Abstract
Let X be a polynomial vector field on \({\mathbb {C}}^{2}\) with at most isolated zeros and whose trajectories are all simply connected. Let us suppose that there is a polynomial \(P\in {\mathbb {C}}[x,y]\) such that (i) \(dP(X)=1\) or (ii) \(dP(X)=a\cdot P,\) with \(a\in {\mathbb {C}}^{*}\). In (Bustinduy and Giraldo, in Adv Math 285:1339–1357, 2015; Bustinduy and Giraldo, in J Differ Equ 264:3933–3939, 2018) the authors determined X and P, up to an algebraic change of coordinates, when \(P\in {\mathbb {C}}[x,y]\) is primitive. In this note, we extend these results for an arbitrary P. Finally, as an application, we show that if a polynomial vector field X on \({\mathbb {C}}^{2}\) with at most isolated zeros has all its trajectories simply connected and there exist \(P\in {\mathbb {C}}[x,y]\) and \(n\in {\mathbb {N}}^{+}\) such that \(X^{n}(P)= 0\) and \(X^{n-1}(P)\ne 0\) or \(X^{n+1}(P)= a\cdot X^{n}(P)\) with \(a\in {\mathbb {C}}^{*}\), X is complete and present some questions on the study of derivations whose image is a Mathieu subspace.
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1 Introduction
1.1 Vector fields and trajectories
A holomorphic vector field X on \({\mathbb {C}}^2\) is a section of the tangent bundle of \({\mathbb {C}}^2\). Set coordinates x, y in \({\mathbb {C}}^{2}\), hence
with \(P_{1}\), \(P_{2}\) holomorphic functions. A point which is a common zero of \(P_{1}\) and \(P_{2}\) is called a singular point of X. Take \(z=(x,y)\) in \({\mathbb {C}}^2\) and the differential equation \(\varphi '_{z}(t)=X(\varphi _z(t))\) with \(\varphi _z(0)=z.\) The local solution \(\varphi _z\) can be extended by analytic continuation along paths from \(t=0\) in \({\mathbb {C}}\) to a maximal connected Riemann surface \(\pi _{z}: \Omega _z \rightarrow {\mathbb {C}}\), which is a Riemann domain over \({\mathbb {C}}\). The solution of X through z is \({\varphi }_z:\Omega _z \rightarrow {\mathbb {C}}^{2}\). The (complex) trajectory \(C_{z}\) of X through z is the Riemann surface \(\varphi _{z}(\Omega _z)\) immersed in \({\mathbb {C}}^{2}\).
If \(\Omega _z={\mathbb {C}}\), as domain in \({\mathbb {C}}\) (then, \(\pi _{z}\) is an analytic isomorphism), X is said to be complete on \(C_{z}\). In this case, \(C_{z}\) is uniformized by \({\mathbb {C}}\), and then analytically isomorphic to (= of type) \({\mathbb {C}}\) or \({\mathbb {C}}^{*}\) (maximum principle).
We say that X is complete if it is complete on \(C_{z}\) for any z. In this case, the flow \(\varphi : {\mathbb {C}}\times {\mathbb {C}}^2 \rightarrow {\mathbb {C}}^2\) of X, \((t,z)\mapsto \varphi (t,z)=\varphi _{z}(t)\), defines a holomorphic action of \(({\mathbb {C}},+)\) on \({\mathbb {C}}^2\) by analytic automorphisms and \(X={\frac{\partial }{\partial t} \varphi (t,z)}_{\mid t=0}\).
There are different types of flows:
\(\varphi \) is algebraic, if \(\varphi \) is a polynomial map.
\(\varphi \) is quasi-algebraic, if \({\varphi }_{t}\), for any \(t\in {\mathbb {C}}\), is a polynomial automorphism.
\(\varphi \) is proper, if the topological closure \({\overline{C}}_{z}\) of \(C_{z}\) in \({\mathbb {C}}^2\), for any z, is an analytic curve.
1.2 Vector fields and simply-connected trajectories
Let X be a polynomial vector field (then \(P_{1},\)\(P_{2}\in {\mathbb {C}}[x,y]\)) with simply-connected trajectories. We addressed in [2, 3] the problem of deciding if X is complete or not, under the assumption that there existed a nonconstant primitive polynomial \(P\in {\mathbb {C}}[x,y]\) such that
- (i):
\(dP(X)=1\), or
- (ii):
\(dP(X)=a\cdot P\) with \(a\ne {0}\).
Concretely, in both cases we proved that \(P=x\) and
after a polynomial automorphism. In fact, it is obtained that \(a=0\) and \(d=1\) in case (i) [2], and \(a\ne 0\), \(d=0\), \(b(0)=0\) and \(c(0)\ne 0\) in case (ii) [3]. In particular, X is complete. There are several motivations to study these problems.
1. If X is a complete polynomial vector field on \({\mathbb {C}}^{2}\) with at most isolated zeros and simply connected trajectories (of type \({\mathbb {C}}\)), by classification of complete polynomial vector fields [1] X is as (1) after a polynomial change of coordinates. Moreover, after performing another polynomial change of coordinates, we can assume that \(d=0\), if \(a\ne {0}\); and \(d=1\), if \(a=0\). Note that these vector fields satisfy one of the following two properties with respect to \(P=x\): either \(dP(X)=ax\), if \(a\ne {0}\); or \(dP(X)=1\), if \(a=0\).
On the other hand, if X has no zeros and flow \(\varphi \), one also knows, according to [13, 16, Théorème 2] and [16, Théorème 4] (see also [2, Introduction]):
\(\text {Algebraic}\,\,\varphi \Rightarrow Quasi-algebraic \,\varphi \Rightarrow Proper \,\,\varphi \,\,\,and trajectories of type \,\, {\mathbb {C}}. \)
In these three situations for \(\varphi \), after a holomorphic automorphism, there is a polynomial \(P=x\) such that \(dP(X)=1\).
Then, it is natural to study if a reciprocal of Brunella’s result is valid:
If a polynomial vector fieldXon\({\mathbb {C}}^{2}\)with at most isolated zeros and simply-connected trajectories satisfies for a primitive polynomialPthat\(dP(X)=aP\), with\(a\ne {0}\), or\(dP(X)=1\), isXcomplete?
The affirmative answer to this question is given in [2] and [3], and it implies that such an X has no trajectories of type \({\mathbb {D}}\), and they are all of type \({\mathbb {C}}\).
Note that in case (i), the trajectories are always proper in \({\mathbb {C}}^{2}\) and X is the constant horizontal vector field after a holomorphic change of coordinates. However, in case (ii), the trajectories are not necessarily proper.
2. Let X be a polynomial vector field on \({\mathbb {C}}^{2}\), and the \({\mathbb {C}}\,\)-derivation \(D_{X}\) of \({\mathbb {C}}[x,y]\) associated to X:
A slices of \(D_{X}\) is a polynomial \(s\in {\mathbb {C}}[x,y]\) such that \(D_{X}(s)=1\). Questions about slices and derivations are related to Cancellation Problem in affine spaces [11, Chapter 10]. Moreover, the Jacobian Conjecture can be formulated as a problem in terms of derivations with a slice [11, Chapter 3]. Furthermore, this famous conjecture has been also formulated by Van de Essen, Wright and Zhao [17] in terms of derivations: it holds if the image of every derivation of \({\mathbb {C}}[x,y]\) with zero divergence and having a slice is a Mathieu subspace (we will recall this notion in the last section).
If \(D_{X}\) is surjective, \(1\in \) Im\((D_{X})\) and \(D_{X}\) has a slice. Surjective derivations in \({\mathbb {C}}[x,y]\) are studied in [5], and they are studied too in affine domains B over \({\mathbb {C}}\) with small dimension in [12]. An important property of a surjective derivation \(D_{X}\) of \({\mathbb {C}}[x,y]\) is that X has simply-connected trajectories [5, Proposition 1.6].
Motivated by these facts, we studied in [2] polynomial vector fields X on \({\mathbb {C}}^{2}\) with simply-connected trajectories such that \(D_{X}\) have a slice, and determined X, modulo a polynomial automorphism. Moreover, we applied this result to the study of surjective derivations. In particular we obtained in [2, Theorem 2] an affirmative answer to a conjecture stated by Cerveau in [6]: If \(D_{X}\)is surjective, then, up to a polynomial change of coordinates,
with\(b\in {\mathbb {C}}.\)
1.3 Main result
In what follows, we will assume that X has at most isolated zeros. We extend the above results to the case of a non-primitive polynomial P with the following theorem:
Theorem 1
LetXbe a polynomial vector field in\({\mathbb {C}}^{2}\). If there is\(P\in {\mathbb {C}}[x,y]\)such that (i) \(dP(X)=1\)or (ii) \(dP(X)= a\cdot P\), with\(a\in {\mathbb {C}}^{*}\); and the trajectories ofXare simply connected, up to a polynomial change of coordinates:
- (1.1)
In case (i), \(P=x\), and
$$\begin{aligned} X=\frac{\partial }{\partial x} + [b(x)y + c(x)] \frac{\partial }{\partial y}, \end{aligned}$$with\(b,\,c\in {\mathbb {C}}[x]\), and
- (1.2)
In case (ii), \(P=x^{n}\), with\(n\in {\mathbb {N}}^{+}\), and
$$\begin{aligned} X=dx\frac{\partial }{\partial x} + [b(x)y + c(x)] \frac{\partial }{\partial y}, \end{aligned}$$with\(d=a/n\), \(b,\,c\in {\mathbb {C}}[x]\), \(b(0)=0\)and\(c(0)\ne {0}\).
In particular,Xis complete and has all its trajectories of type\({\mathbb {C}}\).
2 Proof of Theorem 1
Note that in case (i): \(dP(X)=1\), P is always primitive. Theorem 1, after [2] and [3], follows by this proposition:
Proposition 1
LetXbe a polynomial vector field in\({\mathbb {C}}^{2}\). If there is a non-primitive\(P\in {\mathbb {C}}[x,y]\)such that\(dP(X)= a\cdot P\), with\(a\in {\mathbb {C}}^{*};\)and the trajectories ofXare simply-connected, up to a polynomial change of coordinates,PandXare as in (1.2) with\(n>1\).
Proof
By Stein’s factorization Theorem, we consider a primitive polynomial \(P_{0}\) such that \(P=h(P_{0})\) with h a polynomial in \({\mathbb {C}}[z]\) of degree \(n\ge {2}\).
Lemma 1
The polynomial \(h\in {\mathbb {C}}[z]\) has only one root
Proof
Assume that h(z) has k roots \({\alpha }_{i}\in {\mathbb {C}}\), for i from 1 to k, respectively of multiplicity \(m_{i}\in {\mathbb {N}}^{+}\). Then
with \(\lambda \in {\mathbb {C}}^{*}\), and with n equal to \(\sum _{i=1}^{k} m_{i}\). According to \(dP(X)=a P\), it follows that \(h'(P_{0})d{P}_{0}(X)=a h({P}_{0}).\) Then \(h({P}_{0})/h'(P_{0})\in {\mathbb {C}}[x,y]\), and thus \(h(z)/h'(z)\in {\mathbb {C}}[z]\). Because
it is clear that if \(k\ge {2}\), the polynomial \(\sum _{i=1}^{k} m_{i} (\prod _{j\ne i} (z-\alpha _{j}))\) has other roots different from \({\alpha }_{i}\) and we obtain a contradiction because \(h(z)/h'(z)\) is not a polynomial. \(\square \)
After Lemma 1, Proposition 1 follows easily from [3] .
Assume that \(h(z)=(z-{\alpha }_{1})^{n}\), with \(n\ge {2}\) (\(\lambda =1\)). Condition \(dP(X)=aP\) can be written as
Hence \(dP_{0} (X)=a/n (P_{0}-{\alpha }_{1})\). As \(dP_{0}=d(P_{0}-{\alpha }_{1})\), if \(Q=P_{0}-{\alpha }_{1}\), one obtains that Q is a primitive polynomial such that \(dQ(X)=a/n Q\). According to [3], we can assume that \(Q=x\) and
where \(b,\,c\in {\mathbb {C}}[x]\) with \(b(0)=0\) and \(c(0)\ne {0}\) after a polynomial automorphism. Then \(P= x^{n}\), and Proposition 1 is proved.
\(\square \)
3 An application and some questions on the image of derivations
First, we give an application of Theorem 1.
Theorem 2
Let X be a polynomial vector field in \(\mathbb {C}^{2}\) whose trajectories are all simply connected. If there is a nonconstant \(P\in \mathbb {C}[x,y]\) and \(n\in \mathbb {N}^{+}\) satisfying:
- a)
\(X^{n}(P)= 0\) and \(X^{n-1}(P)\ne 0\), or
- b)
\(X^{n+1}(P)= a\cdot X^{n}(P)\) for \(a\in \mathbb {C}^{*}\)
Then, X is complete.
Proof
In case a), suppose first that \(n=1\); then \(X(P)=0\) and as P is not a constant polynomial, according to [15], after a polynomial change of coordinates, \(X=\partial /\partial x\) which is complete. If \(n \ge 2\), take \(\bar{P}:=X^{n-1}(P)\). If \(\bar{P}\) is not constant, then \(X(\bar{P})=0\) and as before [15] implies that after a polynomial change of coordinates \(X=\partial /\partial x\), that is complete. Otherwise, if \(\bar{P}=\lambda \in \mathbb {C}^{*}\), it is enough to apply Theorem 1 to \(\tilde{X}:=(1/\lambda )X\) and P if \(n=2\), and to \(\tilde{X}\) and \(\tilde{P}:=X^{n-2}(P)\) if \(n>2\), because \(\tilde{X}(P)=1\) and \(\tilde{X}(\tilde{P})=1\) respectively, to conclude that \(\tilde{X}\) , and then X, are complete.
In case b), we note that neither \(X^{n}(P)\) nor \(X^{n+1}(P)\) equals a nonzero constant. Denote \(\hat{P}:= X^{n}(P)\). Since \(X(\hat{P})=a\cdot \hat{P}\), Theorem 1 implies that X is complete.\(\square \)
Consider the natural domain \(\Omega \), containing \(\{0\}\times {\mathbb {C}}^{2}\), in \({\mathbb {C}}\times {\mathbb {C}}^2\), where the local flow \(\varphi : \Omega \rightarrow {\mathbb {C}}^2\) of X is defined as \(\varphi (t,x,y)=\varphi _z(t)\) (see Sect. 1).
Take \(P\in {\mathbb {C}}[x,y]\). Then \(P(\varphi (t,x,y))\) can be expressed according to the Lie series as:
Theorem 2 implies the following:
Corollary 1
LetXbe a polynomial vector field in\({\mathbb {C}}^{2}\)whose trajectories are all simply connected. Consider the local flow\(\varphi : \Omega \rightarrow {\mathbb {C}}^2\)ofX. If there is a nonconstant\(P\in {\mathbb {C}}[x,y]\)and\(n\in {\mathbb {N}}^{+}\)satisfying:
- (a)$$\begin{aligned} P(\varphi (t,x,y))= P+ X(P)t+X^2(P)\frac{t^2}{2!}+\dots +X^{n-1}(P)\frac{t^{n-1}}{(n-1)!},\,\, \end{aligned}$$
with \(X^{n-1}(P)\ne {0}\), or
- (b)$$\begin{aligned} \begin{aligned} P(\varphi (t,x,y))&= P+ X(P)t+X^2(P)\frac{t^2}{2!}+\dots +X^{n-1}(P)\frac{t^{n-1}}{(n-1)!}+ \\&\quad +\,a^{-n} X^n(P)\left[ e^{at}-(1+at+\frac{{(at)}^2}{2!}+\dots + \frac{{(at)}^{n-1}}{(n-1)!})\right] , \end{aligned} \end{aligned}$$
with\(a\in {\mathbb {C}}^{*}\).
Then,\(\varphi \)can be extended to\({\mathbb {C}}\times {\mathbb {C}}^{2}\)andXis complete.
Remark 1
The Lie series is related to r-inflection points of the vector field with respect to curves of degree r, where \(r=\deg (P)\). See [7, 9]; see [10] for an extension of this idea for codimension one foliations.
Finally, we want to point out some ideas related with Mathieu subspaces, recently introduced by Zhao in [18], and the study of derivations of \({\mathbb {C}}[x,y]\). Let us first recall the following
Definition 1
Let R be a commutative k-algebra and M a k-subspace of R. Then M is a Mathieu subspace of R if the following condition holds: if \(a\in R\) is such that \(a^m\in M\) for all \(m\ge 1\), then for any \(b\in R\) there exists and \(N\in {\mathbb {N}}\) such that \(ba^m\in M\) for all \(m\ge N\).
In our situation, \(R={\mathbb {C}}[x,y]\). It is clear that the image of a derivation Im\((D_{X})\) is a \({\mathbb {C}}\)-subspace of \({\mathbb {C}}[x,y]\). However, Im\((D_{X})\) is not necessarily a Mathieu subspace. Indeed, Zhao proved in [18, Lemma 4,5] that if M is a Mathieu subspace of R and \(1\in M\), then \(M=R\). The following example, taken from [17, Example 2.4],
shows that Im\((D_{X})\) is not a Mathieu subspace, as \(1\in \text {Im}\,(D_{X})\) but \(D_{X}\) is not surjective (\(y\notin \text {Im}\,(D_{X})\)).
Recall that \(D_{X}\) is locally finite if for any \(f\in {\mathbb {C}}[x,y]\), the \({\mathbb {C}}\)-vector space spanned by \(\{X^{n}f\,|\,n\ge {0}\}\) has finite dimension. If \(D_{X}\) is locally finite, Im\((D_{X})\) is a Mathieu subspace [17, Theorem 3.1]. In particular, if \(D_{X}\) is locally finite and has a slice, X is surjective, and then of the form (2) after a polynomial automorphism [17, Proposition 3.2].
It would be interesting to determine polynomial vector fieldsXwith all its trajectories simply-connected and such that Im\((D_{X})\)is a Mathieu subspace of\({\mathbb {C}}[x,y]\), up to a polynomial automorphism.
A polynomial vector field X in \({\mathbb {C}}^{2}\) determines a locally finite derivation \(D_{X}\) if and only if its flow \(\varphi : {\mathbb {C}}\times {\mathbb {C}}^2 \rightarrow {\mathbb {C}}^2\) is quasi-algebraic [8, Theorem 3.1]. In particular, X is complete.
As we mentioned before, in [17, Theorem 4.3] it is proved that the Jacobian conjecture in \({\mathbb {C}}^2\) holds if and only if for every derivation D of \({\mathbb {C}}[x,y]\) with zero divergence (where if \(D= p \frac{\partial }{\partial x} + q \frac{\partial }{\partial y}\), \(\text{ Div }\, (D)= \frac{\partial p}{\partial x} + \frac{\partial q}{\partial y}\)) and having a slice, it holds that Im(D) is a Mathieu subspace.
Recall that the jacobian conjecture in \({\mathbb {C}}^2\) affirms that a polynomial map \(F:=(F_1,F_2): {\mathbb {C}}^2 \longrightarrow {\mathbb {C}}^2\) with \(\det J_{F}=1\) is an automorphism. We call a pair of polynomials \(F_1\), \(F_2\in {\mathbb {C}}[x,y]\) with \(\det J_{(F_1,F_2)}=1\) a Jacobian pair. In a joint article with Muciño [4], the authors proved that the invertibility of the map given by the jacobian pair is equivalent to the fact that one of the vector fields
is complete. Hence, the condition that the image of a derivation D with zero divergence and having a slice is a Mathieu subspace is equivalent to the fact that the polynomial vector field inducing D is complete.
Thus, we note that for derivations there is a close relation between having as image a Mathieu subspace and being induced by a complete polynomial vector field. We do not know examples of a derivation\(D_{X}\)determined by a non complete vector fieldXfor which Im\((D_{X})\)is a Mathieu subspace of\({\mathbb {C}}[x,y]\).
Example 1
Let us consider
X is complete with flow \(\varphi (t,x_{0},y_{0})=(t+x_{0},y_{0}e^{\frac{t^2}{2}+x_{0}t})\). Then \(D_{X}\) is not locally finite. Its trajectories are simply-connected. Moreover, \(D_{X}\) has x as slice.
Im\((D_{X})\) is not a Mathieu subspace. Otherwise, as \(1\in \) Im\((D_{X})\), \(D_{X}\) should be surjective as observed above. But this is not possible because \(y\notin \text {Im}\,(D_{X})\), as a simple calculation shows: writing \(Q=a_0(x)+a_1(x)y+\cdots +a_n(n)y^n\), with \(a_i(x)\in {\mathbb {C}}[x]\), \(D_{X}(Q)=y\) implies
hence it should hold that
which is impossible.
Example 2
[14, Theorem 2.6] Let us consider
with \(a,b\ge 1\). Then, Im\((D_{X})\) is a Mathieu subspace if and only if \(a=b\).
Note that, when \(a=b\ge 2\), X is a vector field with non isolated singularities, trajectories of type \({\mathbb {C}}^{*}\), and whose image is a Mathieu subspace.
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The authors want to thank the referees for a careful reading, and for pointing out the references included in Remark 1.
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Bustinduy, A., Giraldo, L. Remarks on vector fields with simply connected trajectories and their associated derivations. RACSAM 113, 4119–4126 (2019). https://doi.org/10.1007/s13398-019-00670-z
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DOI: https://doi.org/10.1007/s13398-019-00670-z
Keywords
- Foliation transverse to a fibration
- Foliation P-complete
- Simply connected trajectories
- Eigenfunctions of derivations