# Importance of Environment Design in Reinforcement Learning: A Study of a
Robotic Environment

Mónika Farsang  Dr. Luca Szegletes

###### Abstract

An in-depth understanding of the particular environment is crucial in
reinforcement learning (RL). To address this challenge, the decision-making
process of a mobile collaborative robotic assistant modeled by the Markov
decision process (MDP) framework is studied in this paper. The optimal state-
action combinations of the MDP are calculated with the non-linear Bellman
optimality equations. This system of equations can be solved with relative
ease by the computational power of Wolfram Mathematica, where the obtained
optimal action-values point to the optimal policy. Unlike other RL algorithms,
this methodology does not approximate the optimal behavior, it gives the
exact, explicit solution, which provides a strong foundation for our study.
With this, we offer new insights into understanding the action selection
mechanisms in RL by presenting various small modifications on the very same
schema that lead to different optimal policies.

Importance of Environment Design in Reinforcement Learning: A Study of a
Robotic Environment

Mónika Farsang and Dr. Luca Szegletes
---

Department of Automation and Applied Informatics
Budapest University of Technology and Economics

<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
---

  * Keywords: Bellman equation; Environment design; Markov decision process; Optimal policy; Reinforcement learning;

##  Section 1 Introduction

A number of approaches exist to find the optimal policy in reinforcement
learning. Even with the best methods, there is a chance that we cannot achieve
significant results. In this case, the explanation is often related to the
insufficient reward design in the given setting. For this reason, shaping the
reward function is increasingly becoming a vital factor beyond building
efficient RL algorithms in order to obtain the desired behavior.

The motivation of this paper is to give novel visual representations of the
importance of environment design in reinforcement learning by showing the
connection between the reward function and the optimal behaviour. We also
analyze the impact of the stochasticity on the action-selection mechanism.
With them, we would like to show that small changes in the functions can
result unexpectedly in new optimal policies. We study a mobile robotic
assistant in a high-level abstraction of the MDP formulation to handle this
challenging task. For finding the optimal policy, the Bellman optimality
equations are computed directly in this paper.

Several methods have been proposed finding an optimal or near-optimal policy.
Policy iteration [1] and value iteration [2] are classical approaches to solve
the Bellman equations. These dynamic programming (DP) techniques [3] use the
combination of policy evaluation and policy improvement. The former refers to
the iterative computation of the value functions for a given policy. The
latter means the computation of an improved policy for a given value function.
Moreover, these techniques give the foundations of temporal difference (TD)
learning proposed by Bertsekas and Tsitsiklis [4] and actor-critic algorithms
by Konda and Borkar [5] as well. The advantage of TD methods over DP is that
the reward function and transition probability distributions are not required
[6].

Many RL approaches estimate the action-value functions from experience. If the
agent performs each action in each state and stores the returns followed by
them, the average return for each state-action pair will converge to the true
values. This kind of technique is used by the Monte Carlo method, where the
values are based on the average over many random samples from the environment
[7]. In the case of complex environments, the values are estimated by
parameterized functions, where the parameters are adjusted in order to match
the observed returns [6].

Watkins and Dayan developed Q-learning [8], which is a simulation-based
tabular method, where arrays or tables are used to store the approximations
for each state-action pair. This widely used TD technique gives the basis of
several algorithms. Later, it was combined with deep learning, resulting in
Deep Q-Networks [9].

Designing the proper reward function for a RL environment is a challenging
task. The difficulty of having the right reward function can be presented by
the credit assignment problem discussed by Minsky [10], where the problem is
the determination of the actions that lead to a specific outcome. As another
aspect, shaping the reward function to accelerate learning often requires
domain-expertise, demonstrated in [11].

Experiments on sparse reward problems were conducted in recent years [12, 13,
14]. Sparse reward problems occur even in simple tasks when the number of non-
zero rewards is limited and encounters far away from each other in time. This
usually leads to an agent behaving randomly for longer periods in the
environment without gaining more experience.

As another approach, MDPs with unknown reward functions were investigated via
inverse reinforcement learning (IRL) by Ng and Russell [15]. IRL showed to be
useful for apprenticeship learning, where the goal is to learn the reward
function of the expert from its behavior, which is a linear combination of
known features [16].

##  Section 2 Preliminaries

A fundamental approach in reinforcement learning is formulating problems as
MDPs [17], which can be used in sequential decision making in stochastic
environments. The formalism of an MDP includes sets of states $S$, actions $A$
and rewards $R$. The agent makes decisions $a_{t}\in A(s_{t})$ based on the
response of the environment at each timestep $t$. With the action $a_{t}$, it
senses the next state $s_{t+1}\in S$ and receives reward $r_{t+1}\in R$. The
transition probability distributions, denoted by $p$, are dependent only on
the preceding state $s_{t}$ and action $a_{t}$ as shown in Eq. (1).
Consequently, these processes have the Markov property, which means that the
future depends only on the present state and the past is not involved.

$\sum_{s^{\prime}\in S}\sum_{r\in R}p(s^{\prime},r\mid s,a)=1,\text{ for all
}s\in S,a\in A(s)$ (1)

The particular way of selecting actions in each state is defined by the policy
$\pi(a\mid s)$, $a\in A(s)$ for each $s\in S$. To measure the performance of
the agent, the action-value function for policy $\pi$ is defined as can be
seen in Eq. (2). It formulates the expected return when starting with the
action $a$ in state $s$ and the present values of the future rewards are
determined by the discount factor $0\leq\gamma\leq 1$.

$q_{\pi}(s,a)=\operatorname{\mathbb{E}}\left[{\sum_{k=0}^{\infty}\gamma^{k}r_{t+k+1}\mid
s,a}\right]$ (2)

Computing the action-value functions can be done by using the Bellman equation
in Eq. (3), which expresses the relation between the current action-value and
its successor action-values [6]. It gives the sum of all possibilities taken
action $a$ in state $s$ weighted by the transition probability $p$.

$q_{\pi}(s,a)=\sum_{s^{\prime},r}p(s^{\prime},r\mid
s,a)\left[r+\gamma\sum_{a^{\prime}}\pi(a^{\prime}\mid
s^{\prime})q_{\pi}(s^{\prime},a^{\prime})\right]$ (3)

A policy is considered to be optimal if and only if $q_{\pi}(s,a)\geq
q_{\pi^{\prime}}(s,a)$, for all $a\in A(s)$ $s\in S$, where $\pi$ is the
current policy and $\pi^{\prime}$ is any other policy [6]. In other words, a
policy is better than any other if its expected return in every state-action
pair situation is greater than or equal to values with the same state-action
combinations of other policies. This also means there could be multiple
optimal policies, which are denoted by $\pi^{*}$. The solutions of the Bellman
optimality equations, which is shown in Eq. (4), share the same value of
optimal action-value functions $q_{*}(s,a)$.

$q_{*}(s,a)=\sum_{s^{\prime},r}p(s^{\prime},r\mid
s,a)\left[r+\gamma\underset{a^{\prime}}{\text{max}}q_{*}(s^{\prime},a^{\prime})\right]$
(4)

Eq. (4) formulate a non-linear equation for each state-action pair. The system
of equations with the known environment behavior can be solved with relatively
small effort. As a result, having the action-values leads to the optimal
policy. Namely, the optimal action of the state maximizes the $q_{*}(s,a)$
value in the given state.

The abstract form of the MDPs provides a wide scale of applications. For
example, low-level states and actions can be torque sensor readings and
voltages applied to motors of a robotic joint. In contrast, high-level
abstractions can be symbolic descriptions of an object, such as the place of
the agent in a grid space with actions turning right or left.

##  Section 3 Methods

This section first outlines the high-level abstraction form of the robotic
environment examined in our study. In the second subsection, we present the
used method of the optimal behavior calculation.

###  3.1 Environment

We now describe the experiment in which we analyze how the RL environment
affects the optimal behavior. In our scenario, the task of a mobile
collaborative robotic assistant is to collect empty soda cans in an office
area. This problem is based on the recycling robot example by Sutton and Barto
[6], where the authors claimed that it was inspired by the mobile robot built
by Connell [18].

This environment is modeled by the MDP framework with high-level abstraction,
where the state set $S=\\{\text{High, Low}\\}$ consists of the rechargeable
battery levels of the robot. Its possible actions are searching actively for
empty cans, waiting for someone to bring it one and recharging its own
battery.

Concerning the action sets $A(s=\text{High})=\\{\text{search\textsubscript{H},
wait\textsubscript{H}}\\}$ and
$A(s=\text{Low})=\\{\text{search\textsubscript{L}, wait\textsubscript{L},
recharge\textsubscript{L}}\\}$, the subscripts refer to the state where the
action is performed. These sets are different in the two states because the
recharging action in the high energy level state is unreasonable. The state-
action combinations with the resulting states, their transition probabilities
and rewards are described in Table 1, where the notation is based on the work
[6] with small modifications.

Table 1: Robotics environment dynamics State-action pairs with next state | Environment
---|---
state | action | next state | transition | reward
$s$ | $a$ | $s^{\prime}$ | $p(s^{\prime}\mid s,a)$ | $r(s,a,s^{\prime})$
High | searchH | High | $\alpha$ | $r_{search}$
Low | $1-\alpha$ | $r_{search}$
waitH | High | $1$ | $r_{wait}$
Low | searchL | High | $1-\beta$ | $r_{rescue}$
Low | $\beta$ | $r_{search}$
waitL | Low | $1$ | $r_{wait}$
rechargeL | High | $1$ | $r_{recharge}$
Figure 1: Transition graph of the MDP in the robotic assistant environment.
The arrows between the selected action (blue) and the resulting state (white)
nodes are labeled by the transition probability and the corresponding reward.

Figure 1 presents the transition graph of the MDP. The states are represented
by the white circles, where each state can be considered as the initial one.
The corresponding actions are indicated by blue color. From the state nodes
black edges connect to the action nodes indicating decision making. In
reverse, blue edges labeled with the transition probability and reward pair
come from the action nodes to the resulting state nodes.

The searching action in the high energy level state can result in two
different outcomes. Meanwhile, the state stays the same with the probability
$\alpha$, due to the consumed energy it switches to the low-power mode with
$1-\alpha$. Similarly, when this action is performed in a low energy state,
the next state will be the same with probability $\beta$ and the battery can
become empty with the probability $1-\beta$, where the office personnel must
rescue the mobile robot and bring it to the recharging station.

The waiting action does not change the state but the number of collected cans
highly depends on the people in the environment. For this reason, it is
important to decide how desirable is to encourage this passive behavior.
Finally, with choosing the recharging action type, the mobile robot assistant
can recharge its battery when it is low, resulting in the high energy level
state.

###  3.2 Calculation of the optimal actions

The aim of this paper is to give reliable results without the possible
divergence of the chosen RL algorithm. Consequently, we calculate the action-
value functions for the optimal policy based on the non-linear Bellman
optimality equation as shown in Eq. (4). This technique was chosen because it
is the most precise way to solve MDP problems.

With this classical method in the described environment, two equations
correspond to the high energy level and three to the low energy level state
with respect to the state-action pairs. To solve the system of equations,
Wolfram Mathematica [19] was used thanks to its computational power.

For the four different rewards ($r_{search}$, $r_{wait}$, $r_{recharge}$,
$r_{rescue}$) and the transition probabilities ($\alpha$, $\beta$), a six-
dimensional graph would show perfectly the impact of the parameters on the
optimal actions in each state. Since it is not achievable, we present several
3D graphs and 2D plots in Figure 2, Figure 3 and 4, which are easy to
understand and give new insights on the optimal behavior.

The not displayed parameters remain constant during each analysis and their
specific values can be seen in the Appendix. To gain more experiments, our
Wolfram Demonstration Project provides a much simpler but more interactive
platform for the same problem at
https://demonstrations.wolfram.com/RecyclingRobotInReinforcementLearning/. The
Wolfram Notebook with the calculation and the graphs is publicly available at
https://www.wolframcloud.com/obj/farsangmonika95/Published/ImportanceofEnvironmentDesigninRL.nb.

##  Section 4 Results and discussion

This section begins by examining how the simultaneous alternation of two
environmental parameters modify the optimal action. In this case, rewards of
two actions and the two transition probabilities are analyzed. The second
subsection investigates the impact of reward function by separately changing
the rewards of the actions.

###  4.1 Experiments of simultaneously changing the reward function of two
actions and the transition probabilities

In our first set of experiments, we investigate the impact of two
environmental attributes on the optimal policy. Figure 2LABEL:sub@fig:3da and
2LABEL:sub@fig:3db present the searching and waiting reward connection to the
optimal action-value with the high-level and the low-level battery
respectively.

The high energy level state has only two possible actions, which are searching
and waiting, and exactly their reward shaping is analyzed here. In this case,
it is self-evident that the action with a higher reward becomes optimal in
this state. Meanwhile, it is not obvious how the optimal action changes in the
low-power mode. The searching reward has an indirect impact on the recharging
action and the searching action never becomes optimal in this scenery.
Although, it is worth mentioning, that with other fixed parameter setting it
has the possibility to get the searching action optimal with the low-level
battery as well. For example, as can be seen in Figure 2LABEL:sub@fig:3dd, the
modified transition probabilities can lead to a completely different optimal
policy.

(a)

(b)

(c)

(d)

Figure 2: Examination of the reward function and the transition probabilities.
(a) Impact of the searching and waiting reward on the high energy level state.
(b) Same inspection as (a), but on the low energy level state. (c) Influence
of the transition probabilities on the high-level battery. (d) Same analysis
as (c), but on the low-level battery.

In Figure 2LABEL:sub@fig:3dc and 2LABEL:sub@fig:3dd, we analyze the transition
probabilities of the environment. They determine the probability of the next
state after performing the searching action in high energy level state
($\alpha$) and low energy level state ($\beta$). Interestingly, we found that
the modification of these values does not change the optimal action with the
high-level battery, where the plane of searching action remains above the
waiting action-values in the whole range. However, the optimal action of the
low-power mode can be immensely influenced simply by the transition
probabilities. Here, the recharging action is optimal in most of the cases,
but if the transition probability $\beta$ has a high enough value, the
searching action exceeds that and become the optimal choice.

###  4.2 Experiments of separately changing the reward function of two actions

The second set of our experiments contain simpler two-dimensional plots, where
we investigated how the minor changes in one specific parameter affect the
chosen optimal actions. Figure 3 and 4 illustrate the effect of the rescuing
and the waiting reward separately, where the red vertical lines present the
boundary when the optimal policy changes in one of the two states.

#### Changing the rescuing reward

The rescuing reward occurs, if the robot starts searching with the low-level
battery. As can be seen in Figure 3LABEL:sub@fig:2d_rescue_a and
3LABEL:sub@fig:2d_rescue_b, positive rewards are not displayed because this
kind of behavior should be avoided. This penalty does influence the action-
values in the high and the low energy level states as well. On one hand, it
does not change the optimal searching action in the high energy level state
throughout the analysis. On the other hand, as the rescuing reward increases,
three different scenarios can be observed in the low energy level state.

(a)

(b)

Figure 3: Changes in optimal actions by modifying the rescue reward in the
high and low energy level states. (a) Effect of the rescue reward on the state
with high-level battery. (b) Same analysis as (a), but with low-level battery.

In the first period, the recharging action is the optimal choice. At one point
around -0.67, the searching action-value becomes identical with the value of
the recharging action, illustrated by the red vertical line. In this case,
both actions are considered to be optimal.

With steadily increasing the rescue reward, the searching action surpasses the
recharging action in the second phase and becomes the only optimal action in
the low-level battery state. The switching point of the optimal behavior can
be unexpected without this kind of analysis.

#### Changing the waiting reward

Figure 4LABEL:sub@fig:2d_wait_a and 4LABEL:sub@fig:2d_wait_b illustrate the
influence of the waiting reward, which is received when the mobile robot is
inactive and constantly waits for the office personnel to bring it cans. In
this case, negative and positive rewards are considered, because it depends on
the expected activity of the robotic assistant. There are five possible
optimal behaviors based on the value of this reward, represented by the
labeled sections and their boundaries.

(a)

(b)

Figure 4: Impact of the waiting reward on the high and low energy level
states. (a) Analyzing how the waiting reward affects the state with high level
battery. (b) Same examination as (a), but the low-power state.

In the first phase, the searching and recharging actions are optimal in the
high and the low energy level state respectively. As the value of the waiting
action increases in the low-power mode, it reaches the recharging value and
both become optimal at around 0.55, presented in Figure
4LABEL:sub@fig:2d_wait_b.

In the second part, the searching action in the high and the waiting action in
the low energy level state are the best choices. Later, the action-value of
the waiting action rapidly increases and both actions become optimal when the
waiting reward equals 1 in the high-level battery state, represented by the
second vertical red line in Figure 4LABEL:sub@fig:2d_wait_a.

With the further increment of the waiting reward, the corresponding waiting
action is the only optimal decision in both states in the third section. This
differs hugely from the optimal actions in the first part, even though there
is only a small growth in the reward of the waiting.

This last part of our experiments demonstrates that the same reward can
influence differently the states caused by the transition probabilities.
Moreover, it shows that the modification of the reward function in a
relatively small range has a huge effect on the optimal behavior in both
states, resulting even in multiple optimal actions.

##  Section 5 Conclusion

In this paper, we have investigated the impact of the environment design on
reinforcement learning problems. For this, we selected an MDP robotic task,
where several adjustments were presented in the environment. The results of
this study support the idea that even small differences in the chosen reward
function and transition probabilities can affect the outcome.

Our research has highlighted the importance of these aspects by the direct
computation of the non-linear Bellman optimality equations and novel
visualization techniques by showing the connection between the optimal action-
values and the environmental parameters. We suggest this method of analysis in
each environment where it is applicable to understand the underlying cause of
the learned behavior.

## Acknowledgments

The author would like to express her thanks to Dr. Luca Szegletes for her
support as a scientific advisor. This work has been supported by the Budapest
University of Technology and Economics.

## Appendix

Table 2: Environment parameters in the different experiments Parameter | Values in the experiments
---|---
Exp. 1. | Exp. 2. | Exp. 3. | Exp. 4.
Figure 2LABEL:sub@fig:3da, | Figure 2LABEL:sub@fig:3dc, | Figure 3LABEL:sub@fig:2d_rescue_a, | Figure 4LABEL:sub@fig:2d_wait_a,
2LABEL:sub@fig:3db |  2LABEL:sub@fig:3dd |  3LABEL:sub@fig:2d_rescue_b |  4LABEL:sub@fig:2d_wait_b
$r_{rescue}$ | -3 | -3 | -2.5$\dots$0 | -3
$r_{recharge}$ | 0 | 0 | 0 | 0
$r_{wait}$ | -5$\dots$5 | 0 | 0 | -2$\dots$2
$r_{search}$ | 0$\dots$5 | 1 | 1 | 1
$\alpha$ | 0.3 | 0$\dots$1 | 0.3 | 0.3
$\beta$ | 0.1 | 0$\dots$1 | 0.6 | 0.1
$\gamma$ | 0.9 | 0.9 | 0.9 | 0.9

## References

  * [1] R. A. Howard, Dynamic Programming and Markov Processes.  Cambridge, MA: MIT Press, 1960.
  * [2] M. L. Puterman and M. C. Shin, “Modified policy iteration algorithms for discounted markov decision problems,” Manage. Sci., vol. 24, pp. 1127–1137, July 1978.
  * [3] R. Bellman, Dynamic Programming.  Princeton University Press, 1 ed., 1957.
  * [4] D. P. Bertsekas and J. N. Tsitsiklis, Neuro-Dynamic Programming.  Athena Scientific, 1996.
  * [5] V. R. Konda and V. S. Borkar, “Actor-critic–type learning algorithms for markov decision processes.,” SIAM J. Control and Optimization, vol. 38, pp. 94–123, June 1999.
  * [6] R. S. Sutton and A. G. Barto, Reinforcement Learning: An Introduction.  The MIT Press, second ed., 2018.
  * [7] S. P. Singh and R. S. Sutton, “Reinforcement learning with replacing eligibility traces,” Mach. Learn., vol. 22, pp. 123–158, Jan. 1996.
  * [8] C. J. C. H. Watkins and P. Dayan, “Q-learning,” Machine Learning, vol. 8, no. 3, pp. 279–292, 1992.
  * [9] V. Mnih, K. Kavukcuoglu, D. Silver, A. Graves, I. Antonoglou, D. Wierstra, and M. A. Riedmiller, “Playing atari with deep reinforcement learning,” CoRR, vol. abs/1312.5602, 2013.
  * [10] M. Minsky, “Steps toward artificial intelligence,” Proc. IRE, Jan. 1961\.
  * [11] M. J. Mataric, “Reward functions for accelerated learning,” in International Conference on Machine Learning, pp. 181–189, Morgan Kaufmann, 1994\.
  * [12] D. Pathak, P. Agrawal, A. A. Efros, and T. Darrell, “Curiosity-driven exploration by self-supervised prediction,” 2017 IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), pp. 488–489, 2017\.
  * [13] Y. Burda, H. Edwards, A. Storkey, and O. Klimov, “Exploration by random network distillation,” arXiv preprint arXiv:1810.12894, 2018.
  * [14] A. Trott, S. Zheng, C. Xiong, and R. Socher, “Keeping your distance: Solving sparse reward tasks using self-balancing shaped rewards,” in NeurIPS, 2019\.
  * [15] A. Ng and S. Russell, “Algorithms for inverse reinforcement learning,” International Conference on Machine Learning, 05 2000.
  * [16] P. Abbeel and A. Y. Ng, “Apprenticeship learning via inverse reinforcement learning,” in International Conference on Machine Learning, 2004.
  * [17] C. J. C. H. Watkins, Learning from Delayed Rewards.  PhD thesis, King’s College, Cambridge, UK, May 1989.
  * [18] R. A. Brooks, J. H. Connell, and P. Ning, “Herbert: A second generation mobile robot,” tech. rep., USA, 1988.
  * [19] Wolfram Research, Inc., “Mathematica, Version 12.2.”  Champaign, IL, 2020.


The purpose of this paper has two fold. The first is to establish a second main theorem for meromorphic functions on annuli and meromorphic function targets (may not be small functions) with truncated counting function (truncation level 1) and with more detailed estimate for error term. The second is to show that if the polynomial

$$P_S(w)=(w-a_1)\cdots (w-a_q)$$

is a uniqueness polynomial for meromorphic functions on an annulus $\A(R_0)$ such that $P'_S(w)$ has exactly $k$ distinct zero and $q>\frac{(5k+3)l}{2l-175}$ then the set $S=\{a_1,\ldots,a_q\}$ is a finiteness range set with truncation level $l$ for admissible meromorphic functions on $\A(R_0)$. This result extends the previous result on the finiteness range set (with truncation level $l=\infty$) for holomorphic functions on $\C$ of H. Fujimoto.

[2010 Mathematics Subject Classification: Primary 30D35.]

[Key words and phrases: Meromorphic function, Nevanlinna theory, the annulus, finiteness range set.]

§ INTRODUCTION

A nonconstant monic polynomial $P(w)$ is called a uniqueness polynomial for meromorphic functions (or holomorphic functions) if, for any nonconstant
meromorphic functions (or holomorphic functions) $f$ and $g$ on $\C$, the equation $P(f)=cP(g)$ implies $f=g$, where $c$ is a nonzero constant which possibly depends on $f$ and $g$.
A fininite subset $S$ of $\C$ is said to be a uniqueness range set for meromorphic functions (or holomorphic functions) if $f^*(S)=g^*(S)$ implies $f=g$ for arbitrary nonconstant meromorphic functions (or holomorphic functions) $f$ and $g$ on $\C$, where $f^*(S)$ and $g^*(S)$ denote the pull-backs of $S$ considered as a divisor. For $S=\{a_1,\ldots,a_q\}$, we consider the polynomial

\begin{align}\label{1.1}
\end{align}

Hence, if $S$ is a uniqueness range set for meromorphic functions (or holomorphic functions), then $P_S(w)$ is a uniqueness polynomial for meromorphic functions (or holomorphic functions).

In [2], Fujimoto extended the notion of unique range set to the following.

Definition A. A finite subset $S$ of $\C$ is called a finiteness range set for holomorphic functions if, for any given nonconstant holomorphic function $g$, there exist only finitely many nonconstant holomorphic functions $f$ such that $f^*(S)=g^*(S)$.

With the above definition, Fujimoto proved a sufficient condition for a finiteness range set for holomorphic functions on $\C$ as follows.

Theorem B. Take a finite set $S=\{a_1,\ldots,a_q\}$ and assume that, for the polynomial $P_S(\omega)$ defined by (<ref>), $P'_S(w)$ has exactly distinct $k$ zeros. If $P_S(w)$ is a uniqueness polynomial for holomorphic functions and $q>k+2$, then $S$ is a finiteness range set for holomorphic functions. More precisely, for an arbitrarily given nonconstant holomorphic function $g$, there exist at most $\frac{2q-2}{q-k-2}$ holomorphic functions $f$ such that $f^*S=g^*S$

Our purpose in this paper is to extend the result of Fujimoto to the case where the meromorphic functions (not only holomorphic) are defined only on annuli and the condition $f^*(S)=g^*(S)$ is replaced by a more general condition that $f$ and $g$ share the finite set $S$ with multiplicity truncated by a certain level. In order to state our result, we give the following definition.

Let $S=\{a_1,\ldots,a_q\}$ be a set of distinct values on $\A(R_0)$ and $l$ be a positive integer (may be $l=+\infty$). Two admissible meromorphic functions (or admissible holomorphic functions) $f$ and $g$ is said to share $S$ with multiplicity truncated to level $l$ if

The set $S$ is said to be a finiteness range set with truncation level $l$ for admissible meromorphic functions if for an arbitrarily given admissible meromorphic function $g$, there exist only many finite admissible meromorphic functions $f$ such that $f$ and $g$ share $S$ with multiplicity truncated to level $l$.

Here, a meromorphic function $f$ on $\A(R_0)$ is said to be admissible if it satisfies

$$ \underset{r\longrightarrow +\infty}{\mathrm{limsup}}\dfrac{T_0(r,f)}{\log r}=+\infty\text{ in the case }R_0=+\infty $$

$$   \underset{r\longrightarrow R_0^-}{\mathrm{limsup}}\dfrac{T_0(r,f)}{-\log (R_0-r)}=+\infty\text{ in the case }1< R_0<+\infty,$$

where $T_0(r,f)$ denotes the characteristic function of the meromorphic function $f$, which is defined in the section 2.
Hence, the condition that $f$ and $g$ share $S$ with multiplicity truncated to level $l=+\infty$ is equivalent to the condition $f^*(S)=g^*(S)$ in the sense of divisors. Therefore, $S$ a finiteness range set with truncation level $+\infty$ for admissible meromorphic functions if and only if it is a finiteness range set for admissible meromorphic functions.

Our main result is stated as follows.

Take a finite set $S=\{a_1,\ldots,a_q\}$ of $q$ distinct values on an annulus $\A(R_0)$ and assume that, for the polynomial $P_S(\omega)$ defined by (<ref>), $P'_S(w)$ has exactly distinct $k$ zeros. Let $l$ be a positive integer (may be $l=+\infty$). If $P_S(w)$ is a uniqueness polynomial for admissible meromorphic functions and $q>\frac{(5k+3)l}{2l-175}$, then $S$ is a finiteness range set with truncation $l$ for admissible meromorphic functions. More precisely, for an arbitrarily given admissible meromorphic function $g$, there exist at most four admissible meromorphic functions $f$ such that $f$ and $g$ share the set $S$ with multiplicity truncated to level $l$.

Here, we would like emphasize that the proof of Fujimoto for Theorem B is based on the sharp second main theorem for meromorphic functions and small functions on $\C$ with counting functions truncated to level $1$ of Yamanoi <cit.>. By the assumption of Theorem B, Fujimoto assumed that $P_S(f_i)=e^{\varphi_i}P_S(g)$ with an holomorphic function $\varphi_i$ for each holomorphic function $f_i$ such that $f_i^*(S)=g^*(S)$ and $\varphi'_i$ is small function, and then he applied the second main theorem of Yamanoi. However, in Theorem <ref>, the holomorphic function $\varphi_i$ does not exists. Therefore we have to establish a new second main theorem for meromorphic functions with meromorphic function targets (may not be small) on annuli where the error term is more detailed estimated. Namely, we will prove the following second main theorem.

Let $g$ be a nonconstant meromorphic function on $\A (R_0)$. Let $a_1,\dots ,a_q$ be $q$ distinct meromorphic functions (may be identity $\infty$). We have

$$\frac{2q}{5}T_0(r,g)\le \sum_{i=1}^q\overline N_0(r,\nu^0_{g-a_i})+35\sum_{i=1}^qT_0(r,a_i)+S(r)\ (1\le r< R_0),$$

where $S(r)=S_g(r)+\sum_{i=1}^qS_{a_i}(r)$ (here we set $T_0(r,a_i)=0$ if $a_i\equiv\infty$).

Here, $\overline N_0(r,\nu)$ denotes the counting function without multiplicity of the divisor $\nu$ and the quantity $S_f(r)$ is the small term with respect to the meromorphic function $f$. These notion are defined in the section 2.

We note that, the weak version of this theorem for the case of small functions $a_i\ (1\le i\le 5)$ is initially stated in [6] without proving (by repeating the method in [8]). Then the proof in this paper will also provide a detail proof for that weak version, although the proof is based on the method of the proofs of Lemma 3.1 and Lemma 3.2 in [8].

§ SOME DEFINITIONS AND RESULTS FROM NEVANLINNA THEORY ON ANNULI

In this section, we will recall some basic notions of Nevanlinna theory for meromorphic functions on annuli from [4] (see also [3]).

For a divisor $\nu$ on $\A (R_0)$, which we may regard as a function on  $\A (R_0)$ with values in $\Z$ whose support is discrete subset of $\A (R_0),$ and  for a positive integer $M$ (maybe $M= \infty$), we define the counting function of $\nu$ as follows

\begin{align*}
\sum\limits_{1\le |z|\le t}\min\{M,\nu (z)\}&\text{ if }1\le t<R_0\\
\sum\limits_{t\le |z|<1}\min\{M,\nu (z)\}&\text{ if }\dfrac{1}{R_0}<t< 1
\end{cases}\\
\text{ and }N_0^{[M]}(r,\nu)&=\int\limits_{\frac{1}{r}}^1 \dfrac {n_0^{[M]}(t)}{t}dt +\int\limits_1^r \dfrac {n_0^{[M]}(t)}{t}dt \quad (1<r<\infty).
\end{align*}

For brevity we set $N_0(r,\nu)=N_0^{[\infty]}(r,\nu)$ and $\overline N_0(r,\nu)=N_0^{[1]}(r,\nu)$.

For a meromorphic function  $\varphi $, we define

* $\nu^0_\varphi$ (resp. $\nu^{\infty}_{\varphi}$) the divisor of zeros (resp. divisor of poles) of $\varphi$,

* $\nu_\varphi=\nu^0_\varphi -\nu^\infty_{\varphi},$

* $\nu^0_{\varphi,\ge k}=\max\{k,\nu^0_{\varphi}\}$.

Note that, by Jensen's formula we have

$$ N_0(r,\nu^0_\varphi)-N_0(r,\nu^\infty_\varphi)=\sum_{t=1/r,r}\dfrac{1}{2\pi}\int\limits_{0}^{2\pi}\log|\varphi (te^{i\theta})| d\theta - \dfrac{1}{\pi}\int\limits_{0}^{2\pi}\log|\varphi(e^{i\theta})| d\theta.$$

Let $f$ be a nonconstant meromorphic function on $\A(R)$. The proximity function of $f$ is defined by

$$ m_0(r,f)=\sum_{t=1/r,r}\dfrac{1}{2\pi}\int\limits_{0}^{2\pi}\log^+|f(te^{i\theta})| d\theta - \dfrac{1}{\pi}\int\limits_{0}^{2\pi}\log^+|f(e^{i\theta})| d\theta $$

and the characteristic function of $f$ is defined by

$$ T_0(r,f)=m_0(r,f)+N_0(r,\nu^\infty_f).$$

Now, we write $f=\frac{f_0}{f_1}$ where $f_0,f_1$ are two holomorphic functions without common zero. It is well-known that

\begin{align*}
\end{align*}

where $\|f\|=\sqrt{|f_0|^2+|f_1|^2}$ and $O(1)$ is a bounded term depending on the choice of $f_0,f_1$.

Throughout this paper, we fix a number $\lambda\ge 0$ and denote by $S_f(r)$ quantities satisfying:

(i) in the case $R_0=+\infty$,

$$ S_f(r)=O(\log (rT_0(r,f)))$$

for $r\in (1,+\infty)$ except for a set $\Delta_R$ such that $\int_{\Delta_R}r^{\lambda -1}dr <+\infty$,

(ii) in the case $R_0<+\infty$,

$$ S_f(r)=O\left (\log\biggl (\dfrac{T_0(r,f)}{R_0-r}\biggl )\right )\text{ as }r\longrightarrow R_0$$

for $r\in (1,R_0)$ except for a set $\Delta'_R$ such that $\int_{\Delta'_R}\dfrac{dr}{(R_0-r)^{\lambda +1}} <+\infty$.

Thus for an admissible meromorphic function $f$ on the annulus $\A(R_0)$, we have $S_f(r) = o(T_0(r, f ))$ as $r\longrightarrow R_0$ for all $1\le r<R_0$ except for the set $\Delta_R$ or the set $\Delta'_R$ mentioned above, respectively (cf. [3]).

A meromorphic function $a$ on $\A (R_0)$ is said to be small with respect to $f$ if

Let $f$ be a nonzero meromorphic function on $\A (R_0)$. Then for each $k\in\N$ we have

$$ m_0\left (r,\dfrac{f^{(k)}}{f}\right )=S_f(r)\ (1\le r <R). $$

Let $f$ be a nonconstant meromorphic function on $\A (R_0)$. Let $a_1,\dots ,a_q$ be $q$ distinct values in $\C\cup\{\infty\}$. We have

$$ (q-2)T_0(r,f)\le \sum_{i=1}^qN_0(r,\nu^0_{f-a_i})+S_f(r)\ (1\le r< R_0).$$

§ UNIQUENESS THEOREM FOR MEROMORPHIC FUNCTIONS SHARING FINITE SETS

Before proving Theorem <ref>, we first prove Theorem <ref>. In order to prove Theorem <ref>, we need the following lemmas.

Let $f_1,f_2$ be two meromorphic function on $\A(R_0)$, $a_1,a_2,a_3$ be three distinct meromorphic functions on $\A(R_0)$ (not identity $\infty$) such that $ f_2=(f_1-a_1)/(f_1-a_2).$ Then we have:

\begin{align*}
\mathrm{(a)}\ &T_0(r,f_2)\ge T_0(r,f_1)-\sum_{i=1}^2T_0(r,a_i)+O(1)\\
\mathrm{(b)}\ &\overline{N}_0(r,\nu^0_{f_2})+\overline{N}_0(r,\nu^0_{f_2-1})+\overline{N}_0(r,\nu^\infty_{f_2})\\
& \le \overline{N}_0(r,\nu^\infty_{f_1})+\sum_{i=1}^2\left(\overline{N}_0(r,\nu^0_{f_1-a_i})+2T_0(r,a_i)\right)+O(1).
\end{align*}

Moreover, if we set $b=\dfrac{a_3-a_1}{a_3-a_2}$ then

$$(c)\ \overline{N}_0(r,\nu^0_{f_2-b})\le\overline{N}_0(r,\nu^0_{f_1-a_3})+T_0(r,a_1)+2T_0(r,a_2)+T_0(r,a_3)+O(1).$$

We write $f_i=\frac{f_{i0}}{f_{i1}}$, where $f_{i0}$ and $f_{i1}$ are holomorphic functions defined on $\A(R_0)$ without common zeros for $i=1,2$. Similarly, we write  $a_i=\frac{a_{i0}}{a_{i1}}$, where $a_{i0}$ and $a_{i1}$ are holomorphic functions defined on $\A(R_0)$ without common zeros for $i=1,2,3$.  We have

\begin{align*}
\frac{f_{10}}{f_{11}}=\frac{f_{10}a_{11}a_{21}-f_{11}a_{10}a_{21}}{f_{10}a_{11}a_{21}-f_{11}a_{11}a_{20}}.
\end{align*}

Then, there is a holomorphic function $h$ and a positive constant $C$ such that

\begin{align*}
&hf_{11}=f_{10}a_{11}a_{21}-f_{11}a_{11}a_{20}, \\
\text{and hence }&|h|\cdot\|f_1\|\le C\|f_2\|\cdot\|a_1\|\cdot \|a_2\|.
\end{align*}

This implies that

$$ T_0(r,f_1)+N_0(r,h)\le T_0(r,f_2)+T_0(r,a_1)+T_0(r,a_1)+O(1).$$

$$T_0(r,f_2)\ge T_0(r,f_1)-T_0(r,a_1)-T_0(r,a_1)+O(1)$$

and we get the inequality (a).

From the definition $f_2=\frac{f_1-a_1}{f_1-a_2}$, we easily have

\begin{align*}
\overline{N}_0(r,\nu^0_{f_2})&+\overline{N}_0(r,\nu^0_{f_2-1})+\overline{N}_0(r,\nu^\infty_{f_2})\\
&\le \overline{N}_0(r,\nu^\infty_{f_1})+\sum_{i=1}^2\overline{N}_0(r,\nu^0_{f_1-a_i})+(\overline{N}_0(r,\nu^0_{a_1-a_2})+\overline{N}_0(r,\nu^\infty_{a_1})+\overline{N}_0(r,\nu^\infty_{a_2}))\\
&\le \overline{N}_0(r,\nu^\infty_{f_1})+\sum_{i=1}^2\left(\overline{N}_0(r,\nu^0_{f_1-a_i})+2T_0(r,a_i)\right)+O(1).
\end{align*}

Then we have the inequality (b).

Similarly we have

$$ \dfrac{1}{f_2-b}=\dfrac{(a_3-a_2)}{(a_1-a_2)}\cdot\left (1+\frac{a_3-a_2}{f_1-a_3}\right).$$

This implies that

$$ \min\{1,\nu^0_{f_2-b}\}\le\min\{1,\nu^\infty_{a_3-a_2}+\nu^0_{a_1-a_2}+\nu^0_{f_1-a_3}\}.$$

It yields that

$$ \overline{N}_0(r,\nu^0_{f_2-b})\le\overline{N}_0(r,\nu^0_{f_1-a_3})+T_0(r,a_1)+2T_0(r,a_2)+T_0(r,a_3),$$

an we get the inequality (c). This completes the proof of the lemma.

Let $f_1,f_2$ be two meromorphic functions on $\A(R_0)$, $a_1,a_2,a_3,a_4$ be four distinct meromorphic function on $\A(R_0)$ such that

$$ f_2=\frac{f_1-a_1}{f_1-a_2}\cdot\frac{a_3-a_2}{a_3-a_1}.$$

Then we have:

\begin{align*}
\mathrm{(a)}\ &T_0(r,f_2)\ge T_0(r,f_1)-\sum_{i=1}^3T_0(r,a_i)+O(1)\\
\mathrm{(b)}\ &\overline{N}_0(r,\nu^0_{f_2})+\overline{N}_0(r,\nu^0_{f_2-1})+\overline{N}_0(r,\nu^\infty_{f_2})\\
&\hspace{30pt} \le \sum_{i=1}^3\left(\overline{N}_0(r,\nu^0_{f_1-a_i})+3T_0(r,a_i)\right)+O(1).
\end{align*}

Moreover, if we set $b=\dfrac{a_4-a_1}{a_4-a_2}\cdot\dfrac{a_3-a_2}{a_3-a_1}$ then

$$\mathrm{(c)}\  \overline{N}_0(r,\nu^0_{f_2-b})\le\overline{N}_0(r,\nu^0_{f_1-a_4})+2T_0(r,a_1)+3T_0(r,a_2)+2T_0(r,a_3)+T_0(r,a_4).$$

We write $f_i=\frac{f_{i0}}{f_{i1}}$ and  $a_i=\frac{a_{i0}}{a_{i1}}\ (1\le i\le 4)$ similarly as in the proof of Lemma <ref> and get

$$ |h|\cdot\|f_1\|\le C\|f_2\|\cdot\prod_{i=1}^3\|a_i\| $$

for some holomorphic function $h$ on $\A(R_0)$.
This implies that

$$ T_0(r,f_1)+N_0(r,h)\le T_0(r,f_2)+\sum_{i=1}^3T_0(r,a_i)+O(1).$$

$$T_0(r,f_2)\ge T_0(r,f_1)-\sum_{i=1}^3T_0(r,a_i)+O(1).$$

and we get the inequality (a).

It also is easy to see that

\begin{align*}
\overline{N}_0(r,\nu^0_{f_2})&+\overline{N}_0(r,\nu^0_{f_2-1})+\overline{N}_0(r,\nu^\infty_{f_2}) \\
&\le \sum_{i=1}^3\overline{N}_0(r,\nu^0_{f_1-a_i})+\sum_{1\le i<j\le 3}\overline{N}_0(r,\nu^0_{a_i-a_j})+\sum_{i=1}^3\overline{N}_0(r,\nu^\infty_{a_i})\\
&\le \sum_{i=1}^3\left(\overline{N}_0(r,\nu^0_{f_1-a_i})+3T_0(r,a_i)\right)+O(1).
\end{align*}

Then we have the inequality (b).

On the other hand, we have

$$ \dfrac{1}{f_2-b}=\dfrac{(a_3-a_1)(a_4-a_2)}{(a_3-a_2)(a_1-a_2)}\cdot\left (1+\frac{a_4-a_2}{f_1-a_4}\right).$$

This implies that

$$ \min\{1,\nu^0_{f_2-b}\}\le\min\{1,\nu^\infty_{a_3-a_1}+\nu^\infty_{a_4-a_2}+\nu^0_{a_3-a_2}+\nu^0_{a_1-a_2}+\nu^0_{f_1-a_4}\}.$$

It yields that

$$ \overline{N}_0(r,\nu^0_{f_2-b})\le\overline{N}_0(r,\nu^0_{f_1-a_4})+2T_0(r,a_1)+3T_0(r,a_2)+2T_0(r,a_3)+T_0(r,a_4).$$

This completes the proof of the lemma.

The proof of Theorem <ref> is straightforward deduced from the below lemma.

Let $g$ be a nonconstant meromorphic function on $\A (R_0)$. Let $a_1,\dots ,a_5$ be distinct meromorphic functions (may be identity $\infty$). We have

$$2T_0(r,g)\le \sum_{i=1}^5\overline N_0(r,\nu^0_{g-a_i})+35\sum_{i=1}^5T_0(r,a_i)+S(r)\ (1\le r< R_0),$$

where $S(r)=S_g(r)+\sum_{i=1}^5S_{a_i}(r)$.

Then, in order to prove Theorem <ref>, it is suffice for us to prove Lemma <ref>.

We split the proof of the lemma into two parts.

Part A. We first consider the case where $a_i\not\equiv\infty$ for all $i=1,\ldots,5$. We set

$$f = \frac{g-a_2}{g-a_1} \cdot \frac{a_3-a_1}{a_3-a_2}, b_1= \frac{a_4-a_2}{a_4-a_1} \cdot \frac{a_3-a_1}{a_3-a_2}, b_2= \frac{a_5-a_2}{a_5-a_1} \cdot \frac{a_3-a_1}{a_3-a_2},b_3=0\text{ and }b_4=1.$$

By Lemma <ref>, we have

\begin{align}\label{3.4}
T_0(r,g)&\le T_0(r,f)+\sum_{i=1}^3T_0(r,a_i)+O(1),\\
\label{3.5}
\label{3.6}
\end{align}

\begin{align}\label{3.7}
\begin{split}
\overline{N}_0(r,\nu^\infty_f)+\sum_{i=1}^4\overline{N}_0(r,\nu^0_{f-b_i})
&\le \sum_{i=1}^5\overline{N}_0(r,\nu^0_{g-a_i})+7T_0(r,a_1)+9T_0(r,a_2)\\
\end{split}
\end{align}

We now prove the following claim.

$2T_0(r,f)\le \overline{N}_0(r,\nu^\infty_f)+\sum_{i=1}^4\overline{N}_0(r,\nu^0_{f-b_i})+18\sum_{i=1,2}T_0(r,b_i)+S(r).$

Indeed, if $b_1$ or $b_{2}$ is constant then the claim directly follows from Theorem <ref>.
Therefore, we may assume that both $b_1$ and $b_2$ are not constant. We define

\begin{equation}\label{3.9}
F= \begin{vmatrix}
ff'&f'&f^2-f \\
b_1b_1'&b_1'&b_1^2-b_1 \\
b_2b_2' &b_2'&b_2^2-b_2
\end{vmatrix}.
\end{equation}

Consider the following two cases.

Case 1: $F(z)\equiv0$. From (<ref>) we get

\begin{align}\label{3.10}
\biggl (\dfrac{b_1'}{b_1}-\dfrac{b_2'}{b_2}\biggl )\biggl (\dfrac{f'}{f-1}-\dfrac{b_2'}{b_2-1}\biggl )\equiv \biggl (\dfrac{b_1'}{b_1-1}-\dfrac{b_2'}{b_2-1}\biggl )\biggl (\dfrac{f'}{f}-\dfrac{b_2'}{b_2}\biggl ).
\end{align}

We distinguish the following four subcases.

Subcase 1: $\dfrac{b_1'}{b_1}\equiv\dfrac{b_2'}{b_2}.$ We have $\dfrac{b_1'}{b_1-1}\not\equiv\dfrac{b_2'}{b_2-1}$, since if otherwise $b_1$ and $b_{2}$ are constants. Therefore, from (<ref>), we have $\dfrac{f'}{f}\equiv\dfrac{b_2'}{b_2}$, and hence $f=cb_2$ with a constant $c$. This implies that $T_0(r,f)=T_0(r,b_2)$ and the claim is proved in this subcase.

Subcase 2: $\dfrac{b_1'}{b_1-1}\equiv\dfrac{b_2'}{b_2-1}.$ By the same arguments in Subcase 1, we get again the inequality of the claim in this subcase.

Subcase 3: $\dfrac{b_1'}{b_1}\not\equiv\dfrac{b_2'}{b_2}, \dfrac{b_1'}{b_1-1}\not\equiv\dfrac{b_2'}{b_2-1},\dfrac{b_1'}{b_1}-\dfrac{b_2'}{b_2} \equiv\dfrac{b_1'}{b_1-1}-\dfrac{b_2'}{b_2-1}.$ The identity (<ref>) implies that

$$\dfrac{f'}{f-1}-\dfrac{f'}{f}\equiv \dfrac{b_2'}{b_2-1}-\dfrac{b_2'}{b_2}.$$

Then, we have

$$\dfrac{f-1}{f}\equiv c \cdot\dfrac{b_2-1}{b_2},$$

where $c$ is a constant. Therefore $f=\frac{b_2}{(c-1)b_2-c}$, and hence $T_0(r,f)=T_0(r,b_2)$. We get again the inequality of the claim in this subcase.

Subcase 4: $\dfrac{b_1'}{b_1}\not\equiv\dfrac{b_2'}{b_2}, \dfrac{b_1'}{b_1-1}\not\equiv\dfrac{b_2'}{b_2-1},\dfrac{b_1'}{b_1}-\dfrac{b_2'}{b_2} \not\equiv\dfrac{b_1'}{b_1-1}-\dfrac{b_2'}{b_2-1}.$ The identity (<ref>) may be rewritten as

\begin{align}\label{3.11}
\biggl (\dfrac{b_1'}{b_1}-\dfrac{b_2'}{b_2}\biggl )\dfrac{f'}{f-1}-\biggl (\dfrac{b_1'}{b_1-1}-\dfrac{b_2'}{b_2-1}\biggl )\dfrac{f'}{f}\equiv \dfrac{b_1'b_2'}{b_1(b_2-1)}-\dfrac{b_2'b_1'}{b_2(b_1-1)}.
\end{align}

From (<ref>), we see that each zero of $(f-1)$ must be a zero or an $1$-point or a pole of $b_j\  (j=1, 2)$ or a zero of $\dfrac{b_1'}{b_1}-\dfrac{b_2'}{b_2}$. Therefore,

\begin{align}\label{3.12}
\min\{1,\nu^0_{f-1}\}\le\sum_{i=1,2}\sum_{a=0,1,\infty}\min\{1,\nu^{0}_{b_i-a}\}+\min\{1,\nu^0_{\frac{b_1'}{b_1}-\frac{b_2'}{b_2}}\} .
\end{align}

Similarly, we have

\begin{align}\label{3.13}
\min\{1,\nu^0_{f}\}\le\sum_{i=1,2}\sum_{a=0,1,\infty}\min\{1,\nu^{0}_{b_i-a}\}+\min\{1,\nu^0_{\frac{b_1'}{b_1-1}-\frac{b_2'}{b_2-1}}\} .
\end{align}

From (2.7), we also see that each pole of $f$ must be a zero or an $1-$point or a pole of $b_j\  (j=1, 2)$ or a zero of $\biggl (\dfrac{b_1'}{b_1}-\dfrac{b_2'}{b_2}\biggl )- \biggl (\dfrac{b_1'}{b_1-1}-\dfrac{b_2'}{b_2-1}\biggl )$. Therefore, by the similar arguments, we have

\begin{align}\label{3.14}
\min\{1,\nu^\infty_{f}\}\le\sum_{i=1,2}\sum_{a=0,1,\infty}\min\{1,\nu^{0}_{b_i-a}\}+\min\{1,\nu^0_{\bigl (\frac{b_1'}{b_1}-\frac{b_2'}{b_2}\bigl )- \bigl (\frac{b_1'}{b_1-1}-\frac{b_2'}{b_2-1}\bigl )}\}.
\end{align}

Combining (<ref>), (<ref>) and (<ref>), we have

\begin{align}\label{3.15}
\begin{split}
\sum_{a=0,1,\infty}\min\{1,\nu^0_{f-a}\}&\le\sum_{i=1,2}\sum_{a=0,1,\infty}\min\{1,\nu^{0}_{b_i-a}\}+\min\{1,\nu^0_{\frac{b_1'}{b_1}-\frac{b_2'}{b_2}}\}\\
&+\min\{1,\nu^0_{\frac{b_1'}{b_1-1}-\frac{b_2'}{b_2-1}}\}+\min\{1,\nu^0_{\bigl (\frac{b_1'}{b_1}-\frac{b_2'}{b_2}\bigl )- \bigl (\frac{b_1'}{b_1-1}-\frac{b_2'}{b_2-1}\bigl )}\}.
\end{split}
\end{align}

Then by using Theorem <ref>, we get

\begin{align*}
T_0( r,f)&\le \overline{N}_0(r,\nu^0_f)+\overline{N}_0(r,\nu^0_{f-1})+\overline{N}_0(r,\nu^\infty_f)+S(r)\\
&\le 2\sum_{i=1}^2\left(T_0\left(r,\frac{b_i'}{b_i}\right)+T_0\left(r,\frac{b_i'}{b_i-1}\right)\right)+S(r)\\
&\le 2\sum_{i=1}^2\left(N_0\left(r,\nu^\infty_{\frac{b_i'}{b_i}}\right)+N_0\left(r,\nu^\infty_{\frac{b_i'}{b_i-1}}\right)\right)+S(r)\\
&\le 2\sum_{i=1}^2(N_0(r,\nu^0_{b_i})+2N_0(r,\nu^\infty_{b_i})+N_0(r,\nu^0_{b_i-1}))\\
&\le 8T_0(r,b_1)+8T_0(r,b_2).
\end{align*}

Then we have the desired inequality of the claim in this subcase.

Case 2: $F(z)\not\equiv0$. For $t=r$ or $t=\frac{1}{r}\ (r>1)$, we set

\begin{align*}
\delta (z)&= \min\{1,|b_1(z)|,|b_2(z)|,|b_1(z)-1|,|b_2(z)-1|,|b_1(z)-b_2(z)|\},\\
\theta _j (t)& = \{\theta:|f(te^{i\theta})-b_j(te^{i\theta})|\le\delta (te^{i\theta})\},(j=1,2),\\
\theta _j(t)& = \{\theta:|f(te^{i\theta})-b_j|\le\delta (te^{i\theta})\},(j=3,4).
\end{align*}

It is clear that the sets $\theta_i(t)\cap\theta_j(t) \ ( i\neq j, i, j=1, 2, 3, 4)$ have at most many finite points. Then, we easily see that

\begin{align}\label{3.16}
\begin{split}
\sum_{t=1/r,r}\dfrac{1}{2\pi}\int\limits_{0}^{2\pi}\log \dfrac{1}{\delta (te^{i\theta})}d\theta
&\le m_0\left( r,\dfrac{1}{b_1}\right)+m_0\left( r,\dfrac{1}{b_2}\right)+m_0\left( r,\dfrac{1}{b_1-1}\right)\\
&+m_0\left( r,\dfrac{1}{b_2-1}\right)+m_0\left( r,\dfrac{1}{b_1-b_2}\right)\\
&\le T(r)+S(r)+O(1),
\end{split}
\end{align}

where $T(r)=3 T_0(r,b_1)+3T_0(r,b_2).$ We also see that

\begin{align*}
\end{align*}

Substituting these functions (on the right hand side) into (<ref>), by the properties of the Wronskian, we have

\begin{equation}\label{3.17}
F= \begin{vmatrix}
\varphi&f'-b_1'&\psi \\
b_2b_2' &b_2'&b_2'-b_2
\end{vmatrix},
\end{equation}

\begin{align*}
\varphi &=(f-b_1)(f'-b_1')+b_1'(f-b_1)+b_1(f'-b_1'),\\
\psi&=(f-b_1)^2+(2b_1-1)(f-b_1).
\end{align*}

From (<ref>) we see that each zero with multiplicity $p$ $(p>1)$ of $f-b_1$ which is neither a pole of $b_1$ nor a pole of $b_2$ must be a zero of  $F$ with multiplicity at least $p-1$. Similarly, each zero with multiplicity $p$ $(p>1)$ of $f-b_2$ which is neither a pole of $b_1$ nor a pole of $b_2$ must be a zero of $F$ with multiplicity at least $p-1$. Moreover, from (<ref>) one see that each zero of multiplicity $p$ $(p>1)$ of $f$ or $f-1$ which is neither a pole of  $b_1$ nor a pole of $b_2$ must be a zero of $F$ with multiplicity at least $p-1$. This implies that

\begin{align}\label{3.18}
\sum_{i=1}^4(N(r,\nu^0_{f-b_i})-\overline{N}_0(r,\nu^0_{f-b_i}))\le N(r,\nu^0_F).
\end{align}

We now estimate the quantities $m_0(r,\frac{1}{f_-b_j})\ (1\le j\le 4)$.

Since $|f(te^{i\theta})-b_1(te^{i\theta})|\le\delta(te^{i\theta})\le 1$ for every $\theta\in\theta_1(t)$,

\begin{align}\label{3.19}
\begin{split}
&\sum_{t=1/r,r}\frac{1}{2\pi }\int\limits_{\theta_1(t)}\log^+\left| \dfrac{F}{f-b_1}\right|d\theta\le\sum_{t=1/r,r} \frac{1}{2\pi }\int\limits_{\theta_1(t)}\log\left (1+\left| \dfrac{F}{f-b_1}\right|\right)d\theta\\
&\le \sum_{t=1/r,r}\frac{1}{2\pi }\int\limits_{\theta_1(t)}\log\left\{(1+|b_1'|)(1+|b_1|)\left(1+\left|\frac{f'-b_1'}{f-b_1}\right|\right)\right\}d\theta\\
&\hspace{30pt}+\sum_{t=1/r,r}\frac{1}{2\pi }\int\limits_{\theta_1(t)}\log\left\{(1+|b_1|)(1+|b_1'|)\right\}d\theta\\
&\hspace{30pt}+\sum_{t=1/r,r}\frac{1}{2\pi }\int\limits_{\theta_1(t)}\log\left\{(1+|b_2|)(1+|b_2'|)\right\}d\theta+O(1)\\
&\le m_0\left( r,\dfrac{f'-b_1'}{f-b_1}\right)+2m_0\left(r,\frac{b_1'}{b_1}\right)+m_0\left(r,\frac{b_2'}{b_2}\right)\\
&\hspace{30pt}+4\sum_{t=1/r,r}\frac{1}{2\pi }\int\limits_{\theta_1(t)}\log^+|b_1|d\theta+2\sum_{t=1/r,r}\frac{1}{2\pi }\int\limits_{\theta_1(t)}\log^+|b_2|d\theta+O(1)\\
&\le 4\sum_{t=1/r,r}\frac{1}{2\pi }\int\limits_{\theta_1(t)}\log^+|b_1|d\theta+2\sum_{t=1/r,r}\frac{1}{2\pi }\int\limits_{\theta_1(t)}\log^+|b_2|d\theta+S(r).
\end{split}
\end{align}

Here, the second inequality comes from the fact that

\begin{align}\label{3.20}
\log^+|\det(x_{ij};1\le i,j\le 3)|\le \sum_{i=1}^3\log^+\max\{|x_{ij}|;1\le j\le 3\}+O(1)
\end{align}

for every $(3\times 3)$-matrix of complex numbers.

Therefore, from (<ref>) and (<ref>) we have

\begin{align}\label{3.21}
\begin{split}
&m_0\left( r,\dfrac{1}{f-b_1}\right)\le \sum_{t=1/r,r}\frac{1}{2\pi }\int\limits_{\theta_1(t)}\log^+\left| \dfrac{1}{f-b_1} \right|d\theta +\dfrac{1}{2\pi}\int\limits_{0}^{2\pi}\log \dfrac{1}{\delta(re^{i\theta})}d\theta\\
&\le \sum_{t=1/r,r}\frac{1}{2\pi }\int\limits_{\theta_1(t)}\log^+\left| \dfrac{F}{f-b_1} \right|d\theta +\frac{1}{2\pi } \int\limits_{\theta_1(r)}\log^+\dfrac{1}{|F|}d\theta+T(r)+S(r)\\
&= \sum_{t=1/r,r}\frac{1}{2\pi }\int\limits_{\theta_1(t)}\biggl(\log^+ \dfrac{1}{|F|}d\theta+4\log^+|b_1|+2\log^+|b_2|\biggl)d\theta+T(r)+S(r).
\end{split}
\end{align}

Similarly, we get

\begin{align}
\label{3.22}
\begin{split}
m_0\left( r,\dfrac{1}{f-b_2}\right)&\le \sum_{t=1/r,r}\frac{1}{2\pi }\int\limits_{\theta_2(t)}\biggl(\log^+ \dfrac{1}{|F|}d\theta+2\log^+|b_1|+4\log^+|b_2|\biggl)d\theta\\
\end{split}
\end{align}

On the other hand, for each $\theta\in\theta_j(t)\ (j=3,4)$ we have $|f(te^{i\theta})-b_j|\le2$ and hence

$$ \log^+\frac{|F(te^{i\theta})|}{|f(te^{i\theta})-b_j|}\le \log^+\frac{|f'(te^{i\theta})|}{|f(te^{i\theta})-b_j|}+\sum_{i=1}^2\left(2\log^+|b_i(te^{i\theta})|+\log^+\frac{|b_i'(te^{i\theta})|}{|b_i(te^{i\theta})|}\right)+O(1).$$

Similarly as above, for $j=3,4$, we get

\begin{align}
\label{3.23}
\begin{split}
m_0\left( r,\dfrac{1}{f-b_j}\right)&\le  \sum_{t=1/r,r}\frac{1}{2\pi }\int\limits_{\theta_j(t)}\biggl(\log^+ \dfrac{1}{|F|}d\theta+2\log^+|b_1|+2\log^+|b_2|\biggl)d\theta\\
\end{split}
\end{align}

Combining (<ref>)-(<ref>), we have

$$\sum_{i=2}^5m_0\left( r,\dfrac{1}{f-a_i}\right)\le m_0\left( r,\dfrac{1}{F}\right)+4m_0(r,b_1)+4m_0(r,b_2)+4T(r)+S(r).$$

\begin{align*}
4T_0( r,f)&\le N_0(r,\nu^0_f)+N_0(r,\nu^0_{f-1})+N_0(r,\nu^0_{f-b_1})+N_0(r,\nu^0_{f-b_2})-N(r,\nu^0_F)\\
\end{align*}

Combining this inequality with (<ref>), we obtain

\begin{align}\label{3.24}
\begin{split}
4T_0( r,f)&\le \overline N_0(r,\nu^0_f)+\overline N_0(r,\nu^0_{f-1})+\overline N_0(r,\nu^0_{f-b_1})+\overline N_0(r,\nu^0_{f-b_2})\\
\end{split}
\end{align}

Also, from (<ref>) and (<ref>), we have

\begin{align*}
m_0(r,F)&\le 2m_0(r,f)+m_0\left(r,\frac{f'}{f}\right)+\sum_{i=1}^2\left(2m_0(r,b_i)+m_0\left(r,\frac{b_1'}{b_1}\right)\right)+S(r)\\
&\le 2m_0(r,f)+2m_0(r,b_1)+2m_0(r,b_2)+S(r)
\end{align*}

$$ N_0(r,\nu^\infty_F)\le 2N_0(r,\nu^\infty_f)+\overline{N}_0(r,\nu^\infty_f)+ 3\sum_{i=1}^2N_0(r,\nu^\infty_{b_i}).$$

These yields that

\begin{align}\label{3.25}
T_0(r,F)\le 2T_0(r,f)+\overline{N}_0(r,\nu^\infty_f)+\sum_{i=1}^2N_0(r,\nu^\infty_{b_i})+\frac{2}{3}T(r)+S(r).
\end{align}

From (<ref>) and (<ref>), we have

\begin{align*}
2T_0( r,f)\le& \overline N_0(r,\nu^0_f)+\overline N_0(r,\nu^0_{f-1})+\overline N_0(r,\nu^\infty_{f})+\overline N_0(r,\nu^0_{f-b_1})\\
&+\overline N_0(r,\nu^0_{f-b_2})+18T_0(r,b_1)+18T_0(r,b_2)+S(r).
\end{align*}

Then, the claim is proved.

Return to the proof of the theorem. Form (<ref>)-(<ref>) and Claim <ref> we get

$$2T_0(r,g)\le \sum_{i=1}^5\overline{N}_0(r,\nu^0_{g-a_i})+45\sum_{i=1,3}T_0(r,a_i)+47T_0(r,a_2)+19\sum_{i=4,5}T_0(r,a_i)+S(r).$$

Then, by the same arguments we have

$$2T_0(r,g)\le \sum_{i=1}^5\overline{N}_0(r,\nu^0_{g-a_i})+45\sum_{j=1,3}T_0(r,a_{i_j})+47T_0(r,a_{i_2})+19\sum_{j=4,5}T_0(r,a_{i_j})+S(r)$$

for any permutation $(i_1,\ldots,i_5)$ of $\{1,\ldots,5\}$. Summing-up both sides of the above inequality over all such permutations, we obtain

$$2T_0(r,g)\le \sum_{i=1}^5\overline{N}_0(r,\nu^0_{g-a_i})+35\sum_{i=1}^5T_0(r,a_i)+S(r).$$

Part B. Now, we suppose that there is a function among $\{a_1,\ldots,a_5\}$ identity with $\infty$, for instance $a_1\equiv\infty$. We set

$$ h=\frac{g-a_2}{g-a_3}, c_1=\frac{a_4-a_2}{a_4-a_3}, c_2=\frac{a_5-a_2}{a_5-a_3},c_3=0\text{ and }c_4=1.$$

By Lemma <ref>, we have

\begin{align}\label{3.26}
T_0(r,g)&\le T_0(r,f)+\sum_{i=2}^3T_0(r,a_i)+O(1)\\
\label{3.27}
\label{3.28}
\end{align}

\begin{align}\label{3.29}
\begin{split}
\overline{N}_0(r,\nu^0_f)&+\overline{N}_0(r,\nu^\infty_f)+\overline{N}_0(r,\nu^0_{f-1})+\overline{N}_0(r,\nu^0_{f-c_1})+\overline{N}_0(r,\nu^0_{f-c_2})\\
&\le \sum_{i=1}^5\overline{N}_0(r,\nu^0_{g-a_i})+4T_0(r,a_2)+8T_0(r,a_3)\\
\end{split}
\end{align}

Applying Claim <ref> for functions $h,c_1,c_2,c_3,c_4$, we have

$$2T_0(r,h)\le \overline{N}_0(r,\nu^\infty_{h})+\sum_{i=1}^4\overline{N}_0(r,\nu^0_{h-c_i})+18\sum_{i=1}^2T_0(r,c_i)+S(r).$$

Combining (<ref>)-(<ref>) with the above inequality, we get

Then, by the same arguments, we have

$$2T_0(r,g)\le \sum_{i=1}^5\overline{N}_0(r,\nu^0_{g-a_i})+42T(r,a_{i_2})+46T(r,a_{i_3})+19\sum_{j=4,5}T(r,a_{i_j})+S(r)$$

for any permutation $(i_2,\ldots,i_5)$ of $\{2,\ldots,5\}$. Summing-up both sides of the above inequality over all such permutations, we obtain

$$2T_0(r,g)\le \sum_{i=1}^5\overline{N}_0(r,\nu^0_{g-a_i})+\frac{33}{2}\sum_{i=2}^5T_0(r,a_i)+S(r).$$

This completes of the proof the lemma.

We now prove the main result of the paper.

Suppose contrarily that there exists a non-constant holomorphic function $g$ and mutually distinct non-constant holomorphic functions $f_i\ (1\le i\le 5)$ such that $f_i$ and $g$ share the set $S$ with multiplicity trucated to level $l$, where $g=f_1$. We note that

$$ P_S(w)=(w-a_1)\cdots (w-a_q).$$

We set $\Psi_i:= P_S(f_i)/P_S(g)\ (1\le i\le 5).$

One has $T_0(r,f_j)=O(T_0(r,g))+S_{f_j}(r)$ and $T_0(r,g)=O(T_0(r,f_j))+S_g(r)$, in particular $S_{f_j}(r)=S_g(r)$ for every $1\le j\le 5$.

Indeed, by Theorem <ref> we have

\begin{align*}
&\le qT_0(r,f_j)+S_g(r).
\end{align*}

Then, $T_0(r,g)=O(T_0(r,f_j))+S_g(r)$. Similarly $T_0(r,f_j)=O(T_0(r,g))+S_{f_j}(r)$ and hence $S_{f_j}(r)=S_g(r)\ (1\le j\le 5)$.

We set $\varphi_j=P'_S(f_j)f_j'/P_S(f_j)$ and $\varphi =\varphi_1$. By the definition of $\Psi_j$, we have

$$\varphi=\alpha_j+\varphi_j,\text{ where }\alpha_j=\frac{\Psi_j'}{\Psi_j}\ (1\le j\le 5).$$

By the lemma on logarithmic derivative, we have

$$ m_0(r,\varphi_j)=S_{P_S(f_j)}(r)=S_g(r)\ \forall 1\le j\le 5\ (\text{note that }T_0(r,P_S(f_j))=qT_0(r,f_j)).$$

Since each pole of $\varphi_j$ is a simple pole and must be either zero or pole of $P_S(f_j)$, we have

$$ N_0(r,\nu^\infty_{\varphi_j})=\overline{N}_0(r,\nu^0_{P_S(f_j)})+\overline{N}_0(r,\nu^\infty_{P_S(f_j)}) \le 2q T_0(r,f_j)).$$

It yields that

$$T_0(r,\varphi_j)=m_0(r,\varphi_j)+N_0(r,\nu^\infty_{\varphi_j})\le 2q T_0(r,f_j))+S_g(r).$$

In particular $T_0(r,\varphi_j)=O(T_0(r,g))$. Moreover, by Theorem <ref> we have

\begin{align*}
N_0(r,\nu^\infty_{\varphi})&\ge \overline{N}_0(r,\nu^0_{P_S(g)})+\overline{N}_0(r,\nu^0_{P_S(g)})\\
&\ge (q-1)T_0(r,f_j)+S_{f_j}(r).
\end{align*}

This implies that

\begin{align}\label{3.31}
(q-1)T_0(r,f_j)\le T_0(r,\varphi).
\end{align}

Also, we note that

\begin{align*}
&\le m(r,\varphi)+m(r,\varphi_j)+\overline{N}_0(r,\nu^0_{\Psi_j})+\overline{N}_0(r,\nu^\infty_{\Psi_j})+O(1)\\
&\le\sum_{i=1}^q\overline{N}_0(r,\nu^0_{f_j-a_i,\ge l})+S_g(r)\le \frac{q}{l}T_0(r,f_j)+S_g(r).
\end{align*}

This also yields that $T_0(r,\alpha_j)=O(T_0(r,g))$.

We now show that $\alpha_j\ (1\le j\le 5)$ are mutually distinct. Indeed, suppose contrarily that $\alpha_i=\alpha_j$  for some $i\ne j$. Then, there exists a non-zero constant $c_0$ with $\Psi_i=c_0\Psi_j$ and hence $c_0P_S(f_i)=P_S(f_j).$ This contradicts to the assumption that $P_S(w)$ is a uniqueness polynomial for meromorphic functions and $f_i\ne f_j$.

Applying Theorem <ref> to the functions $\varphi$ and functions $\alpha_j\ (1\le j\le 5)$, we obtain

On the other hand, we have

$$ \overline{N}_0(r,\nu^0_{\varphi-\alpha_j})=\overline{N}_0(r,\nu^0_{\varphi_j})\le\overline{N}_0(r,\nu^0_{f_j'})+\sum_{t=1}^k\overline{N}_0(r,\nu^0_{f_j-e_t}), $$

where $e_1,\ldots,e_k$ are all of distinct zeros of $P'_S(w)$. Since $\overline{N}_0(r,\nu^0_{f_j-e_t})\le T_0(r,f_j)+O(1)$ and

\begin{align*}
\overline{N}_0(r,\nu^0_{f_j'})\le&T_0(r,f_j')+O(1)=m_0(r,f_j')+O(1)\\
\le&m_0(r,f_j)+m_0\left(r,\frac{f_j'}{f_j}\right)+O(1)\le T_0(r,f_j)+S_g(r),
\end{align*}

we have

\begin{align}\label{3.32}
\sum_{j=1}^5\overline{N}_0(r,\nu^0_{\varphi-\alpha_j})\le \left (k+1+\frac{35q}{l}\right)\sum_{j=1}^5T_0(r,f_j)+S_g(r).
\end{align}

Then, combining (<ref>) and (<ref>) we have

\begin{align*}
2(q-1)T_0(r,f_i)&\le 2T(r,\varphi)+S_g(r)\\
&\le \left (k+1+\frac{35q}{l}\right)\sum_{j=1}^5T_0(r,f_j)+S_g(r).
\end{align*}

Summing up these inequalities over all $i=1,\ldots, 5$, we obtain

$$2(q-1)\sum_{i=1}^5T_0(r,f_i) \le 5\left (k+1+\frac{35q}{l}\right)\sum_{j=1}^5T_0(r,f_j)+S_g(r).$$

Letting $r\rightarrow R_0$ (outside the exceptional set), we get

$$(q-1)\le 5\left (k+1+\frac{35q}{l}\right),\text{ i.e., }q\le\frac{(5k+3)l}{2l-175}.$$

This contradiction completes the proof of the theorem.

§ DISCLOSURE STATEMENT

No potential conflict of interest was reported by the author(s).

[2] H. Fujimoto, Finiteness of entire functions share finite set, Nagoya Math. J.185 (2007), 111–122.

[3] A.Y. Khrystiyanyn and A. A. Kondratyuk On the Nevanlinna theory for meromorphic functions on annuli. II, Mat. Stud. 24 (02), 57–68, 2005.

[4] Lund, M. and Ye, Z. Nevanlinna theory of meromorphic functions on annuli Sci. China Math. 53, 547–554, 2010.

[5] Nevanlinna, R. Einige Eideutigkeitssäte in der theorie der meromorphen funktionen, Acta. Math. 48, 367–391, 1926.

[6] S. D. Quang, T. A. Hai and H. H. Giang, Meromorphic functions on annuli sharing few small functions with truncated multiplicities, Complex Anal. Oper. Theory 13 (2019), no. 4, 1693–1711.

[7] Yamanoi, K. The second main theorem for small functions and related problems, Acta Math. 192, 225–294, 2004.

[8] H. X. Yi and C. C. Yang, Uniqueness theory of meromorphic functions, Pure and Applied Math.Monographs No. 32. Science Press, Beijing (1995).

Si Duc Quang
$^1$Department of Mathematics, Hanoi National University of Education,

136-Xuan Thuy, Cau Giay, Hanoi, Vietnam.
$^2$Thang Long Institute of Mathematics and Applied Sciences,

Nghiem Xuan Yem, Hoang Mai, HaNoi, Vietnam.
E-mail<EMAIL_ADDRESS>


# An Anick type wild automorphism of free Poisson algebras

Ivan Shestakov‡ I.S., Instituto de Matemática e Estatística, Universidade de
São Paulo, Brazil; Sobolev Institute of Mathematics, Novosibirsk, Russia.
<EMAIL_ADDRESS>and  Zerui Zhang∗ Z.Z., School of Mathematical Sciences,
South China Normal University, Guangzhou 510631, P. R. China
<EMAIL_ADDRESS>
###### Abstract.

We construct an Anick type wild automorphism $\delta$ in a 3-generated free
Poisson algebra which induces a tame automorphism in a 3-generated polynomial
algebra. We also show that $\delta$ is stably tame.

Dedicated to the memory of professor V.A.Roman’kov

###### Key words and phrases:

automorphism; Poisson algebra; Jacobian matrix

###### 2010 Mathematics Subject Classification:

08A35, 17A30, 17A50

‡ Supported by grants FAPESP 2018/23690-6, and CNPq 304313/2019-0

∗ Supported by the NNSF of China (Grant No.12101248) and by the Guangdong
Basic and Applied Basic Research Foundation (Grant No. 2024A1515013122)

∗ Corresponding author

##  1\. Introduction

We recall that a _Poisson algebra_ is a vector space $\mathcal{P}$ over a
field $k$ endowed with two bilinear operations: a multiplication denoted by
$x\otimes y\mapsto x\cdot y$ and a Poisson bracket denoted by $x\otimes
y\mapsto[x,y]$, such that $(\mathcal{P},\cdot)$ is a commutative associative
algebra, $(\mathcal{P},[-,-])$ is a Lie algebra, and $\mathcal{P}$ satisfies
the _Leibniz identity_

$[x,y\cdot z]=y\cdot[x,z]+z\cdot[x,y].$

A Poisson algebra $\mathcal{P}$ is called _unital_ if $(\mathcal{P},\cdot)$ is
unital. In this article, we always assume that the Poisson algebra under
consideration is unital. And we usually omit the operation $\cdot$ when no
confusion arises.

It is clear that polynomial algebras are Poisson algebras with trivial Poisson
brackets. On the other hand, assume that $\mathcal{L}$ is the free Lie algebra
generated by an arbitrary set $X$ and assume that $Y$ is a linear basis of
$\mathcal{L}$. Then the Poisson symmetric algebra of $\mathcal{L}$, namely,
the polynomial algebra with free generating set $Y$, is the free Poisson
algebra generated by $X$ [11]. Since research on automorphisms of polynomial
algebras is an interesting subject, it is natural to study automorphisms of
free Poisson algebras.

Let us now review some essential terminology. Let $A_{n}$ be an arbitrary free
algebra with free generating set $X:=\\{x_{1},\dots,x_{n}\\}$ and denote by
$\mathsf{Aut}(A_{n})$ the automorphism group of $A_{n}$. We denote an
automorphism of the form

$\sigma(i,\alpha,f):=(x_{1},\dots,x_{i-1},\alpha
x_{i}+f,x_{i+1},\dots,x_{n}),$

where $1\leq i\leq n$, $0\neq\alpha\in k$, and $f$ lies in the subalgebra of
$A_{n}$ generated by $X\setminus\\{x_{i}\\}$, to be an _elementary
automorphism_ of $A_{n}$. The subgroup $\mathsf{TA}(A_{n})$ of
$\mathsf{Aut}(A_{n})$ generated by all elementary automorphisms is known as
the _tame automorphism group_ of $A_{n}$. We refer to the elements of
$\mathsf{TA}(A_{n})$ as _tame automorphisms_ and those in
$\mathsf{Aut}(A_{n})\setminus\mathsf{TA}(A_{n})$ as _wild automorphisms_. We
denote by

$\varphi=(f_{1},\dots,f_{n})$

the endomorphism $\varphi$ of $A_{n}$ defined by the rule
$\varphi(x_{i})=f_{i}\in A_{n}$ for all $i\leq n$.

Known results have demonstrated close connections between the automorphisms of
polynomial algebras, free associative algebras, and free Poisson algebras. For
instance, automorphisms of polynomial algebras (free associative algebras and
free Poisson algebras, respectively) in two variables are tame [4, 6, 7, 8,
9], where the characteristic of the underlying fields is assumed to be 0 for
Poisson algebras. Moreover, the automorphism groups of the aforementioned two-
generated free algebras are isomorphic.

On the other hand, automorphisms of a polynomial algebra and of a free
associative algebras in three variables can be wild, and the most famous
example are the Nagata automorphism [10, 12, 13] and the Anick automorphism
[16]. Since every automorphism of an $n$-generated polynomial algebra can be
lifted to an automorphism of an $n$-generated free Poisson algebra, it follows
that $3$-generated free Poisson algebras have wild automorphisms as well.
However, to our knowledge, it remains an open question whether every wild
automorphism of an $n$-generated free Poisson algebra induces a wild
automorphism of an $n$-generated polynomial algebra under the projection
sending Poisson brackets to zero. In this article we construct an Anick type
wild automorphism

(1.1)
$\delta:=(x_{1}+[x_{3},x_{1}x_{3}-[x_{3},x_{2}]],x_{2}+(x_{1}x_{3}-[x_{3},x_{2}])x_{3},x_{3})$

of a 3-generated free Poisson algebra, which induces a tame automorphism of a
polynomial algebra in 3 variables. Since
$\delta(x_{1}x_{3}-[x_{3},x_{2}])=x_{1}x_{3}-[x_{3},x_{2}]$, we deduce that
$\delta$ is an epimorphism. By a similar proof as that in [3, p. 343,
Proposition 8.1], we conclude that $\delta$ is an automorphism.

The article is organized as follows: In Section 2, we explore the connections
between the group of tame automorphisms of a free Poisson algebras and that of
a polynomial algebra. In Section 3, we define Jacobian matrices for
endomorphisms of free Poisson algebras and prove the chain rule. Although the
chain rule holds in general [1], for the sake of completeness and for the
readers’ convenience, we still include a proof. In Section 4, we establish our
main result: the automorphism $\delta$ in (1.1) is wild and stably tame.

Our approach is primarily inspired by [16], with modifications to make the
proofs simplified.

##  2\. Tame automorphisms of free Poisson algebras

Our aim in this section is to investigate the connections between tame
automorphisms of free Poisson algebras and those of polynomial algebras.
Denote by $P_{n}$ the free Poisson algebra with free generating set
$X:=\\{x_{1},\dots,x_{n}\\}$. Let $C_{n}$ be the polynomial algebra with
variables $x_{1},\dots,x_{n}$, and we always suppose that $C_{n}$ is endowed
with the trivial Poisson brackets. Then there exists a Poisson algebra
homomorphism

$\pi_{n}:P_{n}\longrightarrow C_{n},\quad x_{i}\mapsto x_{i},\mbox{ for all
}1\leq i\leq n.$

The kernel $\mathsf{Ker}(\pi_{n})$ of $\pi_{n}$ is the ideal of $P_{n}$
generated by $\\{[x_{i},x_{j}]\mid x_{i},x_{j}\in X,i>j\\}$. Finally, for
every endomorphism $\varphi=(f_{1},\dots,f_{n})$ of $P_{n}$, we denote by
$\overline{\varphi}=(\pi_{n}(f_{1}),\dots,\pi_{n}(f_{n}))$ the induced
endomorphism of $\varphi$ in $C_{n}$. If $\varphi$ is an automorphism, then so
is $\overline{\varphi}$. Obviously, every elementary automorphism $\varphi$ of
the form

(2.1) $\sigma(i,1,f),f\in\mathsf{Ker}(\pi_{n}),$

satisfies that $\overline{\varphi}=\mathsf{id}$, the identity map of the
algebra $C_{n}$.

For $p\neq q$, we denote by $\sigma_{pq}$ the automorphism of $A_{n}$ that
interchanges $x_{p}$ and $x_{q}$, while the remaining generators are fixed.
Then $\sigma_{pq}$ is tame and we have

$\sigma_{pq}=\sigma(q,-1,x_{p})\sigma(p,1,-x_{q})\sigma(q,1,x_{p}).$

It is straightforward to see that the following relations [15, 16] between
elementary automorphisms hold for $A_{n}$:

(2.2) $\sigma(i,\alpha,f)\sigma(i,\beta,g)=\sigma(i,\alpha\beta,\beta f+g);$
(2.3)
$\sigma(i,\alpha,f)^{-1}\sigma(j,\beta,g)\sigma(i,\alpha,f)=\sigma(j,\beta,\sigma(i,\alpha,f)^{-1}(g)),$

where $i\neq j$ and $f$ lies in the subalgebra of $A_{n}$ generated by
$X\setminus\\{x_{i},x_{j}\\}$;

(2.4)
$\sigma_{pq}\sigma(i,\alpha,f)\sigma_{pq}=\sigma(j,\alpha,\sigma_{pq}(f)),$

where $1\leq p\neq q\leq n$ and $x_{j}=\sigma_{pq}(x_{i})$.

For $j>1$, by straightforward calculations or by [16], we know that
$\sigma_{1j}$ satisfies the property

(2.5)
$\sigma_{1j}=\sigma(1,-1,0)\sigma(1,1,x_{j})\sigma(j,1,-x_{1})\sigma(1,1,x_{j}).$

And thus, by (2.4), we obtain

(2.6)
$\sigma(j,\alpha,g)=\sigma_{1j}\sigma(1,\alpha,\sigma_{1j}(g))\sigma_{1j}.$

Note that (2.5) and (2.6) are general facts involving only elementary
automorphisms and do not depend on the underlying algebra.

In general, the relations (2.2)-(2.4) do not form a complete set of defining
relations of $\mathsf{TA}(A_{n})$, but they do if $A_{n}=C_{3}$ is the
polynomial algebra with variables $x_{1},x_{2},x_{3}$ over a field of
characteristic 0 [15].

######  Lemma 2.1.

Let $\Pi$ be the group homomorphism as follows:

$\Pi:\mathsf{TA}(P_{n})\longrightarrow\mathsf{TA}(C_{n}),\
\varphi\mapsto\overline{\varphi}.$

Then the kernel of $\Pi$, as a normal subgroup of $\mathsf{TA}(P_{n})$, is
generated by all elementary automorphisms of the form given in (2.1).

###### Proof.

Let $N$ be the normal subgroup of $\mathsf{TA}(P_{n})$ generated by all
elementary automorphisms of the form given in (2.1). Clearly we have
$N\subseteq\mathsf{Ker}(\Pi)$, so we obtain a well-defined group homomorphism

$\Pi_{1}:\mathsf{TA}(P_{n})/N\longrightarrow\mathsf{TA}(C_{n}),\ \varphi
N\mapsto\overline{\varphi}.$

It suffices to show that $\Pi_{1}$ is an isomorphism.

For every element $f\in P_{n}$, we have $f=f_{0}+f_{1}$, where $f_{0}$ does
not involve Poisson brackets and $f_{1}$ lies in $\mathsf{Ker}(\pi_{n})$.
Moreover, for $\sigma(i,\alpha,f)\in\mathsf{TA}(P_{n})$, by (2.2), we have

$\sigma(i,\alpha,f)=\sigma(i,\alpha,f_{0})\sigma(i,1,f_{1}).$

So we deduce $\sigma(i,\alpha,f)N=\sigma(i,\alpha,f_{0})N$ and

(2.7)
$\Pi_{1}(\sigma(i,\alpha,f)N)=\Pi_{1}(\sigma(i,\alpha,f_{0})N)=\overline{\sigma(i,\alpha,f_{0})}=\sigma(i,\alpha,f_{0})\in\mathsf{TA}(C_{n}).$

On the other hand, let $\varphi=(g_{1},\dots,g_{n})$ be a (tame) automorhphism
of $C_{n}$. Considering $g_{1},\dots,g_{n}$ as elements in $P_{n}$, we know
that $\varphi_{{}_{P}}:=(g_{1},\dots,g_{n})\in\mathsf{Aut}(P_{n})$. Define a
map

$\Omega:\mathsf{TA}(C_{n})\longrightarrow\mathsf{TA}(P_{n})/N,\
\varphi\mapsto\varphi_{{}_{P}}N.$

Clearly, $\Omega$ is a group homomorphism and we have

(2.8) $\Omega(\sigma(i,\alpha,f))=\sigma(i,\alpha,f)N.$

Since elementary automorphisms generate the tame automorphism groups, by (2.7)
and (2.8), we have

$\Omega\Pi_{1}=\mathsf{Id}\mbox{ and }\Pi_{1}\Omega=\mathsf{Id^{\prime}},$

where $\mathsf{Id}$ is the identity map of $\mathsf{TA}(P_{n})/N$ and
$\mathsf{Id^{\prime}}$ is the identity map of $\mathsf{TA}(C_{n})$. The result
follows immediately. ∎

##  3\. Jacobian matrices and the chain rule

In this section, we first recall from [17] the universal (multiplicative)
enveloping algebra $\mathcal{P}^{e}$ of an arbitrary Poisson algebra
$\mathcal{P}$ in the variety of all Poisson algebras. Then we define the
Jacobian matrices for endomorphisms of $P_{n}$. Finally, we prove that the
chain rule holds.

Let $M_{\mathcal{P}}=\\{M_{a}\mid a\in\mathcal{P}\\}$ and let
$H_{\mathcal{P}}=\\{H_{a}\mid a\in\mathcal{P}\\}$ be two copies of
$\mathcal{P}$ such that $M:\mathcal{P}\longrightarrow M_{\mathcal{P}}$ sending
$a$ to $M_{a}$ and $H:P\longrightarrow H_{\mathcal{P}}$ sending $a$ to $H_{a}$
are linear isomorphisms. Then $\mathcal{P}^{e}$ is the associative algebra
generated by the spaces $M_{\mathcal{P}}$ and $H_{\mathcal{P}}$, with defining
relations

(3.1) $M_{a}M_{b}=M_{ab},$ (3.2) $H_{[a,b]}=H_{a}H_{b}-H_{b}H_{a},$ (3.3)
$H_{ab}=M_{b}H_{a}+M_{a}H_{b},$ (3.4) $H_{a}M_{b}=M_{[a,b]}+M_{b}H_{a},$ (3.5)
$M_{1}=1$

for all $a,b\in\mathcal{P}$. It follows from (3.3) and (3.5) that $H_{1}=0$.
Moreover, every (unital) $\mathcal{P}$-module $\mathcal{V}$ is a left
$\mathcal{P}^{e}$-module with the actions

$M_{a}(v)=a\cdot v,\mbox{ and }H_{a}(v)=[a,v]$

for all $a\in\mathcal{P}$ and $v\in\mathcal{V}$. Assume that $\mathcal{P}$ is
generated by $X$. Rewriting all the terms on the left-hand side of the above
equations in (3.1)-(3.4), we observe that elements of the form
$M_{a}H_{x_{i_{1}}}...H_{x_{i_{t}}}$ with $t\geq 0$ forms a linear generating
set of $\mathcal{P}^{e}$. Define

$P_{n}^{e}=(P_{n})^{e}\mbox{ and }C_{n}^{e}=(C_{n})^{e}.$

Then we have

######  Lemma 3.1.

[17, Corollary 3] Every nonzero element in $P_{n}^{e}$ can be uniquely written
in the form

$\sum_{1\leq i\leq q}M_{a_{i}}w_{i},$

where each $a_{i}$ lies in $P_{n}$, each $w_{i}$ is of the form
$H_{x_{i_{1}}}...H_{x_{i_{t}}}$, where $t$ depends on $i$, and each $w_{i}$ is
distinct from the others.

Since $C_{n}$ is a Poisson algebra with trivial Poisson brackets, by Equations
(3.1), (3.2) and (3.4), we deduce that $C_{n}^{e}$ is a commutative algebra.
Combining this with Lemma 3.1, we obtain the following corollary:

######  Corollary 3.2.

$C_{1}^{e}=P_{1}^{e}$ is the polynomial algebra with variables $M_{x_{1}}$ and
$H_{x_{1}}$.

We further observe that homomorphisms of Poisson algebras induce homomorphisms
of their corresponding universal enveloping algebras. Let $\mathcal{P}_{1}$
and $\mathcal{P}_{2}$ be two Poisson algebras, and let $\varphi$ be a Poisson
algebra homomorphism from $\mathcal{P}_{1}$ to $\mathcal{P}_{2}$. Since
$(\mathcal{P}_{1})^{e}$ and $(\mathcal{P}_{2})^{e}$ are both defined by
relations of the form (3.1)-(3.5), we can construct an associative algebra
homomorphism

$\varphi^{e}:(\mathcal{P}_{1})^{e}\longrightarrow(\mathcal{P}_{2})^{e},\
M_{a}\mapsto M_{\varphi(a)},\ H_{a}\mapsto H_{\varphi(a)}$

for all $a\in\mathcal{P}_{1}$. In particular, we obtain an algebra homorphism

$\pi_{n}^{e}:P_{n}^{e}\longrightarrow C_{n}^{e}.$

Now we recall that the Fox derivatives $\frac{\partial(-)}{\partial x_{i}}$ is
a linear map from $P_{n}$ to $P_{n}^{e}$ defined by the rule:

$\frac{\partial x_{j}}{\partial x_{i}}=\delta_{i,j},$
$\frac{\partial(ab)}{\partial x_{i}}=M_{a}\frac{\partial b}{\partial
x_{i}}+M_{b}\frac{\partial a}{\partial x_{i}},$ $\frac{\partial[a,b]}{\partial
x_{i}}=H_{a}\frac{\partial b}{\partial x_{i}}-H_{b}\frac{\partial a}{\partial
x_{i}},$

for all $a,b\in P_{n}$, where $\delta_{i,i}=1$ and $\delta_{i,j}=0$ if $i\neq
j$. It is straightforward to show that $\frac{\partial(-)}{\partial x_{i}}$ is
well-defined.

Let $\mathcal{P}$ be a Poisson algebra. For all
$a_{1},\dots,a_{t}\in\mathcal{P}$, define

$[a_{1},\dots,a_{t}]=[\dots[[a_{1},a_{2}],a_{3}],\dots,a_{t}],$

where $[a_{1},\dots,a_{t}]$ means $a_{1}$ if $t=1$. Then we have the following
formula for the Fox derivative acting on elements involving Poisson brackets.

######  Lemma 3.3.

Let $\mathcal{P}$ be a Poisson algebra. Then for all $a_{1},\dots
a_{t}\in\mathcal{P}$ with $t\geq 2$, we have

$\frac{\partial{[a_{1},\dots,a_{t}]}}{\partial x_{r}}=\sum_{1\leq j\leq
t}(-H_{a_{t}})\dots(-H_{a_{j+1}})H_{[a_{1},\dots,a_{j-1}]}\frac{\partial
a_{j}}{\partial x_{r}}.$

###### Proof.

We use induction on $t$. For $t=2$, we have

$\frac{\partial{[a_{1},a_{2}]}}{\partial x_{r}}=(-H_{a_{2}})\frac{\partial
a_{1}}{\partial x_{r}}+H_{a_{1}}\frac{\partial a_{2}}{\partial x_{r}}.$

Now we assume that $t>2$. We define $b_{1}=[a_{1},a_{2}]$, and $b_{j}=a_{j+1}$
for all $2\leq j\leq t-1$. By induction hypothesis, we have

$\displaystyle\frac{\partial{[a_{1},\dots,a_{t}]}}{\partial x_{r}}$
$\displaystyle=$ $\displaystyle\sum_{1\leq p\leq
t-1}(-H_{b_{t-1}})\dots(-H_{b_{p+1}})H_{[b_{1},\dots,b_{p-1}]}\frac{\partial
b_{p}}{\partial x_{r}}$ $\displaystyle=$ $\displaystyle\sum_{2\leq p\leq
t-1}(-H_{a_{t}})\dots(-H_{a_{p+2}})H_{[a_{1},\dots,a_{p}]}\frac{\partial
a_{p+1}}{\partial
x_{r}}+(-H_{a_{t}})\dots(-H_{a_{3}})\frac{\partial[a_{1},a_{2}]}{\partial
x_{r}}$ $\displaystyle=$ $\displaystyle\sum_{3\leq j\leq
t}(-H_{a_{t}})\dots(-H_{a_{j+1}})H_{[a_{1},\dots,a_{j-1}]}\frac{\partial
a_{j}}{\partial
x_{r}}+(-H_{a_{t}})\dots(-H_{a_{3}})((-H_{a_{2}})\frac{\partial
a_{1}}{\partial x_{r}}+H_{a_{1}}\frac{\partial a_{2}}{\partial x_{r}}).$

The proof is completed. ∎

Now we define the Jacobian matrices for endomorphisms of $P_{n}$ in a way that
differs slightly from [17], as it will be more convenient to deal with
matrices over a commutative associative algebra rather than a noncommutative
one. For an arbitrary associative algebra $\mathcal{A}$, we denote by
$M_{n\times n}(\mathcal{A})$ the set of all $n\times n$ matrices over
$\mathcal{A}$. For every endomorphism $\varphi$ of $P_{n}$, we define the
Jacobian matrix $J(\varphi)$ of $\varphi$ to be

$J(\varphi)=\pi_{n}^{e}((\frac{\partial\varphi(x_{i})}{\partial
x_{j}})_{n\times n})=(\pi_{n}^{e}(\frac{\partial\varphi(x_{i})}{\partial
x_{j}}))_{n\times n}\in M_{n\times n}(C_{n}^{e}),$

where $\pi_{n}^{e}(\frac{\partial\varphi(x_{i})}{\partial x_{j}})$ lies in the
$i$-th row and $j$-th column of $J(\varphi)$. Since $C_{n}^{e}$ is
commutative, the determinant $|J(\varphi)|$ of $J(\varphi)$ makes sense.

Recall that every endomorphism $\varphi$ of $P_{n}$ induces an endomorphism
$\overline{\varphi}$ of $C_{n}$ satisfying

$\overline{\varphi}\pi_{n}=\pi_{n}\varphi,$

namely, the diagram

$\textstyle{P_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{n}}$$\scriptstyle{\varphi}$$\textstyle{P_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{n}}$$\textstyle{C_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\varphi}}$$\textstyle{C_{n}}$

is commutative. In particular, it follows immediately that

(3.6)
$\overline{\varphi}^{e}\pi_{n}^{e}=(\overline{\varphi}\pi_{n})^{e}=(\pi_{n}\varphi)^{e}=\pi_{n}^{e}\varphi^{e}.$

We are now ready to study the form of the chain rule under the aforementioned
definition for Jacobian matrices, the proof of which is essentially the same
as that in [1].

######  Lemma 3.4.

(chain rule) Let $\varphi$ and $\psi$ be two endomorphisms of $P_{n}$. Then we
have

$J(\varphi\psi)=\overline{\varphi}^{e}(J(\psi))J(\varphi).$

In particular, if $\psi$ is an automorphism, then $J(\psi)$ is invertible and
$J(\psi)^{-1}=\overline{\psi}^{e}(J(\psi^{-1}))$.

###### Proof.

Assume that $\psi(x_{i})=f=f(x_{1},\dots,x_{n})$. Then we have
$\varphi\psi(x_{i})=f(\varphi(x_{1}),\dots,\varphi(x_{n}))$. It follows that

$\displaystyle\pi_{n}^{e}(\frac{\partial\varphi\psi(x_{i})}{\partial x_{j}})$
$\displaystyle=\sum_{1\leq t\leq n}\pi_{n}^{e}\varphi^{e}(\frac{\partial
f}{\partial x_{t}})\pi_{n}^{e}(\frac{\partial{\varphi(x_{t})}}{\partial
x_{j}})$ $\displaystyle=\sum_{1\leq t\leq
n}\overline{\varphi}^{e}\pi_{n}^{e}(\frac{\partial f}{\partial
x_{t}})\pi_{n}^{e}(\frac{\partial{\varphi(x_{t})}}{\partial x_{j}})$
$\displaystyle=\overline{\varphi}^{e}(\pi_{n}^{e}(\frac{\partial\psi(x_{i})}{\partial
x_{1}}),\dots,\pi_{n}^{e}(\frac{\partial\psi(x_{i})}{\partial
x_{n}}))(\pi_{n}^{e}(\frac{\partial{\varphi(x_{1})}}{\partial
x_{j}}),\dots,\pi_{n}^{e}(\frac{\partial{\varphi(x_{n})}}{\partial
x_{j}}))^{T},$

where $(-)^{T}$ means the transposed matrix of the underlying matrix. It
follows that $J(\varphi\psi)=\overline{\varphi}^{e}(J(\psi))J(\varphi)$.
Consequently, we obtain

$E_{n}=J(\psi^{-1}\psi)=\overline{\psi^{-1}}^{e}(J(\psi))J(\psi^{-1})$

and

$E_{n}=J(\psi\psi^{-1})=\overline{\psi}^{e}(J(\psi^{-1}))J(\psi),$

where $E_{n}$ is the identity matrix. So we have

$E_{n}=\overline{\psi}^{e}(E_{n})=\overline{\psi}^{e}\overline{\psi^{-1}}^{e}(J(\psi))\overline{\psi}^{e}(J(\psi^{-1}))=J(\psi)\overline{\psi}^{e}(J(\psi^{-1})).$

The result follows immediately. ∎

Now we can deduce more properties of Jacobian matrices by applying the chain
rule.

######  Lemma 3.5.

Let $\varphi$ be an endomorphism of $P_{n}$ such that
$\overline{\varphi}=\mathsf{id}$. Then for every automorphism
$\psi\in\mathsf{Aut}(P_{n})$, we have

$J(\psi\varphi\psi^{-1})=J(\psi)^{-1}\overline{\psi}^{e}(J(\varphi))J(\psi)$

and

$|J(\psi\varphi\psi^{-1})|=\overline{\psi}^{e}(|J(\varphi)|).$

###### Proof.

By the chain rule, we have

$J(\psi\varphi\psi^{-1})=\overline{\psi}^{e}(J(\varphi\psi^{-1}))J(\psi)=\overline{\psi}^{e}(\overline{\varphi}^{e}(J(\psi^{-1}))J(\varphi))J(\psi)=J(\psi)^{-1}\overline{\psi}^{e}(J(\varphi))J(\psi).$

The other assertion follows immediately since $\overline{\psi}^{e}$ is an
algebra homomorphism. ∎

Let $P\\{x_{3}\\}$ be the free Poisson algebra generated by $x_{3}$. Then
$P\\{x_{3}\\}$ is the polynomial algebra endowed with trivial Poisson
brackets. In order to prove that the Anick type automorphism $\delta$ in (1.1)
is wild, we shall need the following Poisson algebra homomorphism:

$\eta:C_{3}\longrightarrow P\\{x_{3}\\},x_{1}\mapsto 0,\ x_{2}\mapsto 0,\
x_{3}\mapsto x_{3}.$

Clearly, $\eta$ is well-defined. So for every endomorphism $\varphi$ of
$P_{3}$, we have

$\eta^{e}(J(\varphi))=(\eta^{e}\pi_{3}^{e}(\frac{\partial\varphi(x_{i})}{\partial
x_{j}}))_{n\times n}.$

We shall need the following observation about Fox derivatives acting on
elements of $\mathsf{Ker}(\pi_{3})$ with respect to $x_{3}$.

######  Lemma 3.6.

For every $f\in\mathsf{Ker}(\pi_{3})$, we have
$\eta^{e}\pi_{3}^{e}(\frac{\partial{f}}{\partial x_{3}})=0$.

###### Proof.

For all $x_{i_{1}},\dots,x_{i_{t}}\in X$ such that $x_{i_{2}}<x_{i_{1}}\leq
3$, we have $\eta\pi_{3}(x_{i_{2}})=0=\frac{\partial{x_{i_{2}}}}{\partial
x_{3}}$. And thus by Lemma 3.3, we deduce that

$\eta^{e}\pi_{3}^{e}(\frac{\partial{[x_{i_{1}},\dots,x_{i_{t}}]}}{\partial
x_{3}})=\sum_{1\leq j\leq
t}(-H_{\eta\pi_{3}({x_{i_{t}}})})\dots(-H_{\eta\pi_{3}({x_{i_{j+1}}})})H_{\eta\pi_{3}([x_{i_{1}},\dots,x_{i_{j-1}}])}\eta^{e}\pi_{3}^{e}(\frac{\partial
x_{i_{j}}}{\partial x_{3}})=0.$

Therefore, for every element of the form
$[x_{i_{1}},\dots,x_{i_{t}}]g\in\mathsf{Ker}(\pi_{3})$ ($t\geq 2$), we have

$\eta^{e}\pi_{3}^{e}(\frac{\partial([x_{i_{1}},\dots,x_{i_{t}}]g)}{\partial
x_{3}})=M_{\eta\pi_{3}(g)}\eta^{e}\pi_{3}^{e}(\frac{\partial[x_{i_{1}},\dots,x_{i_{t}}]}{\partial
x_{3}})+M_{\eta\pi_{3}([x_{i_{1}},\dots,x_{i_{t}}])}\frac{\partial g}{\partial
x_{3}}=0.$

Since elements of the form $[x_{i_{1}},\dots,x_{i_{t}}]g$ ($t\geq 2$) linearly
generate $\mathsf{Ker}(\pi_{3})$, the results follows. ∎

We conclude this section with a property of $\eta^{e}(J(\varphi))$ for
endomorphism $\varphi$ of $P_{3}$ such that $\overline{\varphi}=\mathsf{id}$,
the identity map of $C_{3}$.

######  Lemma 3.7.

For all $\varphi,\psi\in\mathsf{Aut}(P_{3})$ satisfying
$\overline{\varphi}=\mathsf{id}$, we have

$\eta^{e}(J(\psi\varphi\psi^{-1}))=\begin{pmatrix}a_{11}&a_{12}&0\\\
a_{21}&a_{22}&0\\\ a_{31}&a_{32}&1\\\ \end{pmatrix}$

for some elements $a_{p,q}$ $(1\leq p\leq 3,1\leq q\leq 2)$ in
$P\\{x_{3}\\}^{e}$ such that

$a_{11}a_{22}-a_{12}a_{21}=\eta^{e}\overline{\psi}^{e}(|J(\varphi)|).$

Consequently, if $\varphi$ is an elementary automorphism of the form
$\sigma(i,1,f)$ with $f\in\mathsf{Ker}(\pi_{3})$, then we have
$a_{11}a_{22}-a_{12}a_{21}=1$.

###### Proof.

Since $\overline{\psi\varphi\psi^{-1}}=\overline{\varphi}=\mathsf{id}$, we may
assume $\psi\varphi\psi^{-1}=(x_{1}+f_{1},x_{2}+f_{2},x_{3}+f_{3})$ and
$f_{1},f_{2},f_{3}\in\mathsf{Ker}(\pi_{3})$. By Lemma 3.6, we obtain the
desired third column of $\eta^{e}(J(\psi\varphi\psi^{-1}))$. Then, by Lemma
3.5, we have

$a_{11}a_{22}-a_{12}a_{21}=|\eta^{e}(J(\psi\varphi\psi^{-1}))|=\eta^{e}(|(J(\psi\varphi\psi^{-1}))|)=\eta^{e}\overline{\psi}^{e}(|J(\varphi)|).$

For the consequence, it suffices to show $|J(\varphi)|=1$ since
$\eta^{e}\overline{\psi}^{e}(1)=1$. Suppose that $J(\varphi)=(c_{pq})_{n\times
n}$. Since $f$ does not involve $x_{i}$, by direct calculation, we have

(3.7) $c_{pq}=\begin{cases}\delta_{p,q},&\mbox{if }p\neq i,\\\ 1,&\mbox{if
}p=q=i,\\\ \pi_{3}^{e}(\frac{\partial f}{\partial x_{q}}),&\mbox{if }p=i,q\neq
i.\end{cases}$

The result follows immediately. ∎

##  4\. The constructed Anick type automorphism is wild

Our aim in this section is to establish our main result: the Anick type
automorphism $\delta$ in (1.1) is wild. For an arbitrary associative algebra
$\mathcal{A}$, we call matrices in $M_{2\times 2}(\mathcal{A})$ of the form

$E_{12}(a):=\begin{pmatrix}1&a\\\ 0&1\end{pmatrix}$

or

$E_{21}(a):=\begin{pmatrix}1&0\\\ a&1\end{pmatrix}$

_elementary $2\times 2$ matrices_ or _elementary matrices_ , where
$a\in\mathcal{A}$. We denote by $E_{2}(\mathcal{A})$ the subgroup of the
general linear group $GL_{2}(\mathcal{A})$ generated by all elementary
$2\times 2$ matrices. The following important and useful result is proved in
[2], see also [16].

######  Lemma 4.1.

[2] Let $k[x,y]$ be the polynomial algebra with variables $x$ and $y$. Then
the matrix

$\begin{pmatrix}1+xy&y^{2}\\\ -x^{2}&1-xy\end{pmatrix}$

does not lie in $E_{2}(k[x,y])$. Consequently, the transpose of the above
matrix,

$\begin{pmatrix}1+xy&-x^{2}\\\ y^{2}&1-xy\end{pmatrix}$

also does not belong to $E_{2}(k[x,y])$.

Let $\varphi$ be an endomorphism of $P_{3}$. Following the notation in [16],
we define

(4.1)
$J_{2}(\varphi)=\begin{pmatrix}\pi_{3}^{e}(\frac{\partial{\varphi(x_{1})}}{\partial
x_{1}})&\pi_{3}^{e}(\frac{\partial{\varphi(x_{1})}}{\partial x_{2}})\\\
\pi_{3}^{e}(\frac{\partial{\varphi(x_{2})}}{\partial
x_{1}})&\pi_{3}^{e}(\frac{\partial{\varphi(x_{2})}}{\partial
x_{2}})\end{pmatrix}.$

######  Theorem 4.2.

Let $\varphi\in\mathsf{Aut}(P_{3})$ be an elementary automorphism of the form
$\sigma(i,1,f)$ with $f\in\mathsf{Ker}(\pi_{3})$. Then for every tame
automorphism $\psi\in\mathsf{TA}(P_{3})$, we have
$\eta^{e}(J_{2}(\psi\varphi\psi^{-1}))\in E_{2}(P\\{x_{3}\\}^{e})$.

###### Proof.

Since $\psi$ is tame, we know that $\psi$ is a product of elementary
automorphisms of the form $\sigma(j,\alpha,g)$. By Equations (2.5) and (2.6),
we may assume that $\psi$ is a product of elementary automorphisms of the form

(4.2) $\sigma(1,\alpha,g)\mbox{ or }\sigma(j,1,-x_{1})\mbox{ with
}j\in\\{2,3\\}.$

So we may assume $\psi=\psi_{1}\psi_{2}\dots\psi_{t}$, where each $\psi_{p}$
is of the form in (4.2).

Now we use induction on $t$ to prove the result. For $t=0$, we have
$\psi\varphi\psi^{-1}=\varphi=\sigma(i,1,f)$. By Lemma 3.6, we have
$\eta^{e}\pi_{3}^{e}(\frac{\partial{f}}{\partial x_{3}})=0$. Since $f$ does
not involve $x_{i}$, we obtain
$\eta^{e}\pi_{3}^{e}(\frac{\partial{f}}{\partial x_{i}})=0$. There are several
cases to consider. If $i=1$, then by (4.1), we have

$\eta^{e}(J_{2}(\varphi))=\begin{pmatrix}1&\eta^{e}\pi_{3}^{e}(\frac{\partial
f}{\partial x_{2}})\\\ 0&1\end{pmatrix}\in E_{2}(P\\{x_{3}\\}^{e});$

If $i=2$, then we have

$\eta^{e}(J_{2}(\varphi))=\begin{pmatrix}1&0\\\
\eta^{e}\pi_{3}^{e}(\frac{\partial f}{\partial x_{1}})&1\end{pmatrix}\in
E_{2}(P\\{x_{3}\\}^{e});$

If $i=3$, then we have

$\eta^{e}(J_{2}(\varphi))=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}\in
E_{2}(P\\{x_{3}\\}^{e}).$

Now we assume $t>0$. Denote
$\psi_{2}\dots\psi_{t}\varphi(\psi_{2}\dots\psi_{t})^{-1}$ by $\varphi_{1}$.
Then by induction hypothesis and by Lemma 3.7, we may assume

$\eta^{e}(J(\varphi_{1}))=\begin{pmatrix}a_{11}&a_{12}&0\\\ a_{21}&a_{22}&0\\\
a_{31}&a_{32}&1\\\ \end{pmatrix}=\begin{pmatrix}A_{2\times 2}&0\\\ B_{1\times
2}&1\end{pmatrix}$

such that $A_{2\times 2}$ lies in $E_{2}(P\\{x_{3}\\}^{e})$. Denote by
$\widetilde{\psi_{1}}$ the automorphism of $P\\{x_{3}\\}$ induced by
$\overline{\psi_{1}}$. Since the following diagram

$\textstyle{C_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{\overline{\psi_{1}}}$$\textstyle{C_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{P\\{x_{3}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\psi_{1}}}$$\textstyle{P\\{x_{3}\\}}$

is commutative, we obtain that

$\eta^{e}\overline{\psi_{1}}^{e}=(\eta\overline{\psi_{1}})^{e}=(\widetilde{\psi_{1}}\eta)^{e}=\widetilde{\psi_{1}}^{e}\eta^{e}.$

Combining the above formula with Lemma 3.5, we have

(4.3)
$\eta^{e}(J(\psi\varphi\psi^{-1}))=\eta^{e}(J(\psi_{1}\varphi_{1}\psi_{1}^{-1}))=\eta^{e}(J(\psi_{1})^{-1})\widetilde{\psi_{1}}^{e}\eta^{e}(J(\varphi_{1}))\eta^{e}(J(\psi_{1})).$

Assume that

$\widetilde{\psi_{1}}^{e}\eta^{e}(J(\varphi_{1}))=\begin{pmatrix}\widetilde{\psi_{1}}^{e}(A_{2\times
2})&0\\\ \widetilde{\psi_{1}}^{e}(B_{1\times
2})&1\end{pmatrix}=\begin{pmatrix}U_{2\times 2}&0\\\ V_{1\times
2}&1\end{pmatrix}.$

Since $\widetilde{\psi_{1}}^{e}$ maps elementary matrices over
$P\\{x_{3}\\}^{e}$ to elementary matrices over $P\\{x_{3}\\}^{e}$, we deduce
that $U_{2\times 2}\in E_{2}(P\\{x_{3}\\}^{e})$.

If $\psi_{1}=\sigma(2,1,-x_{1})$, then we have

$\eta^{e}(J(\psi_{1}))=\begin{pmatrix}1&0&0\\\ -1&1&0\\\
0&0&1\end{pmatrix}\mbox{ and
}\eta^{e}(J(\psi_{1})^{-1})=\begin{pmatrix}1&0&0\\\ 1&1&0\\\
0&0&1\end{pmatrix}.$

By (4.3), we deduce that

$\displaystyle\eta^{e}(J(\psi\varphi\psi^{-1}))=$
$\displaystyle\begin{pmatrix}E_{21}(1)&0\\\
0&1\end{pmatrix}\begin{pmatrix}U_{2\times 2}&0\\\ V_{1\times
2}&1\end{pmatrix}\begin{pmatrix}E_{21}(-1)&0\\\ 0&1\end{pmatrix}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}E_{21}(1)U_{2\times
2}E_{21}(-1)&0\\\ V_{1\times 2}E_{21}(-1)&1\end{pmatrix}.$

So we obtian

$\eta^{e}(J_{2}(\psi\varphi\psi^{-1}))=E_{21}(1)U_{2\times 2}E_{21}(-1)\in
E_{2}(P\\{x_{3}\\})^{e}.$

If $\psi_{1}=\sigma(3,1,-x_{1})$, then we have

$\eta^{e}(J(\psi_{1}))=\begin{pmatrix}1&0&0\\\ 0&1&0\\\
-1&0&1\end{pmatrix}\mbox{ and
}\eta^{e}(J(\psi_{1})^{-1})=\begin{pmatrix}1&0&0\\\ 0&1&0\\\
1&0&1\end{pmatrix}.$

Denote the row matrix $(-1,0)$ by $W_{1\times 2}$. Then by (4.3), we deduce
that

$\displaystyle\eta^{e}(J(\psi\varphi\psi^{-1}))=$
$\displaystyle\begin{pmatrix}E_{2}&0\\\ -W_{1\times
2}&1\end{pmatrix}\begin{pmatrix}U_{2\times 2}&0\\\ V_{1\times
2}&1\end{pmatrix}\begin{pmatrix}E_{2}&0\\\ W_{1\times 2}&1\end{pmatrix}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}U_{2\times 2}&0\\\ -W_{1\times
2}U_{2\times 2}+V_{1\times 2}+W_{1\times 2}&1\end{pmatrix}.$

So we obtian

$\eta^{e}(J_{2}(\psi\varphi\psi^{-1}))=U_{2\times 2}\in
E_{2}(P\\{x_{3}\\})^{e}.$

Finally, if $\psi_{1}=\sigma(1,\alpha,g)$ with $\alpha\neq 0$, say
$w_{1}=\eta^{e}\pi_{3}^{e}(\frac{\partial g}{\partial x_{2}})$ and
$w_{2}=\eta^{e}\pi_{3}^{e}(\frac{\partial g}{\partial x_{3}})$, then we have

$\eta^{e}(J(\psi_{1}))=\begin{pmatrix}\alpha&w_{1}&w_{2}\\\ 0&1&0\\\
0&0&1\end{pmatrix}=\begin{pmatrix}W_{2\times 2}&G_{2\times 1}\\\
0&1\end{pmatrix}$

and

$\eta^{e}(J(\psi_{1})^{-1})=\begin{pmatrix}\alpha^{-1}&-\alpha^{-1}w_{1}&-\alpha^{-1}w_{2}\\\
0&1&0\\\ 0&0&1\end{pmatrix}=\begin{pmatrix}Y_{2\times 2}&Z_{2\times 1}\\\
0&1\end{pmatrix}.$

By (4.3) again, we deduce that

$\displaystyle\eta^{e}(J(\psi\varphi\psi^{-1}))=$
$\displaystyle\begin{pmatrix}Y_{2\times 2}&Z_{2\times 1}\\\
0&1\end{pmatrix}\begin{pmatrix}U_{2\times 2}&0\\\ V_{1\times
2}&1\end{pmatrix}\begin{pmatrix}W_{2\times 2}&G_{2\times 1}\\\
0&1\end{pmatrix}$ $\displaystyle=$ $\displaystyle\begin{pmatrix}Y_{2\times
2}U_{2\times 2}+Z_{2\times 1}V_{1\times 2}&Z_{2\times 1}\\\ V_{1\times
2}&1\end{pmatrix}\begin{pmatrix}W_{2\times 2}&G_{2\times 1}\\\
0&1\end{pmatrix}$ $\displaystyle=$ $\displaystyle\begin{pmatrix}Y_{2\times
2}U_{2\times 2}W_{2\times 2}+Z_{2\times 1}V_{1\times 2}W_{2\times 2}\ &\
Y_{2\times 2}U_{2\times 2}G_{2\times 1}+Z_{2\times 1}V_{1\times 2}G_{2\times
1}+Z_{2\times 1}\\\ V_{1\times 2}W_{2\times 2}&V_{1\times 2}G_{2\times
1}+1\end{pmatrix}$

Then by Lemma 3.7, we have

$1=V_{1\times 2}G_{2\times 1}+1.$

and

$Y_{2\times 2}U_{2\times 2}G_{2\times 1}+Z_{2\times 1}V_{1\times 2}G_{2\times
1}+Z_{2\times 1}=0.$

Assume $V_{1\times 2}=(v_{1},v_{2})$. Then we have

$v_{1}w_{2}=V_{1\times 2}G_{2\times 1}=0.$

And thus

$Z_{2\times 1}=-Y_{2\times 2}U_{2\times 2}G_{2\times 1}.$

Since $P\\{x_{3}\\}^{e}$ is commutative, we also have $w_{2}v_{1}=0$.
Therefore, we deduce that

$\displaystyle\eta^{e}(J_{2}(\psi\varphi\psi^{-1}))$ $\displaystyle=$
$\displaystyle Y_{2\times 2}U_{2\times 2}W_{2\times 2}+Z_{2\times 1}V_{1\times
2}W_{2\times 2}$ $\displaystyle=$ $\displaystyle Y_{2\times 2}U_{2\times
2}W_{2\times 2}-Y_{2\times 2}U_{2\times 2}G_{2\times 1}V_{1\times 2}W_{2\times
2}$ $\displaystyle=$ $\displaystyle Y_{2\times 2}U_{2\times
2}(E_{2}-G_{2\times 1}V_{1\times 2})W_{2\times 2}.$ $\displaystyle=$
$\displaystyle Y_{2\times 2}U_{2\times 2}E_{12}(-w_{2}v_{2})W_{2\times 2}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}\alpha^{-1}&0\\\
0&1\end{pmatrix}E_{12}(-w_{1})U_{2\times
2}E_{12}(-w_{2}v_{2})E_{12}(w_{1})\begin{pmatrix}\alpha&0\\\
0&1\end{pmatrix}.$

Note that for all $u\in P\\{x_{3}\\}^{e}$, we have [2, 16]

$E_{12}(u)\begin{pmatrix}\alpha&0\\\
0&1\end{pmatrix}=\begin{pmatrix}\alpha&0\\\
0&1\end{pmatrix}E_{12}(\alpha^{-1}u)\mbox{ and
}E_{21}(u)\begin{pmatrix}\alpha&0\\\
0&1\end{pmatrix}=\begin{pmatrix}\alpha&0\\\ 0&1\end{pmatrix}E_{21}(\alpha u).$

By the above reasoning, we obtain

$\eta^{e}(J_{2}(\psi\varphi\psi^{-1}))\in\begin{pmatrix}\alpha^{-1}&0\\\
0&1\end{pmatrix}\begin{pmatrix}\alpha&0\\\
0&1\end{pmatrix}E_{2}(P\\{x_{3}\\}^{e})=E_{2}(P\\{x_{3}\\}^{e}).$

The proof is completed. ∎

######  Corollary 4.3.

For $\varphi\in\mathsf{TA}(P_{3})$, if $\overline{\varphi}=\mathsf{id}$, then
$\eta^{e}(J_{2}(\varphi))\in E_{2}(P\\{x_{3}\\}^{e})$.

###### Proof.

By Lemma 2.1, we may assume that
$\varphi=\psi_{1}\varphi_{1}\psi_{1}^{-1}...\psi_{t}\varphi_{t}\psi_{t}^{-1}$
for some $t\geq 1$, where each $\varphi_{j}$ is an elementary automorphism of
the form (2.1) with $n=3$. Define
$\theta_{j}=\psi_{j}\varphi_{j}\psi_{j}^{-1}$ for each $1\leq j\leq t$. Then
$\overline{\theta_{j}}=\mathsf{id}$. By Lemma 3.4 and by induction on $t$, we
have

$J(\varphi)=J(\theta_{1}\dots\theta_{t})=\overline{\theta_{1}\dots\theta_{t-1}}^{e}(J(\theta_{t}))J(\theta_{1}\dots\theta_{t-1})=J(\theta_{t})J(\theta_{1}\dots\theta_{t-1})=J(\theta_{t})\dots
J(\theta_{1}).$

By Lemma 3.7 and by Theorem 4.2, we may assume

$\eta^{e}(J(\theta_{j}))=\begin{pmatrix}A_{j}&0\\\ V_{j}&1\\\ \end{pmatrix}$

such that $A_{j}\in E_{2}(P\\{x_{3}\\}^{e})$. By induction on $t$, it is
straightforward to show that

$\eta^{e}(J(\theta_{1}\dots\theta_{t}))=\eta^{e}(J(\theta_{t}))\dots\eta^{e}(J(\theta_{1}))=\begin{pmatrix}A_{t}A_{t-1}\dots
A_{1}&0\\\ V&1\end{pmatrix}.$

The result follows immediately. ∎

Now we are ready to prove our main result. Since
$\overline{\delta}\neq\mathsf{id}$, we need to construct a tame automorphism
$\psi$ such that $\overline{\delta\psi}=\mathsf{id}$.

######  Theorem 4.4.

The Anick type automorphism $\delta$ in (1.1) is wild.

###### Proof.

Clearly, we have $\overline{\delta}=(x_{1},x_{2}+x_{1}x_{3}^{2},x_{3})$.
Define $\psi=(x_{1},x_{2}-x_{1}x_{3}^{2},x_{3})\in\mathsf{TA}(P_{3})$. Then we
have $\overline{\delta\psi}=\bar{\delta}\bar{\psi}=\mathsf{id}$. Suppose that
$\delta$ is tame. Then by Corollary 4.3, we have

$\eta^{e}(J_{2}(\delta\psi))\in E_{2}(P\\{x_{3}\\}^{e}).$

On the other hand, by Lemma 3.4, we obtain

$\eta^{e}(J(\delta\psi))=\eta^{e}\overline{\delta}^{e}(J(\psi))\eta^{e}(J(\delta))=\begin{pmatrix}1&0&0\\\
-M_{x_{3}}^{2}&1&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}1+M_{x_{3}}H_{x_{3}}&-H_{x_{3}}^{2}&0\\\
M_{x_{3}}^{2}&1-M_{x_{3}}H_{x_{3}}&0\\\ 0&0&1\end{pmatrix}$

So by Lemma 4.1, we deduce a contradiction that

$\eta^{e}(J_{2}(\delta\psi))=\begin{pmatrix}1&0\\\
-M_{x_{3}}^{2}&1\end{pmatrix}\begin{pmatrix}1+M_{x_{3}}H_{x_{3}}&-H_{x_{3}}^{2}\\\
M_{x_{3}}^{2}&1-M_{x_{3}}H_{x_{3}}\\\ \end{pmatrix}\notin
E_{2}(P\\{x_{3}\\}^{e}).$

Therefore, $\delta$ is wild. ∎

We conclude the article with the observation that, as for the case of a
3-generated free associative algebra [5], the Anick type automorphism $\delta$
is stably tame. Suppose that $n=4$ and extend the Anick type automorphism
$\delta$ to

$\varphi=(\delta(x_{1}),\delta(x_{2}),x_{3},x_{4}).$

Then we have $\varphi=\varphi_{8}\dots\varphi_{1}$, where
$\varphi_{1}=\sigma(2,1,x_{3}x_{4})$, $\varphi_{2}=\sigma(1,1,[x_{3},x_{4}])$,
$\varphi_{3}=\sigma(4,1,-[x_{3},x_{2}])$,
$\varphi_{4}=\sigma(4,1,x_{1}x_{3})$, $\varphi_{5}=\sigma(2,1,-x_{3}x_{4})$,
$\varphi_{6}=\sigma(1,1,-[x_{3},x_{4}])$,
$\varphi_{7}=\sigma(4,1,[x_{3},x_{2}])$,
$\varphi_{8}=\sigma(4,1,-x_{1}x_{3})$.

For the convenience of the readers, we list the images below:

(i)
$x_{1}\xrightarrow{\varphi_{1}}x_{1}\xrightarrow{\varphi_{2}}x_{1}+[x_{3},x_{4}]\xrightarrow{\varphi_{3}}x_{1}+[x_{3},x_{4}]-[x_{3},[x_{3},x_{2}]]\xrightarrow{\varphi_{4}}x_{1}+[x_{3},x_{4}]+x_{3}[x_{3},x_{1}]-[x_{3},[x_{3},x_{2}]]\xrightarrow{\varphi_{5}}x_{1}+[x_{3},x_{4}]+x_{3}[x_{3},x_{1}]-[x_{3},[x_{3},x_{2}]]+x_{3}[x_{3},[x_{3},x_{4}]\xrightarrow{\varphi_{6}}x_{1}+x_{3}[x_{3},x_{1}]-[x_{3},[x_{3},x_{2}]]\xrightarrow{\varphi_{8}\varphi_{7}}x_{1}+x_{3}[x_{3},x_{1}]-[x_{3},[x_{3},x_{2}]]=\delta(x_{1})$;

(ii)
$x_{2}\xrightarrow{\varphi_{1}}x_{2}+x_{3}x_{4}\xrightarrow{\varphi_{2}}x_{2}+x_{3}x_{4}\xrightarrow{\varphi_{3}}x_{2}+x_{3}x_{4}-x_{3}[x_{3},x_{2}]\xrightarrow{\varphi_{4}}x_{2}+x_{3}x_{4}+x_{1}x_{3}^{2}-x_{3}[x_{3},x_{2}]\xrightarrow{\varphi_{5}}x_{2}+x_{1}x_{3}^{2}-x_{3}[x_{3},x_{2}]+x_{3}^{2}[x_{3},x_{4}]\xrightarrow{\varphi_{6}}x_{2}+x_{1}x_{3}^{2}-x_{3}[x_{3},x_{2}]\xrightarrow{\varphi_{8}\varphi_{7}}x_{2}+x_{1}x_{3}^{2}-x_{3}[x_{3},x_{2}]=\delta(x_{2})$;

(iii) $x_{3}\xrightarrow{\varphi}x_{3}$;

(iv)
$x_{4}\xrightarrow{\varphi_{2}\varphi_{1}}x_{4}\xrightarrow{\varphi_{4}\varphi_{3}}x_{4}+x_{1}x_{3}-[x_{3},x_{2}]\xrightarrow{\varphi_{5}}x_{4}+x_{1}x_{3}-[x_{3},x_{2}]+x_{3}[x_{3},x_{4}]\xrightarrow{\varphi_{6}}x_{4}+x_{1}x_{3}-[x_{3},x_{2}]\xrightarrow{\varphi_{7}}x_{4}+x_{1}x_{3}\xrightarrow{\varphi_{8}}x_{4}$.

### Disclosure statement

No potential conflict of interest was reported by the authors.

### Acknowledgement

The authors would like to thank Yu Li, Yanhua Wang, Yongjun Xu, James J.
Zhang, and Xiangui Zhao for their valuable discussions and suggestions.

## References

  * [1] R. M. Bryant and V. Drensky, Obstructions to lifting automorphisms of free algebras, Communications in Algebra 21(12) (1993) 4361–4389.
  * [2] P. M. Cohn, On the structure of the $GL_{2}$ of a ring, Inst. Hautes Et. Sci. Publ. Math. 30 (1966) 5–53.
  * [3] P. M. Cohn, Free rings and their relations (Second Edition), London: Academic Press, 1985.
  * [4] A. J. Czerniakiewicz, Automorphisms of a free associative algebra of rank 2, I, II, Trans. Amer. Math. Soc. 160 (1971) 393–401; 171 (1972), 309–315.
  * [5] V. Drensky, J. T. Yu, Automorphisms fixing a variable of $k\langle x,y,z\rangle$, Journal of algebra 291 (2005) 250–258.
  * [6] H. W. E. Jung, Über ganze birationale Transformationen der Ebene, J. reine angew. Math. 184 (1942) 161–174.
  * [7] W. van der Kulk, On polynomial rings in two variables, Nieuw Arch. Wisk. 3(1) (1953) 33–41.
  * [8] L. Makar-Limanov, The automorphisms of the free algebra of two generators, Funks. Anal. Prilozhen. 4(3) (1970) 107–108; English translation: Funct. Anal. Appl. 4 (1970) 262–263.
  * [9] L. Makar-Limanov, U. Turusbekova, U. U. Umirbaev, Automorphisms and derivations of free Poisson algebras in two variables, J. Algebra 322(9) (2009) 3318–3330.
  * [10] M. Nagata, On the automorphism group of $k[x,y]$, Lect. in Math., Kyoto Univ., Kinokuniya, Tokyo, 1972.
  * [11] I. P. Shestakov, Quantization of Poisson superalgebras and speciality of Jordan Poisson superalgebras, Algebra and Logic 32(5) (1993) 309–317.
  * [12] I. P. Shestakov, U. U. Umirbaev, Poisson brackets and two-generated subalgebras of rings of polynomials, J. Amer. Math. Soc. 17 (2004) 181–196.
  * [13] I. P. Shestakov, U. U. Umirbaev, The tame and the wild automorphisms of polynomial rings in three variables, J. Amer. Math. Soc. 17 (2004) 197–227.
  * [14] U. U. Umirbaev, On Schreier varieties of algebras, Algebra i Logika 33(3) (1994) 317–340 [in Russian]: Algebra and Logic 33(3) (1994) 180–193 [Engl. transl.].
  * [15] U. U. Umirbaev, Defining relations of the tame automorphism group of polynomial algebras in three variables, J. reine angew. Math. 600 (2006) 203–235.
  * [16] U. U. Umirbaev, The Anick automorphism of free associative algebras, J. reine angew. Math. 605 (2007) 165–178.
  * [17] U. U. Umirbaev, Universal enveloping algebras and universal derivations of Poisson algebras, Journal of Algebra 354 (2012) 77–94.