not\in\operatorname{LND}(k[x]) }[/math]). However, if [math]\displaystyle{ [\partial_{1},\partial_{2}]=0 }[/math] then [math]\displaystyle{ \partial_{1},\partial_{2}\in \operatorname{LND}(A) }[/math] implies [math]\displaystyle{ \partial_{1}+\partial_{2}\in \operatorname{LND}(A) }[/math] and if [math]\displaystyle{ \partial\in \operatorname{LND}(A) }[/math], [math]\displaystyle{ h\in\ker\partial }[/math] then [math]\displaystyle{ h\partial\in \operatorname{LND}(A) }[/math].

Relation to G_a-actions

Let [math]\displaystyle{ A }[/math] be an algebra over a field [math]\displaystyle{ k }[/math] of characteristic zero (e.g. [math]\displaystyle{ k=\mathbb{C} }[/math]). Then there is a one-to-one correspondence between the locally nilpotent [math]\displaystyle{ k }[/math]-derivations on [math]\displaystyle{ A }[/math] and the actions of the additive group [math]\displaystyle{ \mathbb{G}_{a} }[/math] of [math]\displaystyle{ k }[/math] on the affine variety [math]\displaystyle{ \operatorname{Spec} A }[/math], as follows. A [math]\displaystyle{ \mathbb{G}_{a} }[/math]-action on [math]\displaystyle{ \operatorname{Spec} A }[/math] corresponds to a [math]\displaystyle{ k }[/math]-algebra homomorphism [math]\displaystyle{ \rho\colon A\to A[t] }[/math]. Any such [math]\displaystyle{ \rho }[/math] determines a locally nilpotent derivation [math]\displaystyle{ \partial }[/math] of [math]\displaystyle{ A }[/math] by taking its derivative at zero, namely [math]\displaystyle{ \partial=\epsilon \circ \tfrac{d}{dt}\circ \rho, }[/math] where [math]\displaystyle{ \epsilon }[/math] denotes the evaluation at [math]\displaystyle{ t=0 }[/math]. Conversely, any locally nilpotent derivation [math]\displaystyle{ \partial }[/math] determines a homomorphism [math]\displaystyle{ \rho\colon A\to A[t] }[/math] by [math]\displaystyle{ \rho = \exp (t\partial)=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\partial^{n}. }[/math]

It is easy to see that the conjugate actions correspond to conjugate derivations, i.e. if [math]\displaystyle{ \alpha\in \operatorname{Aut} A }[/math] and [math]\displaystyle{ \partial\in \operatorname{LND}(A) }[/math] then [math]\displaystyle{ \alpha\circ\partial \circ \alpha^{-1}\in \operatorname{LND}(A) }[/math] and [math]\displaystyle{ \exp(t\cdot \alpha\circ\partial \circ \alpha^{-1})=\alpha \circ \exp(t\partial)\circ \alpha^{-1} }[/math]

The kernel algorithm

The algebra [math]\displaystyle{ \ker \partial }[/math] consists of the invariants of the corresponding [math]\displaystyle{ \mathbb{G}_{a} }[/math]-action. It is algebraically and factorially closed in [math]\displaystyle{ A }[/math]. A special case of Hilbert's 14th problem asks whether [math]\displaystyle{ \ker \partial }[/math] is finitely generated, or, if [math]\displaystyle{ A=k[X] }[/math], whether the quotient [math]\displaystyle{ X/\!/\mathbb{G}_{a} }[/math] is affine. By Zariski's finiteness theorem, it is true if [math]\displaystyle{ \dim X\leq 3 }[/math]. On the other hand, this question is highly nontrivial even for [math]\displaystyle{ X=\mathbb{C}^{n} }[/math], [math]\displaystyle{ n\geq 4 }[/math]. For [math]\displaystyle{ n\geq 5 }[/math] the answer, in general, is negative. The case [math]\displaystyle{ n=4 }[/math] is open.

However, in practice it often happens that [math]\displaystyle{ \ker\partial }[/math] is known to be finitely generated: notably, by the Maurer–Weitzenböck theorem, it is the case for linear LND's of the polynomial algebra over a field of characteristic zero (by linear we mean homogeneous of degree zero with respect to the standard grading).

Assume [math]\displaystyle{ \ker \partial }[/math] is finitely generated. If [math]\displaystyle{ A=k[g_1,\dots, g_n] }[/math] is a finitely generated algebra over a field of characteristic zero, then [math]\displaystyle{ \ker\partial }[/math] can be computed using van den Essen's algorithm, as follows. Choose a local slice, i.e. an element [math]\displaystyle{ r\in \ker \partial^{2}\setminus \ker \partial }[/math] and put [math]\displaystyle{ f=\partial r\in \ker\partial }[/math]. Let [math]\displaystyle{ \pi_{r}\colon\, A\to (\ker \partial)_{f} }[/math] be the Dixmier map given by [math]\displaystyle{ \pi_{r}(a)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\partial^{n}(a)\frac{r^{n}}{f^{n}} }[/math]. Now for every [math]\displaystyle{ i=1,\dots, n }[/math], chose a minimal integer [math]\displaystyle{ m_{i} }[/math] such that [math]\displaystyle{ h_{i}\colon = f^{m_{i}}\pi_{r}(g_{i})\in \ker\partial }[/math], put [math]\displaystyle{ B_{0}=k[h_{1},\dots, h_{n},f]\subseteq \ker \partial }[/math], and define inductively [math]\displaystyle{ B_{i} }[/math] to be the subring of [math]\displaystyle{ A }[/math] generated by [math]\displaystyle{ \{h\in A: fh\in B_{i-1}\} }[/math]. By induction, one proves that [math]\displaystyle{ B_{0}\subset B_{1}\subset \dots \subset\ker \partial }[/math] are finitely generated and if [math]\displaystyle{ B_{i}=B_{i+1} }[/math] then [math]\displaystyle{ B_{i}=\ker \partial }[/math], so [math]\displaystyle{ B_{N}=\ker \partial }[/math] for some [math]\displaystyle{ N }[/math]. Finding the generators of each [math]\displaystyle{ B_{i} }[/math] and checking whether [math]\displaystyle{ B_{i}=B_{i+1} }[/math] is a standard computation using Gröbner bases.

Slice theorem

Assume that [math]\displaystyle{ \partial\in\operatorname{LND}(A) }[/math] admits a slice, i.e. [math]\displaystyle{ s\in A }[/math] such that [math]\displaystyle{ \partial s=1 }[/math]. The slice theorem asserts that [math]\displaystyle{ A }[/math] is a polynomial algebra [math]\displaystyle{ (\ker\partial) [s] }[/math] and [math]\displaystyle{ \partial=\tfrac{d}{ds} }[/math].

For any local slice [math]\displaystyle{ r\in\ker\partial \setminus \ker\partial^{2} }[/math] we can apply the slice theorem to the localization [math]\displaystyle{ A_{\partial r} }[/math], and thus obtain that [math]\displaystyle{ A }[/math] is locally a polynomial algebra with a standard derivation. In geometric terms, if a geometric quotient [math]\displaystyle{ \pi\colon\,X\to X//\mathbb{G}_{a} }[/math] is affine (e.g. when [math]\displaystyle{ \dim X\leq 3 }[/math] by the Zariski theorem), then it has a Zariski-open subset [math]\displaystyle{ U }[/math] such that [math]\displaystyle{ \pi^{-1}(U) }[/math] is isomorphic over [math]\displaystyle{ U }[/math] to [math]\displaystyle{ U\times \mathbb{A}^{1} }[/math], where [math]\displaystyle{ \mathbb{G}_{a} }[/math] acts by translation on the second factor.

However, in general it is not true that [math]\displaystyle{ X\to X//\mathbb{G}_{a} }[/math] is locally trivial. For example, let [math]\displaystyle{ \partial=u\tfrac{\partial}{\partial x}+v\tfrac{\partial}{\partial y}+(1+uy^2)\tfrac{\partial}{\partial z}\in \operatorname{LND}(\mathbb{C}[x,y,z,u,v]) }[/math]. Then [math]\displaystyle{ \ker\partial }[/math] is a coordinate ring of a singular variety, and the fibers of the quotient map over singular points are two-dimensional.

If [math]\displaystyle{ \dim X=3 }[/math] then [math]\displaystyle{ \Gamma=X\setminus U }[/math] is a curve. To describe the [math]\displaystyle{ \mathbb{G}_{a} }[/math]-action, it is important to understand the geometry [math]\displaystyle{ \Gamma }[/math]. Assume further that [math]\displaystyle{ k=\mathbb{C} }[/math] and that [math]\displaystyle{ X }[/math] is smooth and contractible (in which case [math]\displaystyle{ S }[/math] is smooth and contractible as well) and choose [math]\displaystyle{ \Gamma }[/math] to be minimal (with respect to inclusion). Then Kaliman proved that each irreducible component of [math]\displaystyle{ \Gamma }[/math] is a polynomial curve, i.e. its normalization is isomorphic to [math]\displaystyle{ \mathbb{C}^{1} }[/math]. The curve [math]\displaystyle{ \Gamma }[/math] for the action given by Freudenburg's (2,5)-derivation (see below) is a union of two lines in [math]\displaystyle{ \mathbb{C}^{2} }[/math], so [math]\displaystyle{ \Gamma }[/math] may not be irreducible. However, it is conjectured that [math]\displaystyle{ \Gamma }[/math] is always contractible.

Examples

Example 1

The standard coordinate derivations [math]\displaystyle{ \tfrac{\partial}{\partial x_i} }[/math] of a polynomial algebra [math]\displaystyle{ k[x_1,\dots, x_n] }[/math] are locally nilpotent. The corresponding [math]\displaystyle{ \mathbb{G}_a }[/math]-actions are translations: [math]\displaystyle{ t\cdot x_{i}=x_{i}+t }[/math], [math]\displaystyle{ t\cdot x_{j}=x_{j} }[/math] for [math]\displaystyle{ j\neq i }[/math].

Example 2 (Freudenburg's (2,5)-homogeneous derivation)

Let [math]\displaystyle{ f_1=x_1x_3-x_2^2 }[/math], [math]\displaystyle{ f_2=x_3f_1^2+2x_1^2x_2f_1+x^5 }[/math], and let [math]\displaystyle{ \partial }[/math] be the Jacobian derivation [math]\displaystyle{ \partial(f_{3})=\det \left[\tfrac{\partial f_{i}}{\partial x_{j}}\right]_{i,j=1,2,3} }[/math]. Then [math]\displaystyle{ \partial\in \operatorname{LND}(k[x_1,x_2,x_3]) }[/math] and [math]\displaystyle{ \operatorname{rank}\partial=3 }[/math] (see below); that is, [math]\displaystyle{ \partial }[/math] annihilates no variable. The fixed point set of the corresponding [math]\displaystyle{ \mathbb{G}_{a} }[/math]-action equals [math]\displaystyle{ \{x_1=x_2=0\} }[/math].

Example 3

Consider [math]\displaystyle{ Sl_2(k)=\{ad-bc=1\}\subseteq k^{4} }[/math]. The locally nilpotent derivation [math]\displaystyle{ a\tfrac{\partial}{\partial b}+c\tfrac{\partial}{\partial d} }[/math] of its coordinate ring corresponds to a natural action of [math]\displaystyle{ \mathbb{G}_a }[/math] on [math]\displaystyle{ Sl_2(k) }[/math] via right multiplication of upper triangular matrices. This action gives a nontrivial [math]\displaystyle{ \mathbb{G}_a }[/math]-bundle over [math]\displaystyle{ \mathbb{A}^{2}\setminus \{(0,0)\} }[/math]. However, if [math]\displaystyle{ k=\mathbb{C} }[/math] then this bundle is trivial in the smooth category

LND's of the polynomial algebra

Let [math]\displaystyle{ k }[/math] be a field of characteristic zero (using Kambayashi's theorem one can reduce most results to the case [math]\displaystyle{ k=\mathbb{C} }[/math]) and let [math]\displaystyle{ A=k[x_1,\dots, x_n] }[/math] be a polynomial algebra.

n = 2 (G_a-actions on an affine plane)

Rentschler's theorem — Every LND of [math]\displaystyle{ k[x_1,x_2] }[/math] can be conjugated to [math]\displaystyle{ f(x_1)\tfrac{\partial}{\partial x_2} }[/math] for some [math]\displaystyle{ f(x_1)\in k[x_1] }[/math]. This result is closely related to the fact that every automorphism of an affine plane is tame, and does not hold in higher dimensions.

n = 3 (G_a-actions on an affine 3-space)

Miyanishi's theorem — The kernel of every nontrivial LND of [math]\displaystyle{ A=k[x_1,x_2,x_3] }[/math] is isomorphic to a polynomial ring in two variables; that is, a fixed point set of every nontrivial [math]\displaystyle{ \mathbb{G}_{a} }[/math]-action on [math]\displaystyle{ \mathbb{A}^{3} }[/math] is isomorphic to [math]\displaystyle{ \mathbb{A}^{2} }[/math].

In other words, for every [math]\displaystyle{ 0\neq \partial \in \operatorname{LND}(A) }[/math] there exist [math]\displaystyle{ f_{1},f_{2}\in A }[/math] such that [math]\displaystyle{ \ker\partial=k[f_{1},f_{2}] }[/math] (but, in contrast to the case [math]\displaystyle{ n=2 }[/math], [math]\displaystyle{ A }[/math] is not necessarily a polynomial ring over [math]\displaystyle{ \ker \partial }[/math]). In this case, [math]\displaystyle{ \partial }[/math] is a Jacobian derivation: [math]\displaystyle{ \partial(f_{3}) = \det\left[\tfrac{\partial f_{i}}{\partial x_{j}}\right]_{i,j=1,2,3} }[/math].

Zurkowski's theorem — Assume that [math]\displaystyle{ n=3 }[/math] and [math]\displaystyle{ \partial\in \operatorname{LND}(A) }[/math] is homogeneous relative to some positive grading of [math]\displaystyle{ A }[/math] such that [math]\displaystyle{ x_1,x_2,x_3 }[/math] are homogeneous. Then [math]\displaystyle{ \ker\partial=k[f,g] }[/math] for some homogeneous [math]\displaystyle{ f,g }[/math]. Moreover, if [math]\displaystyle{ \deg x_{1},\deg x_{2},\deg x_{3} }[/math] are relatively prime, then [math]\displaystyle{ \deg f,\deg g }[/math] are relatively prime as well.

Bonnet's theorem — A quotient morphism [math]\displaystyle{ \mathbb{A}^{3}\to \mathbb{A}^{2} }[/math] of a [math]\displaystyle{ \mathbb{G}_{a} }[/math]-action is surjective. In other words, for every [math]\displaystyle{ 0\neq \partial \in \operatorname{LND}(A) }[/math], the embedding [math]\displaystyle{ \ker\partial\subseteq A }[/math] induces a surjective morphism [math]\displaystyle{ \operatorname{Spec}A\to \operatorname{Spec}\ker\partial }[/math].

This is no longer true for [math]\displaystyle{ n\geqslant 4 }[/math], e.g. the image of a quotient map [math]\displaystyle{ \mathbb{A}^{4}\to\mathbb{A}^{3} }[/math] by a [math]\displaystyle{ \mathbb{G}_{a} }[/math]-action [math]\displaystyle{ t\cdot (x_1,x_2,x_3,x_4)=(x_1,x_2,x_3-tx_2,x_4+tx_1) }[/math] (which corresponds to a LND given by [math]\displaystyle{ x_1\tfrac{\partial}{\partial x_4}-x_2\tfrac{\partial}{\partial x_3}) }[/math] equals [math]\displaystyle{ \mathbb{A}^{3}\setminus \{(x_1,x_2,x_3): x_{1}=x_{2}=0,x_{3}\neq 0\} }[/math].

Kaliman's theorem — Every fixed-point free action of [math]\displaystyle{ \mathbb{G}_{a} }[/math] on [math]\displaystyle{ \mathbb{A}^{3} }[/math] is conjugate to a translation. In other words, every [math]\displaystyle{ \partial \in \operatorname{LND}(A) }[/math] such that the image of [math]\displaystyle{ \partial }[/math] generates the unit ideal (or, equivalently, [math]\displaystyle{ \partial }[/math] defines a nowhere vanishing vector field), admits a slice. This results answers one of the conjectures from Kraft's list.

Again, this result is not true for [math]\displaystyle{ n\geqslant 4 }[/math]: e.g. consider the [math]\displaystyle{ x_1\tfrac{\partial}{\partial x_2}+ x_2\tfrac{\partial}{\partial x_3}+(x_2^2-2x_1 x_3-1)\tfrac{\partial}{\partial x_{4}}\in \operatorname{LND}(\mathbb{C}[x_{1},x_{2},x_{3},x_{4}]) }[/math]. The points [math]\displaystyle{ (x_1,1,0,0) }[/math] and [math]\displaystyle{ (x_1,-1,0,0) }[/math] are in the same orbit of the corresponding [math]\displaystyle{ \mathbb{G}_a }[/math]-action if and only if [math]\displaystyle{ x_{1}\neq 0 }[/math]; hence the (topological) quotient is not even Hausdorff, let alone homeomorphic to [math]\displaystyle{ \mathbb{C}^{3} }[/math].

Principal ideal theorem — Let [math]\displaystyle{ \partial\in\operatorname{LND}(A) }[/math]. Then [math]\displaystyle{ A }[/math] is faithfully flat over [math]\displaystyle{ \ker\partial }[/math]. Moreover, the ideal [math]\displaystyle{ \ker \partial \cap \operatorname{im}\partial }[/math] is principal in [math]\displaystyle{ A }[/math].

Triangular derivations

Let [math]\displaystyle{ f_1,\dots,f_n }[/math] be any system of variables of [math]\displaystyle{ A }[/math]; that is, [math]\displaystyle{ A=k[f_1,\dots, f_n] }[/math]. A derivation of [math]\displaystyle{ A }[/math] is called triangular with respect to this system of variables, if [math]\displaystyle{ \partial f_1\in k }[/math] and [math]\displaystyle{ \partial f_{i} \in k[f_1,\dots,f_{i-1}] }[/math] for [math]\displaystyle{ i=2,\dots,n }[/math]. A derivation is called triangulable if it is conjugate to a triangular one, or, equivalently, if it is triangular with respect to some system of variables. Every triangular derivation is locally nilpotent. The converse is true for [math]\displaystyle{ \leq 2 }[/math] by Rentschler's theorem above, but it is not true for [math]\displaystyle{ n\geq 3 }[/math].

Bass's example

The derivation of [math]\displaystyle{ k[x_1,x_2,x_3] }[/math] given by [math]\displaystyle{ x_1\tfrac{\partial}{\partial x_2}+2x_2x_1\tfrac{\partial}{\partial x_3} }[/math] is not triangulable. Indeed, the fixed-point set of the corresponding [math]\displaystyle{ \mathbb{G}_{a} }[/math]-action is a quadric cone [math]\displaystyle{ x_2x_3=x_2^2 }[/math], while by the result of Popov, a fixed point set of a triangulable [math]\displaystyle{ \mathbb{G}_{a} }[/math]-action is isomorphic to [math]\displaystyle{ Z\times \mathbb{A}^{1} }[/math] for some affine variety [math]\displaystyle{ Z }[/math]; and thus cannot have an isolated singularity.

Freudenburg's theorem — The above necessary geometrical condition was later generalized by Freudenburg. To state his result, we need the following definition:

A corank of [math]\displaystyle{ \partial\in \operatorname{LND}(A) }[/math] is a maximal number [math]\displaystyle{ j }[/math] such that there exists a system of variables [math]\displaystyle{ f_1,\dots, f_n }[/math] such that [math]\displaystyle{ f_1,\dots, f_j\in\ker\partial }[/math]. Define [math]\displaystyle{ \operatorname{rank}\partial }[/math] as [math]\displaystyle{ n }[/math] minus the corank of [math]\displaystyle{ \partial }[/math].

We have [math]\displaystyle{ 1\leq \operatorname{rank}\partial \leq n }[/math] and [math]\displaystyle{ \operatorname{rank}(\partial)=1 }[/math] if and only if in some coordinates, [math]\displaystyle{ \partial=h\tfrac{\partial}{\partial x_{n}} }[/math] for some [math]\displaystyle{ h\in k[x_1,\dots,x_{n-1}] }[/math].

Theorem: If [math]\displaystyle{ \partial\in \operatorname{LND}(A) }[/math] is triangulable, then any hypersurface contained in the fixed-point set of the corresponding [math]\displaystyle{ \mathbb{G}_{a} }[/math]-action is isomorphic to [math]\displaystyle{ Z\times \mathbb{A}^{\operatorname{rank} \partial} }[/math].

In particular, LND's of maximal rank [math]\displaystyle{ n }[/math] cannot be triangulable. Such derivations do exist for [math]\displaystyle{ n\geq 3 }[/math]: the first example is the (2,5)-homogeneous derivation (see above), and it can be easily generalized to any [math]\displaystyle{ n\geq 3 }[/math].

Makar-Limanov invariant

The intersection of the kernels of all locally nilpotent derivations of the coordinate ring, or, equivalently, the ring of invariants of all [math]\displaystyle{ \mathbb{G}_{a} }[/math]-actions, is called "Makar-Limanov invariant" and is an important algebraic invariant of an affine variety. For example, it is trivial for an affine space; but for the Koras–Russell cubic threefold, which is diffeomorphic to [math]\displaystyle{ \mathbb{C}^{3} }[/math], it is not.

Further reading

• A Nowicki, the fourteenth problem of hilbert for polynomial derivations