In this paper we show that the image of any locally finite $k$-derivation of
the polynomial algebra $k[x, y]$ in two variables over a field $k$ of
characteristic zero is a Mathieu subspace. We also show that the
two-dimensional Jacobian conjecture is equivalent to the statement that the
image $Im D$ of every $k$-derivation $D$ of $k[x, y]$ such that $1\in Im D$ and
$div D=0$ is a Mathieu subspace of $k[x, y]$.Comment: Minor changes and improvements. Latex, 9 pages. To appear in J. Pure
  Appl. Algebr

Arno van den Essen

Bass

Cerveau

David Wright

Duistermaat

Freudenburg

Mathieu

Ostrowski

Stein

Sturmfels

van den Essen

Wenhua Zhao

Zhao

English

arXiv

Crossref

Journal of Pure and Applied Algebra

Images of locally finite derivations of polynomial algebras in two variables

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Essen, Arno van den

Wright, D.

Zhao, W.H.

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In this paper we show that the image of any locally finite k-derivation of the polynomial algebra k[x,y] in two variables over a field k of characteristic zero is a Mathieu subspace. We also show that the two-dimensional Jacobian conjecture is equivalent to the statement that the image of every k-derivation D of k[x,y] such that and is a Mathieu subspace of k[x,y]

van den Essen, Arno

Wright, David

Zhao, Wenhua

Washington University St. Louis: Open Scholarship

Washington University in St. LouisWashington University Open ScholarshipMathematics Faculty Publications Mathematics and Statistics2011Images of locally finite derivations of polynomialalgebras in two variablesArno van den EssenRadboud UniversityDavid WrightWashington University in St LouisWenhua ZhaoIllinois State UniversityFollow this and additional works at: https://openscholarship.wustl.edu/math_facpubsThis Article is brought to you for free and open access by the Mathematics and Statistics at Washington University Open Scholarship. It has beenaccepted for inclusion in Mathematics Faculty Publications by an authorized administrator of Washington University Open Scholarship. For moreinformation, please contact digital@wumail.wustl.edu.Recommended Citationvan den Essen, Arno; Wright, David; and Zhao, Wenhua, "Images of locally finite derivations of polynomial algebras in two variables"(2011). Mathematics Faculty Publications. 1.https://openscholarship.wustl.edu/math_facpubs/1arXiv:1004.0521v2  [math.AC]  9 Dec 2010IMAGES OF LOCALLY FINITE DERIVATIONS OFPOLYNOMIAL ALGEBRAS IN TWO VARIABLESARNO VAN DEN ESSEN, DAVID WRIGHT AND WENHUA ZHAOAbstract. In this paper we show that the image of any locallyfinite k-derivation of the polynomial algebra k[x, y] in two variablesover a field k of characteristic zero is a Mathieu subspace. We alsoshow that the two-dimensional Jacobian conjecture is equivalentto the statement that the image ImD of every k-derivation D ofk[x, y] such that 1 ∈ ImD and divD = 0 is a Mathieu subspace ofk[x, y].1. IntroductionKernels of derivations have been studied in many papers. On theother hand, only a few results are known concerning images of deriva-tions.In this paper we consider the question if the image of a derivationof a polynomial algebra in two variables over a field k is a Mathieusubspace of the polynomial algebra.The notion of the Mathieu subspaces was introduced recently by thethird-named author in [Z2] in order to study the Mathieu conjecture[M], the image conjecture [Z1] and the Jacobian conjecture (see [BCW]and [E1]). We will recall its definition in Section 2 below.Throughout this paper we fix the following notation: k is a field ofcharacteristic zero and x, y are two free commutative variables. Wedenote by A the polynomial algebra k[x, y] over the field k.The contents of the paper are arranged as follows.In Section 2 we recall some facts concerning Mathieu subspaces andshow that the image of a k-derivation of A needs not be a Mathieusubspace (see Example 2.4).Date: December 10, 2010.2000 Mathematics Subject Classification. 13N15, 14R10, 14R15.Key words and phrases. The Mathieu subspaces, locally finite derivations, theJacobian conjecture.The third-named author has been partially supported by the NSA Grant H98230-10-1-0168.12 ARNO VAN DEN ESSEN, DAVID WRIGHT AND WENHUA ZHAOIn Section 3 we prove in Theorem 3.1 that for every locally finitek-derivation D of A, the image ImD is a Mathieu subspace. Finally inSection 4 we show in Theorem 4.3 that the two-dimensional Jacobianconjecture is equivalent to the following: if D is a k-derivation of Awith divD = 0 such that 1 ∈ ImD, then ImD is a Mathieu subspaceof A.2. PreliminariesWe start with the following notion introduced in [Z2].Definition 2.1. Let R be any commutative k-algebra and M a k-subspace of R. Then M is a Mathieu subspace of R if the followingcondition holds: if a ∈ R is such that am ∈ M for all m ≥ 1, then forany b ∈ R, there exists an N ∈ N such that bam ∈M for all m ≥ N .Obviously every ideal of R is a Mathieu subspace of R. However notevery Mathieu subspace of R is an ideal of R. Before we give someexamples, we first recall the following simple lemma proved in Lemma4.5, [Z2], which will be very useful for our later arguments. For thesake of completeness, we here also include a proof.Lemma 2.2. If M is a Mathieu subspace of R and 1 ∈M , then M =R.Proof: Since 1 ∈ M , it follows that 1m = 1 ∈ M for all m ≥ 1.Then for every a ∈ R, a = a1m ∈ M for all large m. Hence R ⊆ Mand R = M . 2Example 2.3. Let R := k[t, t−1] be the algebra of Laurent polynomialsin the variable t. For each c ∈ k, let Dc be the differential operatorddt+ ct−1 of R. Then ImDc := DcR is a Mathieu subspace of R if andonly if c 6∈ Z or c = −1.Note that the conclusion above follows directly by applying Lef-schetz’s principle to Proposition 2.6 [Z2]. Since Proposition 2.6 in [Z2]is for multi-variable case and its proof is quite involved, we here includea self-contained proof for the one variable case.Proof: Note first that for any m ∈ Z, Dctm = (m + c)tm−1. So, ifc 6∈ Z, then ImDc = R. Hence a Mathieu subspace of R.If c ∈ Z but c 6= −1, then Dct = (1 + c) 6= 0. So 1 ∈ ImDc. SinceDct−c = (−c+ c)t−c−1 = 0, it is easy to see that t−c−1 6∈ ImDc. HenceImDc 6= R. Then by Lemma 2.2, ImDc is not a Mathieu subspace ofR.IMAGES OF LOCALLY FINITE DERIVATIONS 3Finally, assume c = −1. Since D−1tm = (m− 1)tm−1 for all m ∈ Z,it is easy to see that ImD−1 is the subspace of the Laurent polynomialsin R without constant term. Then by the Duistermaat-van der Kallentheorem [DK], M is a Mathieu subspace of R. 2Note that when c = −1, ImD−1 is a Mathieu subspace of R. But itclearly is not an ideal of R. For more examples of Mathieu subspaceswhich are not ideals, see Section 4 in [Z2].When c = 0, we see that Im d/dt is not a Mathieu subspace of R.Now observe that k[t, t−1] ≃ k[x, y]/(xy − 1), where t corresponds tothe class of x and t−1 to the class of y. Then the derivation d/dt of Rcan be lifted to a k-derivation D of k[x, y], which maps x to ddtt = 1and y to ddtt−1 = −t−2, i.e., −y2. This leads to the following example.Example 2.4. Let D = ∂x − y2∂y. Then ImD is not a Mathieusubspace of k[x, y].Proof: Note that 1 = Dx ∈ ImD. However y 6∈ ImD since for anyg ∈ k[x, y] the y-degree of Dg can not be 1. So by Lemma 2.2, ImD isnot a Mathieu subspace of k[x, y]. 2The following lemma will also be needed in Section 3.Lemma 2.5. Let R be any k-algebra, L a field extension of k and Ma k-subspace of R. Assume that L⊗k M is a Mathieu subspace of theL-algebra L⊗k R. Then M is a Mathieu subspace of the k-algebra R.Proof: We view L ⊗k R as a k-algebra in the obvious way. SinceL⊗kM is a Mathieu subspace of the L-algebra L⊗kR, from Definition2.1 it is easy to see that L ⊗k M (as a k-subspace) is also a Mathieusubspace of the k-algebra L⊗k R.Now we identify R with the k-subalgebra 1 ⊗k R of the k-algebraL ⊗k R. Then from Definition 2.1 again, it is easy to check that theintersection (L⊗k M) ∩R = M is a Mathieu subspace of R. 2Note that by the lemma above, when we prove that a k-subspaceof a polynomial algebra over k is a Mathieu subspace of the polyno-mial algebra, we may freely replace k by any field extension of k. Forinstance, we may assume that k is algebraically closed.To conclude this section we recall a result from [EWZ] which will beused in Section 3 below.Let z = (z1, z2, ..., zn) be n commutative free variables and k[z, z−1]the algebra of Laurent polynomials in zi (1 ≤ i ≤ n). For any non-zerof(z) =∑α∈Zn cαzα ∈ k[z, z−1], we denote by Supp (f) the support of4 ARNO VAN DEN ESSEN, DAVID WRIGHT AND WENHUA ZHAOf(z), i.e., the set of all α ∈ Zn such that cα 6= 0, and Poly (f) the(Newton) polytope of f(z), i.e., the convex hull of Supp (f) in Rn.Theorem 2.6. ([EWZ]) Let 0 6= f ∈ k[z, z−1] and u any rational point,i.e., a point with all coordinates being rational, of Poly (f). Then thereexists m ≥ 1 such that (R+u) ∩ Supp (fm) 6= ∅.3. Images of Locally Finite Derivations of k[x, y]Let D be any k-derivation of A(= k[x, y]). Then D is said to belocally finite if for every a ∈ A the k-vector space spanned by theelements Dia (i ≥ 1) is finite dimensional.The main result of this section is the following theorem.Theorem 3.1. Let D be any locally finite k-derivation of A. ThenImD is a Mathieu subspace of A.To prove this theorem, we need the following result, which is Corol-lary 4.7 in [E2].Proposition 3.2. Let D be any locally finite k-derivation of A. Thenup to the conjugation by a k-automorphism of A, D has one of thefollowing forms:i) D = (ax+ by)∂x + (cx+ dy)∂y for some a, b, c, d ∈ k;ii) D = ∂x + by∂y for some b ∈ k;iii) D = ax∂x + (xm + amy)∂y for some a ∈ k and m ≥ 1;iv) D = f(x)∂y for some f(x) ∈ k[x].Lemma 3.3. With the same notations as in Proposition 3.2, the fol-lowing statements hold.(a) If D is of type ii), then D is surjective.(b) If D is of type iii), thenImD ={(xm) if a = 0.(x, y) if a 6= 0.(3.1)(c) If D is of type iv), then ImD = (f(x)).Proof: (a) is well-known, see [C] or [F] (p. 96). (c) is obvious, so itremains to prove (b).If a = 0, then D = xm∂y, and hence ImD = (xm). So assumea 6= 0. Replacing D by a−1D (without changing the image ImD), wemay assume that D = (x∂x +my∂y) + bxm∂y for some nonzero b ∈ k.Observe that for any i, j ∈ N, we haveD(xiyj) = (i+mj)xiyj + jbxm+iyj−1.(3.2)IMAGES OF LOCALLY FINITE DERIVATIONS 5Next we use induction on j ≥ 0 to show that xiyj ∈ ImD wheneveri+ j > 0.First, assume j = 0. Then by Eq. (3.2), we have Dxi = ixi, andhence xi ∈ ImD for all i ≥ 1.Now assume j ≥ 1. Since m ≥ 1, we have m + i ≥ 1 for all i ≥ 0.Then by the induction assumption, jbxm+iyj−1 ∈ ImD for all i ≥ 0.Combining this fact with Eq. (3.2), we get xiyj ∈ ImD since i+mj 6= 0for all i ≥ 0. Hence we have proved that xiyj ∈ ImD if i+ j > 0. Notethat 1 does not lie in ImD since this space is contained in the idealgenerated by x and y. Therefore we have ImD = (x, y). 2Lemma 3.4. Let z = (z1, z2, ..., zn) be n free commutative variablesand D :=∑ni=1 aizi∂zi for some ai ∈ k (1 ≤ i ≤ n). Then ImD is aMathieu subspace of k[z].Note that D in the lemma is a locally finite derivation of the poly-nomial algebra k[z]. To show the lemma, let’s first recall the followingwell-known results.Lemma 3.5. For any polynomials f, g ∈ k[z] and a positive integerm ≥ 1, we havePoly (fg) = Poly (f) + Poly (g),(3.3)Poly (fm) = mPoly (f),(3.4)where the sum in the first equation above denotes the Minkowski sumof polytopes.Proof: Eq. (3.3) is well-known, which was first proved by A. M. Os-trowski [O1] in 1921 (see also Theorem VI, p. 226 in [O2] or Lemma 2.2,p. 11 in [Stu]). To show Eq. (3.4), one can first check easily that thepolytopemPoly (f) and the polytope obtained by taking the Minkowskisum of m copies of Poly (f) actually share the same set of extremalvertices, namely, the set of the vertices mvi, where vi runs through allextremal vertices of Poly (f). Consequently, these two polytopes coin-cide. Then from this fact and Eq. (3.3), we see that Eq. (3.4) follows.2Proof of Lemma 3.4: If all ai’s are zero, then D = 0 and ImD = 0.Hence the lemma holds in this case. So, we assume that not all ai’s arezero.Let S be the set of integral solutions β ∈ Zn of the linear equation∑ni=1 aiβi = 0. Note that S 6= ∅ (since 0 ∈ S) and is a finitely generatedZ-module. Let V be the subspace of Rn spanned by elements of S over6 ARNO VAN DEN ESSEN, DAVID WRIGHT AND WENHUA ZHAOR. Then V is a R-subspace of Rn with r := dimR V < n. Furthermore,V can be described as the set of common solutions of some linearequations with rational coefficients, since clearly the Q-vector spacegenerated by the Z-generators of S can.Note also that for any β = (β1, β2, ..., βn) ∈ Nn, we have Dzβ =(∑ni=1 aiβi)zβ . Hence, for any β ∈ Nn, the monomial zβ ∈ ImD iffβ 6∈ S, or equivalently, β 6∈ V . Consequently, for any 0 6= h(z) ∈ C[z],we haveh(z) ∈ ImD ⇔ Supp (h) ∩ V = ∅.(3.5)Now, let 0 6= f(z) ∈ C[z] such that fm ∈ ImD for all m ≥ 1. Weclaim Poly (f) ∩ V = ∅.Assume otherwise. Since all vertices of the polytope Poly (f) arerational (actually integral), every face of Poly (f) can be described asthe set of common solutions of some linear equations with rationalcoefficients. Since this is also the case for V (as pointed above) andPoly (f)∩V 6= ∅ (by our assumption), it is easy to see that there existsat least one rational point u ∈ Poly (f) ∩ V . Then by Theorem 2.6,there exists m ≥ 1 such that (R+u)∩ Supp (fm) 6= ∅, and by Eq. (3.5),fm 6∈ ImD. Hence, we get a contradiction. Therefore, the claim holds.Finally, we show that ImD is a Mathieu subspace as follows.Let f(z) be as above and d the distance between V and Poly (f).Then by the claim above and the fact that Poly (f) is a compact subsetof Rn, we have d > 0. Furthermore, for anym ≥ 1, by Eq. (3.4) we havePoly (fm) = mPoly (f). Hence, the distance between V and Poly (fm)is given by dm.Now let h(z) be an arbitrary element of k[z]. Note that by Eqs. (3.3)and (3.4) we have Poly (fmh) = mPoly (f) + Poly (h) for all m ≥ 1.Hence, for large enough m, the distance between V and Poly (fmh) ispositive, whence Poly (fmh)∩V = ∅. In particular, Supp (fmh)∩V =∅, and by Eq. (3.5), fmh ∈ ImD when m≫ 0. Then by Definition 2.1,we see that ImD is indeed a Mathieu subspace of k[z]. 2Now we can prove the main theorem of this section as follows.Proof of Theorem 3.1: First, by Proposition 3.2, we only need toshow that ImD is a Mathieu subspace of A in each of the four cases inProposition 3.2. Furthermore, by Lemma 3.3 it only remains to provecase i). So assume D = (ax+by)∂x+(cx+dy)∂y for some a, b, c, d ∈ k.Second, by Lemma 2.5, we may assume that k is algebraically closed.Third, note that D preserves the subspace H := kx + ky ⊂ A, soits restriction D|H on H is a linear endomorphism of H . Since k isIMAGES OF LOCALLY FINITE DERIVATIONS 7algebraically closed, there exists a linear automorphism σ of H suchthat the conjugation σ(D|H)σ−1 gives the Jordan form of D|H . Let σ˜be the unique extension of σ to an automorphism of A. Then it is easyto see that σ˜Dσ˜−1 is also a k-derivation of A.Note that Im σ˜Dσ˜−1 = σ˜(ImD) and in general Mathieu subspacesare preserved by k-algebra automorphisms. Therefore, we may replaceD by σ˜Dσ˜−1, if necessary, and assume that D = a(x∂x + y∂y) + x∂y(in case that the Jordan form of D|H is an 2 × 2 Jordan block) orD = ax∂x + by∂y (in case that the Jordan form of D|H is diagonal).For the former case, by Lemma 3.3, (b) with m = 1, we see thatImD is an ideal, and hence a Mathieu subspace of A. For the lattercase, it follows from Lemma 3.4 that ImD also a Mathieu subspace ofA. Therefore, the theorem holds. 24. Connection with the Two-Dimensional JacobianConjectureIn the previous section we showed that the image of every locallyfinite k-derivation of A is a Mathieu subspace of A. However, as wehave shown in Example 2.4, ImD needs not to be a Mathieu subspaceof A for every k-derivation D of A. This leads to the question of whichk-derivationsD of A have the property that ImD is a Mathieu subspaceof A. More precisely, we can askQuestion 4.1. Let D be any k-derivation of A such that divD = 0,where for any D = p∂x + q∂y (p, q ∈ A), divD := ∂xp+ ∂yq. Is ImD aMathieu subspace of A?Adding one more condition, we getQuestion 4.2. Let D be any k-derivation of A such that divD = 0.If 1 ∈ ImD, is ImD a Mathieu subspace of A?Note that by Lemma 2.2, this question is equivalent to asking if Dis surjective under the further condition 1 ∈ ImD.The motivation of the two questions above come from the followingtheorem.Theorem 4.3. Question 4.2 has an affirmative answer iff the two di-mensional Jacobian conjecture is true.Proof: (⇒) Assume that Question 4.2 has an affirmative answer.Let F = (f, g) ∈ k[x, y]2 with det JF = 1. Consider the k-derivationD := gy∂x − gx∂y. Then divD = 0 and 1 = det JF = Df ∈ ImD.Since by our hypothesis ImD is a Mathieu subspace of A, it follows8 ARNO VAN DEN ESSEN, DAVID WRIGHT AND WENHUA ZHAOfrom Lemma 2.2 that ImD = A, i.e., D is surjective. Then it followsfrom a theorem of Stein [Ste] (see also [C]) that D is locally nilpotent.Since D = ∂/∂f , kerD = ker ∂/∂f = k[g] by Proposition 2.2.15in [E1]. Since D has a slice f , it follows that A = k[g][f ], i.e., F isinvertible over k. So the two-dimensional Jacobian conjecture is true.(⇐) Assume that the two-dimensional Jacobian conjecture is true.Let D = p∂x+q∂y (p, q ∈ A) be a k-derivation of A such that divD = 0and 1 ∈ ImD.Since divD = 0, we have ∂xp = ∂y(−q). Then by Poincare´’s lemma,there exists g ∈ A such that p = ∂yg and q = −∂xg. SoD = gy∂x−gx∂y.Since 1 ∈ ImD, we get 1 = Df for some f ∈ A. Let F := (f, g) ∈k[x, y]2. Then we have det JF = Df = 1. Since by our hypothesis Fis invertible, it follows that k[x, y] = k[f, g]. Hence, we haveImD = Im∂∂f=∂∂f(k[f, g]) = k[f, g] = A.In particular, ImD is a Mathieu subspace of A. 2References[BCW] D. Bass, E. Connell, D. Wright, The Jacobian Conjecture, Reduction ofDegree and Formal Expansion of the Inverse. Bull. Amer. Math. Soc. 7,(1982), 287–330. [MR83k:14028].[C] D. Cerveau, De´rivations surjectives de l’anneau C[x, y]. (French) [Surjec-tive derivations of the ring C[x, y]] J. Algebra 195 (1997), no. 1, 320–335.[MR1468896].[DK] J. J. Duistermaat and W. van der Kallen, Constant Terms in Powersof a Laurent Polynomial. Indag. Math. (N.S.) 9 (1998), no. 2, 221–231.[MR1691479].[E1] A. van den Essen, Polynomial Automorphisms and the Jacobian Con-jecture. Progress in Mathematics, 190. Birkha¨user Verlag, Basel, 2000.[MR1790619].[E2] A. van den Essen, Locally Finite and Locally Nilpotent Derivations withApplications to Polynomial Flows and Polynomial Morphisms. Proc. AMS,116, No. 3 (1992), 861–871. [MR1111440].[EWZ] A. van den Essen, R. Willems and W. Zhao, Some Results on the Van-ishing Conjecture of Differential Operators with Constant Coefficients.arXiv:0903.1478v1 [math.AC]. Under submission.[F] G. Freudenburg, Algebraic Theory of Locally Nilpotent Derivations. Ency-clopaedia of Mathematical Sciences, 136. Invariant Theory and AlgebraicTransformation Groups, VII. Springer-Verlag, Berlin, 2006. [MR2259515].[M] O. Mathieu, Some Conjectures about Invariant Theory and Their Ap-plications. Alge`bre non commutative, groupes quantiques et invariants(Reims, 1995), 263–279, Se´min. Congr., 2, Soc. Math. France, Paris, 1997.[MR1601155].IMAGES OF LOCALLY FINITE DERIVATIONS 9[O1] A. M. Ostrowski, U¨ber die Bedeutung der Theorie der konvexen Polyederfu¨r die formale Algebra. Jahresberichte Deutsche Math. Verein 30 (1921),98–99.[O2] A. M. Ostrowski, On multiplication and factorization of polynomials I.Lexicographic ordering and extreme aggregates of terms. Aequationes Math.13 (1975), no. 3, 201-228. [MR0399060].[Ste] Y. Stein, On the Density of Image of Differential Operators Generated byPolynomials. J. Analyse Math. 52 (1989), 291–300. [MR0981505].[Stu] B. Sturmfels, Gro¨bner bases and convex polytopes. University Lecture Se-ries, 8. American Mathematical Society, 1996. [MR1363949].[Z1] W. Zhao, Images of Commuting Differential Operators of Order One withConstant Leading Coefficients. J. Alg. 324 (2010), no. 2, 231–247. See alsoarXiv:0902.0210 [math.CV].[Z2] W. Zhao, Generalizations of the Image Conjecture and the MathieuConjecture. J. Pure Appl. Algebra. 214 (2010), 1200-1216. See alsoarXiv:0902.0212 [math.CV].A. van den Essen, Department of Mathematics, Radboud UniversityNijmegen, Postbus 9010, 6500 GL Nijmegen, The Netherlands. E-mail:essen@math.ru.nlD. Wright, Department of Mathematics, Washington University inSt. Louis, St. Louis, MO. 63130, USA. E-mail: wright@math.wustl.eduW. Zhao, Department of Mathematics, Illinois State University,Normal, IL 61790-4520, USA. E-mail: wzhao@ilstu.edu.

Images of Locally Finite Derivations of Polynomial Algebras in Two
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