---
abstract: |
    The Dixmier Conjecture says that every endomorphism of the (first) Weyl algebra $A_1$ (over a field of characteristic zero) is an automorphism, i.e., if $PQ-QP=1$ for some $P, Q \in A_1$ then $A_1 = K \langle P, Q \rangle$. The Weyl algebra $A_1$ is a $\Z$-graded algebra. We prove that the Dixmier Conjecture holds if the elements $P$ and $Q$ are sums of no more than two homogeneous elements of $A$ (there is no restriction on the total degrees of $P$ and $Q$).\
    [*Key Words: the Weyl algebra, the Dixmier Conjecture, automorphism, endomorphism, a $\Z$-graded algebra.*]{}

    [*Mathematics subject classification 2010: 16S50, 16W20, 16S32, 16W50.*]{}
address:
- |
    Department of Pure Mathematics\
    University of Sheffield\
    Hicks Building\
    Sheffield S3 7RH, UK
- |
    Lehrstuhl D für Mathematik\
    RWTH Aachen University\
    52062 Aachen, Germany
author:
- 'V. V. Bavula'
- 'V. Levandovskyy'
title: A remark on the Dixmier Conjecture
---

Introduction
============

In the paper, $K$ is a field of characteristic zero and $K^*:=K \setminus\{0\}$. The algebra $A_1 := K \langle X, Y \mid [Y, X]=1 \rangle$ is called the [*first Weyl algebra*]{} where $[Y, X]= YX - XY$. The $n$’th tensor power of $A_1$, $A_n := A_1^{\otimes n} =
\underbrace{A_1 \otimes \cdots \otimes A_1}_{n \text{ times}}$, is called the [*$n$’th Weyl algebra*]{}. The algebra $A_n$ is a simple Noetherian domain of Gel’fand-Kirillov dimension $\GK(A_n)=2n$, it is canonically isomorphic to the algebra of polynomial differential operators $K \langle X_1, \ldots, X_n, \partial_1, \ldots, \partial_n \rangle$ (where $\partial_i = \tfrac{\partial}{\partial x_i}$) via $X_i \mapsto X_i$, $Y_i \mapsto \partial_i$ for $i=1,\ldots,n$.

In his seminal paper [@Dix], Dixmier (1968) found explicit generators for the group $G = \Aut_K(A_1)$ of $K$-automorphisms of the Weyl algebra $A_1$. Namely, the group $G$ is generated by the obvious automorphisms: $$(X, Y) \mapsto (X, Y + \l X^n), \quad (X, Y) \mapsto (X + \l Y^n, Y), \quad  (X, Y) \mapsto (\mu X, \mu^{-1} Y)$$ where $\l \in K$, $\mu \in K^*$ and $n \in \Nat_{+}:=\{1, 2, \ldots\}$.

In [@Dix], Dixmier posed six problems: The first problem of Dixmier (in the list) asks if [*every endomorphism of the Weyl algebra $A_1$ is an automorphism*]{}, i.e., given elements $P, Q$ of $A$ such that $[P, Q]=1$, do they generate the algebra $A_1$? A similar problem but for the $n$’th Weyl algebra is called the [*Dixmier Conjecture*]{}. Problems 3 and 6 have been solved by Joseph [@josclA1] (1975), Problem 5 and Problem 4 (in the case of homogeneous elements) have been solved by Bavula [@Bav-DixPr5] (2005).

The Dixmier Conjecture implies the [*Jacobian Conjecture*]{} (see [@BCW]) and the inverse implication is also true (see [@Tsuchimoto-2005] and [@Belov-Konts-2007]); a short proof is given in [@Bav-DCJC-2005]; see also [@Adjam-vdEs-2007]).

In [@Bav-RenQn], it is shown that for each $K$-endomorphism $\phi: A_n \to A_n$ its image is very large, i.e., the left $A_{2n}$-module ${}^{\phi}{A_n}^{\phi}$ is a holonomic $A_{2n}$-module (where for all $a, b \in A_n$ and $c \in {}^{\phi}{A_n}^{\phi}$, $a \cdot c \cdot b := \phi(a) c \phi(b)$). In particular, it has finite length with simple holonomic factors over $A_{2n}$ (see [@Bav-RenQn] for details). To prove that the Dixmier Conjecture holds for the Weyl algebra $A_n$ it remains to show that the length is 1. Note, that the Gel’fand-Kirillov dimension of a simple $A_{2n}$-module can be $2n, 2n+1, \ldots, 4n-1$, and the last case is the generic case.

In [@Bav-T1DC], it is shown that every algebra endomorphism of the algebra $\CI_1 = K\langle x, \partial, \int \rangle$ of polynomial integro-differential operators is an automorphism and it is conjectured that the same result holds for $\CI_n := \CI_1^{\otimes n}=K\langle x_1, \ldots , x_n, \der_1, \ldots , \der_n , \int_1, \ldots , \int_n\rangle$.

The Weyl algebra $A = \oplus_{i\in\Z} A_{1,i}$ is a $\Z$-graded algebra ($A_{1,i} A_{1,j} \subseteq A_{1,i+j}$ for all $i, j\in \Z$) where $A_{1,0}=K[H]$, $H=YX$ and, for $i\geq 1$, $A_{1,i} = K[H]X^i$ and $A_{1,-i} = K[H]Y^i$. For a nonzero element $a$ of $A_1$, the number of [*nonzero homogeneous*]{} components is called the [*mass*]{} of $a$, denoted by $m(a)$. For example, $m(\alpha X^i)=1$ for all $\alpha \in K[H]\setminus\{0\}$ and $i\geq 1$. The aim of this paper is to prove the following theorem.

\[20Sep16\] Let $P, Q$ be elements of the first Weyl algebra $A_1$ with $m(P)\leq 2$ and $m(Q)\leq 2$. If $[P,Q]=1$ then $P = \tau(Y)$ and $Q=\tau(X)$ for some automorphism $\tau\in\Aut_K (A_1)$.

Proof of Theorem \[20Sep16\]
============================

[**The Weyl algebra is a generalized Weyl algebra**]{}. Let $D$ be a ring with an automorphism $\s $ and a central element $a$. The [**generalized Weyl algebra**]{} $A=D(\sigma, a)$ of degree 1, is the ring generated by $D$ and two indeterminates $X$ an $Y$ subject to the relations [@Bav-FA-1991]: $$X\alpha=\sigma(\alpha)X \ {\rm and}\ Y\alpha=\sigma^{-1}(\alpha)Y,\; {\rm for \; all }\;
\alpha \in D, \ YX=a \ {\rm and}\ XY=\sigma(a).$$ The algebra $A={\oplus}_{n\in \mathbb{Z}}\, A_n$ is a $\mathbb{Z}$-graded algebra where $A_n=Dv_n$, $v_n=X^n\,\, (n>0), \,\,v_n=Y^{-n}\,\, (n<0), \,\,v_0=1.$ It follows from the defining relations that $$v_nv_m=(n,m)v_{n+m}=v_{n+m}<n,m>$$ for some elements $(n,m)=\s^{-n-m}(<n,m>)\in D$. If $n>0$ and $m>0$ then $$\begin{aligned}
n\geq m & : & (n,-m)=\sigma^n(a)\cdots \sigma^{n-m+1}(a), \,\,  (-n,m)=\sigma^{-n+1}(a)\cdots \sigma^{-n+m}(a), \\
n\leq m & : & (n,-m)=\sigma^{n}(a)\cdots \sigma(a),\,\,\, (-n,m)=\sigma^{-n+1}(a)\cdots a,\end{aligned}$$ in other cases $(n,m)=1$.

Let $K[H]$ be a polynomial ring in a variable $H$ over the field $K$, $\s :H\ra H-1$ be the $K$-automorphism of the algebra $K[H]$ and $a=H$. The first Weyl algebra $A_1=K\langle X, Y \mid YX-XY=1\rangle$ is isomorphic to the generalized Weyl algebra $$A_1\simeq K[H](\s , H),\; X \mapsto X,\; Y \mapsto Y,\; YX  \mapsto H.$$ We identify both these algebras via this isomorphism, that is $A_1=K[H](\s , H)$ and $H=YX$.

If $n>0$ and $m>0$ then $$\begin{aligned}
n\geq m & : & (n,-m)= (H-n)\cdots (H-n+m-1),\; (-n,m)=(H+n-1)\cdots (H+n-m), \\
n\leq m & : & (n,-m)=(H-n)\cdots (H-1), \; (-n,m)=(H+n-1)\cdots H,\end{aligned}$$ in other cases $(n,m)=1$.

The localization $B=S^{-1}A_1$ of the Weyl algebra $A_1$ at the Ore subset $S=K[H]\backslash \{ 0\}$ of $A_1$ is the [*skew Laurent polynomial ring*]{} $B=K(H)[X, X^{-1}; \s ]$ with coefficients from the field $K(H)=S^{-1}K[H]$ of rational functions where $\s \in \Aut_K \, K(H)$ and $\s (H)=H-1$. The map $A_1\ra B$, $a  \mapsto a/1$ is an algebra monomorphism. We identify the algebra $A_1$ with its image in the algebra $B$ via $A_1 \ra  B, \;\; X \mapsto  X, \;\; Y \mapsto HX^{-1}$. The algebra $B=\oplus_{i\in \mathbb{Z}}\, B_i$ is a $\mathbb{Z}$-graded algebra where $B_i=K(H)X^i$. The algebra $A_1$ is a $\mathbb{Z}$-graded subalgebra of $B$.

A polynomial $f(H)=\l_nH^n+\l_{n-1}H^{n-1}+\cdots +\l_0\in K[H]$ of degree $n$ is called a [*monic*]{} polynomial if the [*leading coefficient*]{} $\l_n$ of $f(H)$ is $1$. A rational function $h\in
K(H)$ is called a [*monic*]{} rational function if $h=f/g$ for some monic polynomials $f,g$. A homogeneous element $u=\alpha x^n$ of $B$ is called [*monic*]{} if $\alpha $ is a monic rational function. We can extend the concept of degree of polynomial to the field of rational functions by the rule $\deg \, h = \deg\, f - \deg\, g$ where $h=f/g\in K[H]$. If $h_1, h_2 \in K(H)$ then $\deg\, h_1h_2=\deg\,h_1 +\deg\, h_2$ and $\deg (h_1+h_2)\leq \max \{ \deg\,h_1 , \deg\, h_2\}$. We denote by ${\rm sign} (n)$ and by $|n|$ the [*sign*]{} and the [*absolute value*]{} of $n\in \mathbb{Z}$, respectively.

Let $A$ be an algebra and $a\in A$. The subalgebra of $A$, $C_A(a)=\{ b \in A \mid ab=ba \}$, is called the [*centralizer*]{} of the element $a$ in $A$.

\[cheab\] [*(Centralizer of a Homogeneous Element of the Algebra $B$)*]{}

1.  Let $u=\alpha X^n$ be a monic element of $B_n$ with $n\neq 0$. Then the centralizer $C_B(u)=K[v,v^{-1}]$ is a Laurent polynomial ring for a unique element $v=\beta X^{{\rm sign} (n) s}$ where $s$ is the least positive divisor of $n$ for which there exists an element $\beta =\beta_s \in K(H)$, necessarily monic and uniquely defined, such that $$\label{bap}
    \beta \,\s^s (\beta )\, \s^{2s}(\beta ) \cdots \s^{(n/s -1)s}(\beta )=\alpha, \;\; {\rm if}\;\; n>0,$$ $$\label{bam}
    \beta \, \s^{-s} (\beta ) \,\s^{-2s}(\beta ) \cdots \s^{-(|n|/s -1)s}(\beta )=\alpha, \;\; {\rm if}\;\; n<0.$$

2.  Let $u\in K(H)\backslash K$. Then $C_B(u)=K(H)$.

Let $A_{1,+} := K[H][X; \sigma]$ and $A_{1,-} := K[H][Y; \sigma^{-1}]$. The algebras $A_{1,+}$ and $A_{1,-}$ are (skew polynomial) subalgebras of $A_1$.

\[a20Sep16\] If $u \in A_{1,\pm}\setminus\{0\}$ then $C_A(u) \subseteq A_{1,\pm}$.

The $K$-automorphism of the Weyl algebra $A_1$, $$\label{xiaut}
\xi: A_1 \to A_1, \; X \mapsto Y, \; Y \mapsto -X,$$ reverses the $\Z$-grading of the Weyl algebra $A_1$, that is $$\label{xiaut1}
\xi(A_{1,i}) = A_{1,-i} \text{ for all } z\in\Z.$$

By the [*degree*]{} of an element of $A_1$ we mean its [*total degree*]{} with respect to the canonical generators $X$ and $Y$ of $A_1$. Let $A_{1, \leq i} := \{ p\in A \mid \deg(p)\leq i\}$ for $i\in\Nat$. Then $\{A_{1, \leq i}\}_{i\in\Nat}$ is the standard filtration of the algebra $A_1$ associated with the generators $X$ and $Y$. For all $i\in\Z\setminus\{0\}$ and $f\in K[H]\setminus K$, $$\label{degsf}
\deg \s^i(f) = \deg f \text{ and } \deg (1 - \s^i)(f)  = \deg f - 1.$$

[**Proof of Theorem \[20Sep16\]**]{}: [*(i)*]{} [*If $P, Q\in A_{1,\leq 1}$ then $P = \tau(Y)$ and $Q=\tau(X)$ for some $\tau\in\Aut_K (A_1)$*]{}: Clearly, $P = aY + bX + \l$ and $Q = cY +dX + \mu$ for some $a, b, c, d, \l, \mu \in K$. Then $1 = [P,Q]=ad-bc$. So, the automorphism $\tau$ can be chosen of the form $\tau(Y) = aY + bX + \l$ and $\tau(X) = cY +dX + \mu$.

So, till the end of the proof we assume that at least one of the polynomials $P$ or $Q$ does not belong to the space $A_{1, \leq 1}$. In view of the relation $1 = [P,Q] = [-Q,P]$, we can assume that $P \notin A_{1, \leq 1}$. In view of Equation (\[xiaut1\]), we can assume that the highest homogeneous part of $P$, say $P_p \in A_{1,p}$, satisfies the condition that $p \geq 2$. Since $m(P)\leq 2$, either $P = P_p$ (if $m(P)=1$) or otherwise $P = P_r + P_p$ for some nonzero $P_r \in A_{1,r}$ where $r<p$.

[*(ii)*]{} $(m(P), m(Q)) \neq (1,1)$: Suppose that $m(P)= m(Q)=1$, we seek a contradiction. Then $P =\alpha X^p$ and $Q=\beta Y^p$ for some nonzero polynomials $\alpha, \beta\in K[H]$. Then $$1 = [P,Q] = \alpha \s^p(\beta) (p,-p) - \beta \s^{-p}(\alpha) (-p,p)$$ $$=
\alpha \s^p(\beta) (p,-p) - \beta \s^{-p}(\alpha) \s^{-p}((p,-p)) =
(1-\s^{-p})(\alpha \s^p(\beta) (p,-p)).$$ Since $p \geq 2$ (or $P\notin A_{1, \leq 1}$), $0 = \deg 1 = \deg\, (1-\s^{-p})(\alpha \s^p(\beta) (p,-p)) =
\deg \alpha + \deg \beta +\deg\, (p,-p) -1$ (by Equation (\[degsf\])) $\geq 0+0+p-1 \geq 2-1=1$, a contradiction.

[*(iii)*]{} $(m(P), m(Q)) \neq (1,2)$: Suppose that $m(P)=1$ and $m(Q)=2$. Then $P = \alpha X^p$ for some $p\geq 2$ and $Q=Q_s + Q_q$ where $Q_s \in A_{1,s}$, $Q_q \in A_{1,q}$ and $s<q$. By Lemma \[a20Sep16\], the equality $[P,Q]=1$ implies that $[P, Q_s]=1$ and $[P, Q_q]=0$. By the case (ii), this is not possible.

[*(iv)*]{} Suppose that $m(P)=2$ and $m(Q)=1$. Then $P = P_r + P_p$ and $Q=Q_q$. By Lemma \[a20Sep16\] the equality $[P,Q]=1$ implies that $[P_p, Q_q]=0$ and $[P_r, Q_q]=1$. Then, $q\geq 0$, by Lemma \[a20Sep16\]. The case $q=0$ is not possible since then both $P_r, Q_q\in K[H]$ and this would contradict the equality $[P_r, Q_q]=1$. Therefore, $q>0$. Then $P_r = \beta Y^q$ and $Q_q = \alpha X^q$ for some nonzero elements $\beta, \alpha \in K[H]$. Then $$-1 = [Q_q, P_r] = (1-\s^{-q}) (\alpha \s^p(\beta) (q,-q) )$$ implies that $0 = \deg (-1) = \deg\, (1-\s^{-q}) (\alpha \s^p(\beta) (q,-q) ) = \deg \alpha + \deg \beta +q -1$, by Equation (\[degsf\]). Hence, $q=1$, $\alpha, \beta \in K^*$ and $\beta = -\alpha^{-1}$. Then $P, Q \in A_{1,\leq 1}$, and, by the statement (i), the pair $(P,Q)$ is obtained from the pair $(Y,X)$ by applying an automorphism of $A_1$. [*(v)*]{} $(m(P), m(Q)) \neq (2,2)$: Since $m(P)=m(Q)=2$, we can write $P = P_r + P_p$ and $Q = Q_s + Q_q$ as sums of homogeneous elements where $r < p$, $P_r \in A_{1,r}$, $P_p \in A_{1,p}$ and $s<q$, $Q_s \in A_{1,s}$, $Q_q \in A_{1,q}$. The equality $[P,Q]=1$ implies that $[P_r, Q_s]=0$ and $[P_p, Q_q]=0$ (see Lemma \[a20Sep16\]). By Lemma \[a20Sep16\], the elements $r$ and $s$ have the same sign (i.e., either $r<0, s<0$ or $r=s=0$ or $r>0, s>0$) and also the elements $p$ and $q$ have the same sign. Since $p\geq 2$, we must have $q>0$.

Suppose that $r\geq 0$, we seek a contradiction. Then $s\geq 0$ and so the elements $P$ and $Q$ are elements of the subring $A_{1,+}= \oplus_{i \geq 0} K[H] X^i$. Now, $$K[H] \ni 1 = [P, Q] \in [A_{1,+}, A_{1,+}] \subseteq \oplus_{i \geq 1} K[H] X^i,$$ a contradiction. Therefore, $r<0$ and $s<0$.

The equality $1=[P,Q]=[P_r, Q_q] + [P_p, Q_s]$ and Lemma \[a20Sep16\] imply that $r+q=0$ and $p+s=0$, that is $r=-q$ and $s=-p$. So, $$P = P_{-q} + P_p \text{ and } Q = Q_{-p} + Q_q.$$ The elements $P_p$ and $P_{-q}$ are homogeneous elements of the Weyl algebra $A_1$. The Weyl algebra $A_1$ is a homogeneous subalgebra of the algebra $K(H)[X, X^{-1}; \s] = K(H)[Y, Y^{-1}; \s^{-1}]$ where $K(H)$ is the field of rational functions in the variable $H$ and the automorphism $\s$ of $K(H)$ is given by the rule $\s(H) = H-1$. By [@Bav-DixPr5 Proposition 2.1(1)], the centralizer $C_B(P_p)$ of the element $P_p$ in $B$ is a Laurent polynomial algebra $K[\alpha X^n, (\alpha X^n)^{-1}]$ for some nonzero element $\alpha\in K(H)$ and $n \geq 1$. In general, $\alpha \notin K[H]$. Similarly, $C_B(P_{-q}) = K[\beta Y^m, (\beta Y^m)^{-1}]$ for some nonzero element $\beta \in K(H)$ and $m \geq 1$.

Since $[P_p, Q_q]=0$, $Q_q \in C_B(P_p)$ and $$P_p = \lambda(P_p) (\alpha X^n)^i =
\lambda(P_p) \alpha  \s^n(\alpha) \cdots \s^{n(i-1)}(\alpha) X^{ni} = \alpha_{n,i} X^p,$$ $$Q_q = \lambda(Q_q) (\alpha X^n)^j = \lambda(Q_q) \alpha \s^n(\alpha) \cdots \s^{n(j-1)}(\alpha) X^{nj} = \alpha'_{n,j} X^q,$$ for some nonzero scalars $\lambda(P_p),  \lambda(Q_q) \in K^*$ and some $i \geq 1$ and $j\geq 1$ where $$\alpha_{n,i} = \lambda(P_p) \alpha  \s^n(\alpha) \cdots \s^{n(i-1)}(\alpha) \in K[H], \; p=ni,$$ $$\alpha'_{n,j}  = \lambda(Q_q) \alpha  \s^n(\alpha) \cdots \s^{n(j-1)}(\alpha) \in K[H], \; q=nj.$$

Since $[P_{-p}, Q_{-p}]=0$, $Q_{-p} \in C_B(P_{-q})$ and $$P_{-q} = \lambda(P_{-q}) (\beta Y^m)^s =
\lambda(P_{-q}) \beta \s^{-m}(\beta)\cdots \s^{-m(s-1)}(\beta) Y^{ms} = \beta_{m,s} Y^p,$$ $$Q_{-p} = \lambda(Q_{-p}) (\beta Y^m)^t =
\lambda(Q_{-p}) \beta\s^{-m}(\beta)\cdots \s^{-m(t-1)}(\beta) Y^{mt} = \beta'_{m,t} Y^q,$$ for some nonzero scalars $\lambda(P_{-q}),  \lambda(Q_{-p}) \in K^*$ and some $s \geq 1$ and $t\geq 1$ where $$\beta_{m,s} = \lambda(P_{-q}) \beta \s^{-m}(\beta) \cdots \s^{-m(s-1)}(\beta) \in K[H],\;  p=ms,$$ $$\beta'_{m,t}  = \lambda(Q_{-p})\beta \s^{-m}(\beta) \cdots \s^{-m(t-1)}(\beta) \in K[H], \; q=mt.$$

Now, $$1 = [P,Q] = [P_p, Q_{-p}] + [P_{-q}, Q_q]
 = [\alpha_{n,i} X^p, \beta_{m,t}' Y^p ] + [\beta_{m,s} Y^q , \alpha_{n,j}' X^q]$$ $$= \alpha_{n,i} \s^p (\beta'_{m,t}) (p,-p) - \beta'_{m,t} \s^{-p}(\alpha_{n,i}) (-p,p)$$ $$+
\beta_{m,s} \s^{-q}(\alpha'_{n,j}) (-q,q) - \alpha'_{n,j} \s^{q}(\beta_{m,s}) (q,-q).$$

Using the equalities $(-p,p)=\s^{-p}((p,-p))$ and $(-q,q)=\s^{-q}((q,-q))$, the last equality above can be rewritten as follows $$\label{1=ab}
1 = (1-\s^{-p})(a) + (1-\s^{-q})(b)$$ where $a = \alpha_{n,i} \s^{p}(\beta'_{m,t}) (p,-p) \in K[H]$ and $b = \alpha'_{n,j} \s^{q}(\beta_{m,s}) (q,-q) \in K[H]$.

Recall that $P = P_{-q} + P_p$, $Q = Q_{-p} + Q_q$, $$\label{2=ab}
p = mt=ni \geq 2 \text{ and } q=ms=nj \geq 1.$$

Suppose that $p=q$, and so $P = P_{-p} + P_p$, $Q = Q_{-p} + Q_p$. Then $Q=\lambda P_p$ for some $\lambda\in K^*$. Since $1=[P,Q]=[P, Q-\lambda P]$, $m(P)=2$ and $m(Q-\lambda P)=1$. By the case (iv), the pair $(P, Q-\lambda P)$ is obtained from the pair $(Y, X)$ by applying an automorphism of the Weyl algebra $A_1$. So, either $p<q$ or $p>q$. In view of $(P,Q)$-symmetry ($1=[P,Q]=[-Q,P]$), it suffices to consider, say, the first case only. Since $p<q$, the equalities (\[2=ab\]) imply that $i<j$ and $t<s$. Then, using Equation (\[degsf\]) and the fact that $\deg (p,-p) = p$ for all $p \geq 1$, we see that $$\deg a = \deg \alpha_{n,i} + \deg \beta'_{m,t} + p -1,$$ $$\deg b = \deg \alpha'_{n,j} + \deg \beta_{m,s} + q -1.$$ Since $i<j$ and $t<s$, $\deg \alpha_{n,i} < \deg \alpha'_{n,j}$ and $\deg \beta'_{m,t} < \deg \beta_{m,s}$. In particular, $\deg a < \deg b$. This equality contradicts Equation (\[1=ab\]) since, by Equation (\[degsf\]), $$0=\deg 1 = \deg a - 1 - \deg b +1 = \deg a - \deg b >0.$$ This means that the cases $p<q$ and $p>q$ are impossible. The proof of the theorem is complete. $\Box$

\[24Sep16\] Let $P, Q$ be elements of the first Weyl algebra $A_1$ with $m(P)=1$ or $m(Q)=1$. If $[P,Q]=1$ then $P = \tau(Y)$ and $Q=\tau(X)$ for some automorphism $\tau\in\Aut_K (A_1)$.

[**Proof:**]{} Without loss of generality we may assume $m(Q)=1$ and $m(P)\geq 3$. That is $Q = Q_q$ and $P = \sum_{i\in I} P_i$, where $I \subset \Z$ is a finite set, $q\in \Z\setminus\{0\}$ and the elements $Q_q$ and $P_i$ are homogeneous in $A_1$. By Equation (\[xiaut1\]), we may assume that $q>0$. Then $1 = [P,Q]=\sum_i [P_i, Q_q]$ implies that $-q\in I$, $[P_{-q}, Q_q]=1$ and $[P_j, Q_q]=0$ for all $j\in I$ such that $j\neq -q$. By Theorem \[20Sep16\], $$q=1, Q_1 = \l X \text{ and } P_{-1} = \l^{-1}Y \text{ for some }\l\in K^*.$$ By Lemma \[a20Sep16\], $C := P - P_{-1} \in C_A(X)=K[X]$. Then $P = \tau(Y)$ and $Q=\tau(X)$ where $\tau: A_1 \to A_1$, $X \mapsto \l X$, $Y \mapsto \l^{-1} Y + C$, is an automorphism.

Acknowledgements {#acknowledgements .unnumbered}
================

This paper has been written during the visit of V. V. Bavula to Aachen in 2016, which was supported by the Graduiertenkolleg “Experimentelle und konstruktive Algebra" of the German Research Foundation (DFG). The second author has been supported by Project II.6 of SFB-TRR 195 “Symbolic Tools in Mathematics and their Applications” of the DFG.

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