# Machine-learning parameter tracking with partial state observation

Zheng-Meng Zhai  School of Electrical, Computer and Energy Engineering,
Arizona State University, Tempe, AZ 85287, USA  Mohammadamin Moradi  School of
Electrical, Computer and Energy Engineering, Arizona State University, Tempe,
AZ 85287, USA  Bryan Glaz  Army Research Directorate, DEVCOM Army Research
Laboratory, 2800 Powder Mill Road, Adelphi, MD 20783-1138, USA  Mulugeta Haile
Army Research Directorate, DEVCOM Army Research Laboratory, 6340 Rodman Road,
Aberdeen Proving Ground, MD 21005-5069, USA  Ying-Cheng Lai  Ying-
<EMAIL_ADDRESS>School of Electrical, Computer and Energy Engineering,
Arizona State University, Tempe, AZ 85287, USA  Department of Physics, Arizona
State University, Tempe, Arizona 85287, USA

###### Abstract

Complex and nonlinear dynamical systems often involve parameters that change
with time, accurate tracking of which is essential to tasks such as state
estimation, prediction, and control. Existing machine-learning methods require
full state observation of the underlying system and tacitly assume adiabatic
changes in the parameter. Formulating an inverse problem and exploiting
reservoir computing, we develop a model-free and fully data-driven framework
to accurately track time-varying parameters from partial state observation in
real time. In particular, with training data from a subset of the dynamical
variables of the system for a small number of known parameter values, the
framework is able to accurately predict the parameter variations in time. Low-
and high-dimensional, Markovian and non-Markovian nonlinear dynamical systems
are used to demonstrate the power of the machine-learning based parameter-
tracking framework. Pertinent issues affecting the tracking performance are
addressed.

The behavior of a nonlinear dynamical system is controlled by its parameters.
In a real-world environment, the parameters typically change or drift with
time. For example, when an optical sensor system is deployed to an outdoor
environment, climatic disturbances such as temperature and humidity
fluctuations can cause the geometrical and material parameters of the system
to change with time. In an ecological system, seasonal fluctuations and human
influences on the environment can induce changes in the parameters underlying
the population dynamics such as the carrying capacity and species decay rates.
Often, due to the complex interactions between the system and the environment,
the simplistic assumption that the parameters drift linearly with time is not
valid. Rather, the variations of the parameters with time can be complicated.
A generic feature of nonlinear dynamical systems is that even a small
parameter change can lead to characteristically different and even
catastrophic behaviors. For example, a nonlinear system can typically exhibit
a variety of bifurcations including a crisis [1] at which a chaotic attractor
is destroyed and replaced by transient chaos [2], leading to system collapse.
Being able to predict or forecast how some key system parameters change with
time into the future can lead to control strategies to prevent system
collapse.

The problem of tracking parameter variations is an inverse problem, which is
difficult even if an accurate mathematical model of the system is known. Our
assumption is that the parameter of interest cannot be directly accessed or
measured, so tracking its variations will need to be done indirectly using the
measurements of some accessible dynamical variables of the system. Suppose
that a key parameter will change with time in the future but, at present the
system is stationary so that a few distinct values of this parameter can be
measured, together with the time series of a subset of the dynamical
variables. A scenario is that an instrument or device is to be deployed in
certain missions where the harsh and nonstationary environment will cause the
key parameter to change with time. Before deployment, the device can be tested
in a controlled laboratory environment where the values of the parameter and
the corresponding time series can be obtained. Assuming in the real
environment the parameter cannot be measured but some time series from partial
state observation still can be, we ask the question of whether it is possible
to extract the parameter variations. In this Letter, we demonstrate that
machine learning can be exploited to provide an affirmative answer.

The idea of exploiting machine learning for parameter tracking has been
investigated recently [3, 4, 5, 6, 7, 8] (see Supporting Information [9] for
an extensive background review of previous works on parameter identification
and tracking). In existing machine-learning works, the time series from all
dynamical variables and the time variations of the parameter is required for
training. Here, we articulate a machine-learning framework with the following
two main features that go beyond the existing methods: (1) only the
measurements from a partial set of the dynamical variables are needed, and (2)
observation of the state from a small number of parameter values suffices.
More specifically, let $\mathbf{x}\in\mathcal{R}^{D}$ be the $D$-dimensional
state vector of the dynamical system and let
$\mathbf{y}\in\mathcal{R}^{D^{\prime}}$ be the measurement or observation
vector: $\mathbf{y}=\mathbf{g}(\mathbf{x})$, where $D^{\prime}<D$ and
$\mathbf{g}:\mathcal{R}^{D}\rightarrow\mathcal{R}^{D^{\prime}}$ is the
measurement function. We choose reservoir computing [10, 11, 12, 13] as the
machine-learning architecture, which in recent years has been applied to
predicting nonlinear dynamical systems [14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], and propose the
following parameter tracking scheme. Suppose the goal is to track a single
parameter $p$ (for simplicity), so the output of the neural network is a
scalar quantity $o(t)$. In a well-controlled laboratory environment, vector
time series of dimension $D^{\prime}$ from a small number of parameter values
can be measured. We construct the input data by first breaking the measured
time series into a number of segments of equal length and recombining them to
form an integrated vector time series $\mathbf{y}(t)$. Corresponding to each
segment in $\mathbf{y}(t)$, there is an exact value of the parameter, thereby
generating a piecewise constant function of the parameter with time, denoted
as $p(t)$. The goal of training is to minimize the error between $o(t)$ and
$p(t)$, as shown in Fig. 1(a). This arrangement ensures that the neural
network learns the dynamical “climate” of the underlying system and how it
changes with time through alternating exposure to the measurements taking from
different parameter values. During the testing phase, e.g., when the system is
deployed to a real application environment, the parameter varies with time and
it is no longer accessible to observation or measurement. What can be observed
is vector time series $\mathbf{y}(t)$. When the well-trained neural network
takes in $\mathbf{y}(t)$ as the input, its output should give the time
variation of the parameter, realizing accurate parameter tracking, as
illustrated in Fig. 1(b).

Figure 1: Proposed reservoir-computing based parameter tracking scheme. The
goal is to track a parameter of the system from which only partial state
observation is available. The output of the reservoir computer is $o(t)$. (a)
In the training or laboratory calibration phase, the input to the neural
network is an integrated vector time series $\mathbf{y}_{T}(t)$ recombining
segments of the time series from partial state observation for a small number
of parameter values. The training goal is the corresponding piecewise constant
function of the parameter values versus time, denoted as $p(t)$, i.e.,
minimizing the error between $p(t)$ and $o(t)$. (b) In the testing or
deployment phase where the parameters are inaccessible, the input to the
neural network is the current observation vector time series $\mathbf{y}(t)$
and the output is the parameter as a function of time.

We consider three prototypical nonlinear dynamical systems: a three-species
chaotic food chain system [38], the chaotic Rössler oscillator [39], and the
Mackey-Glass delay-differential equation system [40]. The first two systems
are three-dimensional while the Mackey-Glass system is non-Markovian with an
infinite-dimensional phase space. For each system, three types of parameter
variations are considered: frequency modulation (FM), sawtooth wave, and
amplitude modulation (AM), with different numbers of parameter sampling in
different ranges for training. The machine-learning performance is
characterized by the root-mean-square error (RMSE). Here we present the
results from the chaotic food-chain system given by (results for the other two
systems can be found in SI [9]): $dR/dt=R(1-R/{\rm K})-{\rm
x_{c}y_{c}}CR/(R+{\rm R_{0}})$, $dC/dt={\rm x_{c}}C[{\rm y_{c}}R/(R+{\rm
R_{0}})-1]-{\rm x_{p}y_{p}}PC/(C+{\rm C_{0}})$, $dP/dt={\rm x_{p}}P[{\rm
y_{p}}C/(C+{\rm C_{0}})-1]$, where $R$, $C$, and $P$ are the population
densities of the resource, consumer, and predator species, respectively. The
system has seven parameters: ${\rm K,x_{c},y_{c},x_{p},y_{p},R_{0},C_{0}}>0$.
To illustrate the process of parameter tracking, we assume that the following
three parameters: ${\rm K,y_{c},y_{p}}$ are time-varying, and fix all other
parameters at constant values: $\rm x_{c}=0.4$, $x_{p}=0.08$, $C_{0}=0.5$, and
$R_{0}=0.16129$, according to some bioenergetics argument [38]. We focus on
tracking a single parameter: $\rm{K}$, $\rm{y_{c}}$, or $\rm{y_{p}}$, whose
respective nominal values are 0.94, 1.7, and 5.0. When tracking one of the
three parameters, the other two are fixed at their respective nominal values.

Figure 2: Tracking time-varying parameters of the chaotic food-chain system.
Different combinations of the parameter waveforms and partial state
observation are illustrated: the top, middle, and bottom row correspond to
three types of parameter variations (AM, FM, and sawtooth waveform), while the
gray shaded region in the left column illustrates the partial state
observation. The right column gives the results of parameter tracking in
comparison with the ground truth. (a,b) One-dimensional observational state
variable $\mathbf{y}=(R)$ for tracking AM parameter $\rm{K}$; (c,d) Two-
dimensional observation $\mathbf{y}=(R,C)$ for tracking FM parameter
$\rm{y_{c}}$; (e,f) Two-dimensional observation $\mathbf{y}=(C,P)$ for
tracking the sawtooth-wave parameter $\rm{y_{p}}$. The five horizontal dashed
lines indicate the parameter values from which training data are generated,
i.e., $s_{n}=5$.

Figure 2 exemplifies the results of tracking the three different types of
parameter variations for different combinations of the bifurcation parameter
and state observation. The computational setting is as follows. Time series
are generated by integrating the system model using the time step $dt=0.01$.
The initial states of both the dynamical process and the reservoir neural
network are randomly chosen from a uniform distribution. The training and
testing data are obtained by sampling the time series at the interval
$\Delta_{s}$. We set $\Delta_{s}=250dt=2.5$, corresponding to approximately
$1/25$ cycles of oscillation in the chaotic food-chain system. Let $\Delta
T_{s}$ be the switching time interval in which the target parameter is a
constant. The training time is $900\Delta T_{s}$. The testing length is chosen
to be slightly longer than the training length, enough for tracking several
cycles of the parameter variation. The size of the reservoir network is
$D_{r}=500$ and the bias number is set to one. The other six hyperparameters
are determined by Bayesian optimization (SI [9]), which are fixed during
training and testing. Often, a constant bias may arise between the machine
predicted parameter variation and the ground truth, which can be removed by
calibrating using two parameter values (SI [9]). The results in Fig. 2
indicate that the machine-learning framework is capable of accurate parameter
tracking based on partial state observation only.

Two characteristics of the bifurcation parameter values used in the training
which can affect significantly the performance of tracking are the number of
such parameter values (denoted as $s_{n}$) and the relative range of the
parameter variation (denoted as $s_{w}$) from which the training time series
data are generated. Intuitively, if the training data come from only one value
of the bifurcation parameter, it will not be possible for the reservoir
computer to learn the features of parameter variation, resulting in a large
tracking error. As $s_{n}$ increases from one, we expect the error to
decrease. What is the minimum number of bifurcation parameter values required
for accurate parameter tracking? Likewise, if the bifurcation parameter values
are taken from the full range of the actual parameter variation, accurate
tracking is likely, resulting in a small error. As the range $s_{w}$ is
reduced, the error will increase. How tolerant can the machine-learning
parameter tracking scheme be with respect to this range?

Figure 3: Effects of the training bifurcation-parameter values on tracking.
The two relevant quantities are $s_{n}$ \- the number of distinct bifurcation-
parameter values for training, and $s_{w}$ \- the relative parameter range
from which time series are taken for training. (a,c,e) Testing RMSE versus
$s_{n}$ for parameters $\rm{K}$, $\rm{y_{c}}$, and $\rm{y_{p}}$, respectively.
(b,d,f) Testing RMSE versus $s_{w}$ for $\rm{K}$, $\rm{y_{c}}$, and
$\rm{y_{p}}$, respectively. Each panel has results from FM, sawtooth, and AM
types of parameter variations, and each point is the result of averaging over
50 independent realizations. The results in (a,c,e) indicates that using time
series from three distinct values of the bifurcation parameter suffices to
guarantee accurate parameter tracking. The results in (b,d,f) suggests
acceptable parameter-tracking performance for $s_{w}>20\%$.

Figures 3(a,c,e) demonstrate the effect of varying $s_{n}$ on the parameter-
tracking performance for different combinations of the bifurcation parameter
and its time variations. It can be seen that, for all cases illustrated, the
testing error decreases dramatically as $s_{n}$ increases from one to three,
and remains approximately constant afterwards, indicating that using the time
series from as few as three values of the bifurcation parameter suffices for
accurate tracking of the actual parameter variations. Figures 3(b,d,f) show
the effect of varying the relative range $s_{w}$ on the tracking performance.
As $s_{w}$ decreases from $100\%$, the RMSE increases, but it does so in a
slow manner until $s_{w}$ falls below about $20\%$, indicating that the
machine-learning parameter tracking scheme is remarkably tolerant to the range
of the bifurcation parameter from which the time series for training are
generated.

A key feature of our work is that machine-learning parameter tracking can be
done using only partial state observation. But how partial can the observation
be while still delivering reasonable parameter tracking performance? For
$\mathbf{x}\equiv(x_{1},x_{2},x_{3})^{T}$, full state observation is
represented by $\mathbf{y}=\mathbf{x}$. There are six cases of partial state
observation: $\mathbf{y}=(x_{1})$, $\mathbf{y}=(x_{2})$, $\mathbf{y}=(x_{3})$,
$\mathbf{y}=(x_{1},x_{2})$, $\mathbf{y}=(x_{1},x_{3})$, and
$\mathbf{y}=(x_{2},x_{3})$. Altogether, there are seven distinct cases of
state observation. For each case, we conduct a reasonable of number (e.g., 50)
of tests. For each test, we define an error threshold. If the resulting
testing RMSE is below the threshold, the test is deemed successful. The
fraction of successful cases gives the success probability. We then calculate,
for different types of parameter variations, the success rate $\rm P_{s}$ and
use a bar graph to represent the rate for the seven cases of state
observation. Some representative results for the chaotic food-chain system are
shown in Fig. 4. It can be seen that in most cases, close to $100\%$ success
rate can be achieved by observing two state variables. Depending on the
specific parameter, in some cases observing even one state variable can lead
to satisfactory success rate.

Figure 4: Success probability for different partial observation scenarios for
the chaotic food-chain system. (a-c) Bar-graph representation of the success
rate for the seven distinct cases of state observation for the three different
bifurcation parameters, for three types of parameter variation (FM: green;
sawtooth: blue; AM: yellow). The success rate is calculated from 50
independent realizations. In most cases, observing two state variables can
lead to nearly $100\%$ success rate.

In summary, we developed a reservoir-computing based framework to continuously
track parameter variations with time based on observation from a subset of the
full state variables. Relying on partial observation is necessary in real
applications where not all the dynamical variables of the system are
accessible and there can be unknown hidden variables. Our unique training
scheme enables the reservoir computer to learn the correspondence between the
input time series and the underlying parameter value so that it is able to
forecast the parameter variations with time based in input time series from
partial state observation. We demonstrated the working of the proposed
parameter-tracking framework using chaotic systems subject to three distinct
types of parameter variations. The effects of a number of factors on the
tracking performance were investigated: minimally required bifurcation
parameter values for training, the number of state vectors associated with
partial-state observation, measurement and dynamical noises, network size and
training time, and the minimum switching time required of the training data
(SI [9]). The developed reservoir-computing based scheme represents a general
and robust parameter-tracking framework that can be deployed in real-world
applications.

A possible application is in epidemiology. For example, some virus has
seasonal behaviors: its spreading rate varies in different seasons. Our
framework can be used to track the spreading rate change with time based on
state evolution, thereby enabling accurate prediction of the infection scale
and period. Another potential application is predicting if a dynamical system
is about to approach a tipping point at which a transition from a normal to a
collapsing (e.g., large-scale extinction in an ecosystem) steady state occurs.
With inevitable noises, we can use the noisy time series from partial state
observation as the input to our machine-learning scheme to generate the time-
varying behavior of some key bifurcation parameters of the system. Based on
the predicted trend of the parameter variation, the chance of a tipping point
occurring in the near future can be assessed.

This work was supported by the Air Force Office of Scientific Research through
Grant No. FA9550-21-1-0438.

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# On polynomial automorphisms of Nagata type

Huarcaya, Jorge A. C. 1 and  Palacios, Joe 2,3 1 Universidad Nacional Mayor de
San Marcos, UNMSM  2 Universidad Nacional de Ingeniería, UNI  3 Instituto de
Matemática y Ciencias Afines, IMCA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We define a family of polynomial ring homomorphisms generalising the Nagata
automorphism, and relate the condition of non-vanishing of their Jacobian
determinant with a certain linear partial differential equation. We also give
a condition on that family to get wild automorphisms.

###### Key words and phrases:

polynomial automorphisms, Nagata automorphism, wild automorphisms, linear PDE

###### 2010 Mathematics Subject Classification:

14R10, 13B25, 13F20, 35F05

###### Contents

  1. 1 Introduction
  2. 2 Nagata polynomial map
  3. 3 Statement of the main results
  4. 4 Proof of the main results
  5. 5 Conclusions

##  1\. Introduction

In what follows $F$ will be a field of characteristic $0$ unless otherwise
stated. The polynomials:

$\displaystyle f(x,y,z)$ $\displaystyle=x-2y(zx+y^{2})-z(zx+y^{2})^{2},$
$\displaystyle g(x,y,z)$ $\displaystyle=y+z(zx+y^{2}),$ $\displaystyle
h(x,y,z)$ $\displaystyle=z.$

define an automorphism on $F[x,y,z]$ called Nagata automorphism [1]. The
determinant of the Jacobian matrix of this automorphism es $1$.

In the following paragraph we recall the notion of tame automorphisms of
polynomials. Let $1\leq i\leq n$, the $i$th elementary automorphism
$\Phi^{(i)}\colon F^{n}\to F^{n}$ is the automorphism defined as

$\Phi^{(i)}(x_{1},\dots,x_{n})=\Big{(}x_{1},\dots,x_{i-1},x_{i}+f(x_{1},\dots,x_{i-1},x_{i+1},\dots,x_{n}),x_{i+1},\dots,x_{n}\Big{)}$

and its inverse is the automorphism

${\Phi^{(i)}}^{-1}(x_{1},\dots,x_{n})=\Big{(}x_{1},\dots,x_{i-1},x_{i}-f(x_{1},\dots,x_{i-1},x_{i+1},\dots,x_{n}),x_{i+1},\dots,x_{n}\Big{)}\,.$

One may notice that the Jacobian matriz of $\Phi^{(i)}$ has determinant $1$.

An automorphism is tame if it is a finite composition of elementary and linear
automorphisms. An automorphism that is not tame is called wild.

In 1972, Nagata conjectured that the Nagata automorphism is wild. In [2,
Corollary 9], the authors proved this conjecture.

In this work, we define ring homomorphisms on $F[x,y,z]$, given in (2),
extending the Nagata automorphism which are wild automorphisms under some
assumptions, see Corollary 1). We also relate the condition of non-vanishing
of their Jacobian determinant with a certain linear partial differential
equation, see Theorem 1. As a consequence, we prove that those ring
homomorphisms on $F[x,y,z]$ with nonzero Jacobian determinant, are in fact
automorphisms, see Theorem 2.

##  2\. Nagata polynomial map

For a polynomial $\varphi\in F[x,y,z]$, we set

(1) $\begin{split}f(x,y,z)&=x-2y\varphi-z\varphi^{2}\,,\\\
g(x,y,z)&=y+z\varphi\,,\\\ h(x,y,z)&=z\,.\end{split}$

The triplet $(f,g,h)$ induces a ring homomorphism on $F[x,y,z]$. Note that
$(f,g,h)$ is the identity when one takes $\varphi=0$, so $(f,g,h)$ may be seen
as a perturbation of the identity. If $\varphi=xz+y^{2}$ then $(f,g,h)$ is the
Nagata automorphism.

The Milnor number of $(f,g,h)$ is the dimension

$\mu(f,g,h)=\dim_{F}\frac{F[x,y,z]}{\langle f,g,h\rangle}$

as vector space over $F$.

######  Lemma 1.

If $(f,g,h)$ is a ring homomorphism as in (2), then $\mu(f,g,h)=1$.

###### Proof.

One has,

$\displaystyle y$ $\displaystyle=(y+z\varphi)-\varphi z=g-\varphi z,$
$\displaystyle x$
$\displaystyle=(x-2y\varphi-z\varphi^{2})+2(y+z\varphi)\varphi-\varphi^{2}z=f+2\varphi
g-\varphi^{2}z,$

hence we get an equality $\langle f,g,h\rangle=\langle x,y,z\rangle$ of
polynomial ideals, and this implies that $\mu(f,g,h)=1$. ∎

######  Remark 1.

Note that $(f,g,h)$ is not always an automorphism for any $\varphi$ despite
the fact that $\mu(f,g,h)=1$ by Lemma 1. For instance, if $\varphi=x$ or
$\varphi=y$, then $(f,g,h)$ is not an automorphism. In the first case,
$(f,g,h)$ sends both points $(0,0,1)$ and $(-1,1,1)$ to $(0,0,1)$. Theorem 2
provides a necessary and sufficient condition on $\varphi$ such that $(f,g,h)$
is an automorphism.

######  Proposition 1.

The Jacobian matrix $J(f,g,h)$ has nonzero determinant in $F$ if and only if
$\varphi$ satisfies the linear PDE:

$-2y\varphi_{x}+z\varphi_{y}=0\,.$

###### Proof.

We have

$\displaystyle f_{x}$ $\displaystyle=1-2y\varphi_{x}-2z\varphi\varphi_{x},$
$\displaystyle f_{y}$
$\displaystyle=-2\varphi_{x}-2y\varphi_{y}-2z\varphi\varphi_{y},$
$\displaystyle g_{x}$ $\displaystyle=z\varphi_{x},$ $\displaystyle g_{y}$
$\displaystyle=1+z\varphi_{y}\,.$

Hence the Jacobian matrix $J(f,g,h)$ is

$\begin{pmatrix}1-2y\varphi_{x}-2z\varphi\varphi_{x}&-2\varphi_{x}-2y\varphi_{y}-2z\varphi\varphi_{y}&f_{z}\\\
z\varphi_{x}&1+z\varphi_{y}&g_{z}\\\ 0&0&1\end{pmatrix}$

whose determinant is

$\begin{vmatrix}1-2y\varphi_{x}-2z\varphi\varphi_{x}&-2\varphi_{x}-2y\varphi_{y}-2z\varphi\varphi_{y}\\\
z\varphi_{x}&1+z\varphi_{y}\end{vmatrix}=1-2y\varphi_{x}+z\varphi_{y}\,.$

If $-2y\varphi_{x}+z\varphi_{y}=0$ then the determinant of $J(f,g,h)$ is $1$.
Conversely, if the determinant of $J(f,g,h)$ is a constant $c\in F-\\{0\\}$,
then

$1-2y\varphi_{x}+z\varphi_{y}=c\,.$

However this equality is only possible when $c=1$, therefore
$-2y\varphi_{x}+z\varphi_{y}=0$. ∎

######  Proposition 2.

Let $p\in F[t_{1},t_{2}]$ and set $\varphi=p(xz+y^{2},z)$ in $F[x,y,z]$. Then
the ring homomorphism $(f,g,h)$ given in (2) is an automorphism on $F[x,y,z]$
with inverse:

(2) $\begin{split}f^{\prime}(x,y,z)&=x+2y\varphi-z\varphi^{2},\\\
g^{\prime}(x,y,z)&=y-z\varphi,\\\ h^{\prime}(x,y,z)&=z\,.\end{split}$

###### Proof.

We have

$\displaystyle\varphi(f^{\prime},g^{\prime},h^{\prime})$
$\displaystyle=p(f^{\prime}z+g^{\prime 2},z)$
$\displaystyle=p\Big{(}(x+2y\varphi-z\varphi^{2})z+(y-z\varphi)^{2},z\Big{)}$
$\displaystyle=p(xz+y^{2},z)$ $\displaystyle=\varphi\,,$

thus $\varphi(f^{\prime},g^{\prime},h^{\prime})=\varphi$. Hence,

$\displaystyle f(f^{\prime},g^{\prime},h^{\prime})$
$\displaystyle=f^{\prime}-2g^{\prime}\varphi(f^{\prime},g^{\prime},h^{\prime})-z\varphi^{2}(f^{\prime},g^{\prime},h^{\prime})$
$\displaystyle=(x+2y\varphi-z\varphi^{2})-2(y-z\varphi)\varphi-z\varphi^{2}$
$\displaystyle=x\,.$

On the other hand, we have

$\displaystyle g(f^{\prime},g^{\prime},h^{\prime})$
$\displaystyle=g^{\prime}+z\varphi(f^{\prime},g^{\prime},h^{\prime})$
$\displaystyle=(y-z\varphi)+z\varphi$ $\displaystyle=y\,.$

Therefore, the composite $(f,g,h)\circ(f^{\prime},g^{\prime},h^{\prime})$ is
the identity. Similarly, one verifies that
$(f^{\prime},g^{\prime},h^{\prime})\circ(f,g,h)$ is also the identity. ∎

For a polynomial $\psi$, we denote by $\overline{\psi}$ the highest degree
homogeneous component of $\psi$.

######  Proposition 3.

Let $\varphi\in F[x,y,z]$ and suppose that $(f,g,h)$ is an automorphism on
$F[x,y,z]$ of the form (2). If one the partial derivatives
$\overline{\varphi}_{x},\overline{\varphi}_{y}$ is nonzero, then the
automorphism $(f,g,h)$ is wild.

###### Proof.

Let $\overline{f},\overline{g},\overline{h}$ be the highest homogeneous
component of $f,g,h$, respectively. Since $\overline{\varphi}_{x}$ or
$\overline{\varphi}_{y}$ is nonzero, we have $\deg(\overline{\varphi})>0$, so

$\begin{split}\overline{f}&=-z\overline{\varphi}^{2}\,,\\\
\overline{g}&=z\overline{\varphi}\,,\\\ \overline{h}&=z\,.\end{split}$

Hence,

$\displaystyle\overline{f}_{x}\overline{g}_{y}-\overline{f}_{y}\overline{g}_{x}$
$\displaystyle=0$
$\displaystyle\overline{f}_{y}\overline{g}_{z}-\overline{f}_{z}\overline{g}_{y}$
$\displaystyle=-z\,\overline{\varphi}^{2}\overline{\varphi}_{y}$
$\displaystyle\overline{f}_{x}\overline{g}_{z}-\overline{f}_{z}\overline{g}_{x}$
$\displaystyle=-z\,\overline{\varphi}^{2}\overline{\varphi}_{x}$
$\displaystyle\overline{g}_{x}\overline{h}_{y}-\overline{g}_{y}\overline{h}_{x}$
$\displaystyle=0$
$\displaystyle\overline{g}_{y}\overline{h}_{z}-\overline{g}_{z}\overline{h}_{y}$
$\displaystyle=z\,\overline{\varphi}_{y}$
$\displaystyle\overline{g}_{x}\overline{h}_{z}-\overline{g}_{z}\overline{h}_{x}$
$\displaystyle=z\,\overline{\varphi}_{x}$
$\displaystyle\overline{f}_{x}\overline{h}_{y}-\overline{f}_{y}\overline{h}_{x}$
$\displaystyle=0$
$\displaystyle\overline{f}_{y}\overline{h}_{z}-\overline{f}_{z}\overline{h}_{y}$
$\displaystyle=-2z\,\overline{\varphi}_{y}$
$\displaystyle\overline{f}_{x}\overline{h}_{z}-\overline{f}_{z}\overline{h}_{x}$
$\displaystyle=-2z\,\overline{\varphi}_{x}\,.$

By [3, Corollary 2] or [2, Lemma 1], we deduce that
$\overline{f},\overline{g},\overline{h}$ are mutually algebraically
independent. On the other hand, from the hypothesis $\overline{\varphi}\in
F[x,y,z]$ is a homogeneous polynomial of positive degree such that
$\overline{\varphi}\not\in F[z]$, it is not difficult to see that none of the
polynomials $\overline{f},\overline{g},\overline{h}$ belongs to the subalgebra
generated by the other two polynomials. Therefore, by [2, Corollary 8]
$(f,g,h)$ is wild. ∎

We give the following definition in order to determine the highest weighted
homogeneous component of the polynomial $\varphi=p(xz+y^{2},z)$.

######  Definition 1.

Let us fix algebraically independent indeterminates $t_{1},\dots,t_{n}$ over
$F$. For $k\in\mathbb{Z}^{n}_{\geq 1}$, we write $t^{k}$ to denote the
monomial $t_{1}^{k_{1}}\cdots t_{n}^{k_{n}}$. A polynomial $h\in
F[t_{1},\dots,t_{n}]$ can be written as $h=\sum_{k}a_{k}t^{k}$.

  1. (1)

The support of $h$, denoted by $\mathrm{supp}(h)$, is defined as the set

$\mathrm{supp}(h)=\\{k\in\mathbb{Z}^{n}_{\geq 0}:a_{k}\neq 0\\}\,.$

  2. (2)

Let $v=(v_{1},\dots,v_{n})\in\mathbb{Z}^{n}_{\geq 1}$. The degree of $h$ with
respect to $v$, denoted by $d_{v}(h)$, is the maximum of the scalar product
$\langle v,k\rangle$ such that $a_{k}\neq 0$. We denote by $h^{v}$ the sum of
those terms $a_{k}t^{k}$ such that $\langle v,k\rangle=d_{v}(p)$. In
particular, if $v=(1,\dots,1)$ then $\deg(h)=d_{v}(h)$ and
$h^{v}=\overline{h}$.

  3. (3)

We say that $h$ is weighted homogeneous with respect to $v$ when $h^{v}=h$.

######  Lemma 2.

Let $p\in F[t_{1},t_{2}]$ and $v=(2,1)$. If $\varphi=p(xz+y^{2},z)\in
F[x,y,z]$, then $\deg(\varphi)=d_{v}(p)$ and
$\overline{\varphi}=p^{v}(xz+y^{2},z)$. In particular, if $p$ is weighted
homogeneous with respect to $v=(2,1)$, then $\varphi$ is homogeneous.

###### Proof.

Let us write the polymomial $p$ as $p=\sum_{k}a_{k}t^{k}$. One has

$\displaystyle\varphi=\sum_{k}a_{k}(xz+y^{2})^{k_{1}}z^{k_{2}}=\sum_{k}a_{k}\sum_{n=0}^{k_{1}}\binom{k_{1}}{n}x^{k_{1}-n}y^{2n}z^{k_{1}+k_{2}-n}\,.$

From the last equality, we observe that

(3)
$\mathrm{supp}(\varphi)=\Big{\\{}(k_{1}-n,2n,k_{1}+k_{2}-n):k\in\mathrm{supp}(p)\,\textnormal{and}\,\,0\leq
n\leq k_{1}\Big{\\}}\,.$

Therefore,

$\displaystyle\deg(\varphi)$
$\displaystyle=\max\Big{\\{}k_{1}-n+2n+k_{1}+k_{2}-n:(k_{1},k_{2})\in\mathrm{supp}(p)\,\textnormal{and}\,0\leq
n\leq k_{1}\Big{\\}}$
$\displaystyle=\max\Big{\\{}2k_{1}+k_{2}:(k_{1},k_{2})\in\mathrm{supp}(p)\Big{\\}}$
(4) $\displaystyle=d_{v}(p)\,.$

Hence, by the equalities given in (3) and (4), the highest degree homogeneous
component of $\varphi$ is computed as follows

$\displaystyle\overline{\varphi}=$
$\displaystyle\sum_{\begin{subarray}{c}k\in\mathrm{supp}(p)\\\ 0\leq n\leq
k_{1}\\\
2k_{1}+k_{2}=d_{v}(p)\end{subarray}}a_{k}\sum_{n=0}^{k_{1}}\binom{k_{1}}{n}x^{k_{1}-n}y^{2n}z^{k_{1}+k_{2}-n}$
$\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}k\in\mathrm{supp}(p)\\\ 0\leq n\leq
k_{1}\\\
2k_{1}+k_{2}=d_{v}(p)\end{subarray}}a_{k}\left(\sum_{n=0}^{k_{1}}\binom{k_{1}}{n}x^{k_{1}-n}z^{k_{1}-n}y^{2n}\right)z^{k_{2}}$
$\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}k\in\mathrm{supp}(p)\\\ 0\leq n\leq
k_{1}\\\
2k_{1}+k_{2}=d_{v}(p)\end{subarray}}a_{k}\left(\sum_{n=0}^{k_{1}}\binom{k_{1}}{n}(xz)^{k_{1}-n}y^{2n}\right)z^{k_{2}}$
$\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}k\in\mathrm{supp}(p)\\\
2k_{1}+k_{2}=d_{v}(p)\end{subarray}}\left(xz+y^{2}\right)^{k_{1}}z^{k_{2}}$
$\displaystyle=$ $\displaystyle p^{v}(xz+y^{2},z)\,,$

and it proves the assertion. ∎

######  Example 1.

If $\varphi(x,y,z)=p(xz+y^{2},z)$, then it is not always true that
$\overline{\varphi}(x,y,z)=\overline{p}(xz+y^{2},z)$. Indeed, consider
$p(t_{1},t_{2})=t_{1}^{2}-t_{2}^{3}+t_{1}t_{2}^{2}\in F[s,t]$, then by a
immediate computation one has

$\overline{\varphi}=\left(xz+y^{2}\right)^{2}+\left(xz+y^{2}\right)z^{2}\neq-z^{3}+\left(xz+y^{2}\right)z^{2}=\overline{p}(xz+y^{2},z)\,.$

However, it is easy to verify that if $v=(2,1)$, then $d_{v}(p)=4$ and
$p^{v}=t_{1}^{2}+t_{1}t_{2}^{2}$. Moreover, by direct computation one has

$\overline{\varphi}=\left(xz+y^{2}\right)^{2}+\left(xz+y^{2}\right)z^{2}=p^{v}(xz+y^{2},z)\,.$

######  Corollary 1.

Let $p\in F[t_{1},t_{2}]$ and set $\varphi=p(xz+y^{2},z)$ in $F[x,y,z]$. If
the partial derivative $(p^{v})_{t_{1}}$ is nonzero, then the corresponding
ring homomorphism $(f,g,h)$ given in (2) is a wild automorphism.

###### Proof.

By Proposition 2, $(f,g,h)$ is an automorphism and Lemma 2 establishes the
equality $\overline{\varphi}=p^{v}(xz+y^{2},z)$. Hence the fact that
$(p^{v})_{t_{1}}\neq 0$ implies that $\overline{\varphi}_{x}\neq 0$ and
$\overline{\varphi}_{y}\neq 0$. By Proposition 3, the automorphism $(f,g,h)$
is wild. ∎

######  Example 2.

Let $\varphi=(xz+y^{2})^{n}$ with $n\geq 1$. Corollary 1 implies that

$\displaystyle f(x,y,z)$ $\displaystyle=x-2y(xz+y^{2})^{n}-z(xz+y^{2})^{2n},$
$\displaystyle g(x,y,z)$ $\displaystyle=y+z(xz+y^{2})^{n},$ $\displaystyle
h(x,y,z)$ $\displaystyle=z,$

is a wild automorphism.

######  Example 3.

Let $\varphi(x,y,z)=p(xz+y^{2},z)$, where
$p(t_{1},t_{2})=t_{1}^{2}-t_{2}^{3}+t_{1}t_{2}^{2}\in F[s,t]$ as in Example 1.
Since $p^{v}=t_{1}^{2}+t_{1}t_{2}^{2}$ and
$(p^{v})_{t_{1}}=2t_{1}+t_{1}^{2}\neq 0$, Corollary 1 implies that
corresponding ring homomorphism $(f,g,h)$ is a wild automorphism.

##  3\. Statement of the main results

The main results in this work are the following:

######  Theorem 1.

The linear PDE

(5) $-2y\varphi_{x}+z\varphi_{y}=0$

has as general solution in $F[x,y,z]$, polynomials that can be written as:

$\varphi=p(xz+y^{2},z)\,,$

where $p\in F[t_{1},t_{2}]$.

######  Theorem 2.

Let $\varphi\in F[x,y,z]$ and set

$\displaystyle f(x,y,z)$ $\displaystyle=x-2y\varphi-z\varphi^{2}\,,$
$\displaystyle g(x,y,z)$ $\displaystyle=y+z\varphi\,,$ $\displaystyle
h(x,y,z)$ $\displaystyle=z\,.$

Then, the determinant of the Jacobian matrix $J(f,g,h)$ is a nonzero constant
in $F$ if and only if $(f,g,h)$ is an automorphism on $F[x,y,z]$. In this
case, there exists a polynomial $p\in F[t_{1},t_{2}]$ such that
$\varphi=p(xz+y^{2},z)$, moreover the inverse of $(f,g,h)$ is given in (2).

The proofs of these results are given after the following previous results.

######  Lemma 3.

Let $\varphi=\varphi^{0}+\varphi^{1}+\cdots+\varphi^{n}$ be a solution of the
linear PDE:

(6) $A(x,y,z)\varphi_{x}+B(x,y,z)\varphi_{y}+C(x,y,x)\varphi_{z}=0\,,$

where each $\varphi^{k}\in F[x,y.z]$ is the homogeneous component of $\varphi$
of degree $k$ for $0\leq k\leq n$, and $A,B,C$ are homogeneous polynomials of
the same degree. Then each homogeneous component $\varphi^{k}$ is also a
solution of the equation (6).

###### Proof.

Replacing $\varphi=\sum_{k=0}^{n}\varphi^{k}$ into the linear PDE (5), we
obtain:

$\sum_{k=0}^{n}\Big{(}A(x,y,z)\varphi^{k}_{x}+B(x,y,z)\varphi^{k}_{y}+C(x,y,x)\varphi^{k}_{z}\Big{)}=0$

which is a sum of homogeneous polynomials of different degrees, therefore:

$A(x,y,z)\varphi^{k}_{x}+B(x,y,z)\varphi^{k}_{y}+C(x,y,x)\varphi^{k}_{z}=0\,\,\,\textnormal{for
all}\,\,k=0,1,\dots,n\,.$

∎

By the previous lemma, to find the general solution of the equation (5), it is
enough to deal with its homogeneous polynomials components.

Let $\varphi\in F[x,y,z]$ be a homogeneous polynomial of degree $n$, if we
consider the canonical isomorphism of rings $F[x,y,z]\cong F[x,y][z]$, then we
can suppose that $\varphi$ is written as:

(7)
$\varphi=\varphi^{n}+\varphi^{n-1}z+\varphi^{n-2}z^{2}+\dots+\varphi^{1}z^{n-1}+\varphi^{0}z^{n}=\sum_{k=0}^{n-1}\varphi^{n-k}z^{k}+\varphi^{0}z^{n}$

with $\varphi^{k}\in F[x,y]$ homogeneous polynomials of degree $k$ for
$k=0,1,\dots,n$.

######  Lemma 4.

Let $\varphi$ be a homogeneous polynomial in $F[x,y,z]$ of degree $n$ that is
a solution of the linear PDE (5). Then, $\varphi^{n}=ay^{n}$ and
$\varphi^{1}=bx$, where $a$ and $b$ are constants in $F$, and moreover

$2y\varphi^{n-k-1}_{x}=\varphi^{n-k}_{y}$

for all $k=0,1,\dots,n-2$.

###### Proof.

Let us derivate $\varphi$ with respect to $x$ and $y$:

$\displaystyle\varphi_{x}$
$\displaystyle=\sum_{k=0}^{n-1}\varphi^{n-k}_{x}z^{k}=\varphi_{x}^{n}+\sum_{k=1}^{n-1}\varphi^{n-k}_{x}z^{k}\,,$
$\displaystyle\varphi_{y}$
$\displaystyle=\sum_{k=0}^{n-1}\varphi^{n-k}_{y}z^{k}=\sum_{k=0}^{n-2}\varphi^{n-k}_{y}z^{k}+\varphi^{1}_{y}z^{n-1}\,.$

Substituting these equalities in the equation (5), we have

$-2y\varphi_{x}^{n}+\sum_{k=1}^{n-1}-2y\varphi^{n-k}_{x}z^{k}+\sum_{k=0}^{n-2}\varphi^{n-k}_{y}z^{k+1}+\varphi^{1}_{y}z^{n}=0\,,$

and by index rescaling, we get

$-2y\varphi_{x}^{n}+\sum_{k=0}^{n-2}-2y\varphi^{n-k-1}_{x}z^{k+1}+\sum_{k=0}^{n-2}\varphi^{n-k}_{y}z^{k+1}+\varphi^{1}_{y}z^{n}=0\,.$

Hence,

$\displaystyle-2y\varphi_{x}^{n}+\sum_{k=0}^{n-2}\left(-2y\varphi^{n-k-1}_{x}+\varphi^{n-k}_{y}\right)z^{k+1}+\varphi^{1}_{y}z^{n}=0\,.$

This equation gives the equalities $\varphi^{n}_{x}=0$ and
$\varphi^{1}_{y}=0$, so $\varphi^{n}=ay^{n}$ and $\varphi^{1}=bx$ for some
constants $a,b\in F$. Moreover,

(8) $2y\varphi^{n-k-1}_{x}=\varphi^{n-k}_{y}\,\,\,\textnormal{para
todo}\,\,\,k=0,1,\dots,n-2\,.$

∎

The following result, will enable us to know how a solution can be written of
the equation (5).

######  Proposition 4.

Let $\varphi\in F[x,y,z]$ be a homogeneous solution of the linear PDE (5). We
have the following assertions:

  1. ($a$)

If $\varphi$ has degree $2n$, then

$\varphi=a(xz+y^{2})^{n}+z\psi$

where $a\in F$ and $\psi\in F[x,y,z]$ is a homogeneous polynomial which is
also a solution of the equation (5) of degree $2n-1$.

  2. ($b$)

If $\varphi$ has degree $2n+1$, then

$\varphi=z\psi$

where $\psi\in F[x,y,z]$ is homogeneous of degree $2n$ which is also a
solution of (5) of degree $2n$.

###### Proof.

($a$). Let us suppose that:

(9)
$\varphi=\varphi^{2n}+\varphi^{2n-1}z+\cdots+\varphi^{1}z^{2n-1}+\varphi^{0}z^{2n}$

with $\varphi^{2n-k}$ be a homogeneous polynomial in $F[x,y]$ of degree $2n-k$
for $0\leq k\leq 2n$. By Lemma 4, we get the following equalities:

(10) $\displaystyle\varphi^{2n}$
$\displaystyle=ay^{2n}\,\,\,\,\textnormal{and}\,\,\,\,\varphi^{1}=bx$ (11)
$\displaystyle 2y\varphi^{2n-k-1}_{x}$
$\displaystyle=\varphi^{2n-k}_{y}\,\,\,\textnormal{para
todo}\,\,\,k=0,1,\dots,2n-2\,.$

We claim that there are constants $a,a^{2n-k}_{j}$, with $1\leq k\leq n$ y
$0\leq j\leq k-1$, such that:

(12) $\displaystyle\varphi^{2n-k}$
$\displaystyle=a\binom{n}{k}y^{2n-2k}x^{k}+\sum_{j=0}^{k-1}a^{2n-k}_{k-1-j}y^{2n-2k+1+j}x^{k-1-j}$

Indeed, taking $k=0$, from these equalities (10) and (11) we find that:

$2y\varphi^{2n-1}_{x}=\varphi^{2n}_{y}=2nay^{2n-1}$

so $\varphi_{x}^{2n-1}=nay^{2n-2}$. Hence, by integrating with respect to $x$,
we obtain

$\varphi^{2n-1}=nay^{2n-2}x+a^{2n-1}_{0}y^{2n-1}\,,$

which can be written as

$\displaystyle\varphi^{2n-1}=a\binom{n}{1}y^{2n-2}x+a^{2n-1}_{0}y^{2n-1}\,,$

thus (12) is true for $k=1$. Suppose we have (12). Taking the derivative with
respect to $y$, we have

$\varphi^{2n-k}_{y}=2a\binom{n}{k}(n-k)y^{2n-2k-1}x^{k}+\sum_{j=0}^{k-1}\,.(2n-2k+1+j)a^{2n-k}_{k-1-j}y^{2n-2k+j}x^{k-1-j}$

From the equality (11), we get

$\varphi^{2n-k-1}_{x}=a\binom{n}{k}(n-k)y^{2n-2k-2}x^{k}+\sum_{j=0}^{k-1}\frac{(2n-2k+1+j)a^{2n-k}_{k-1-j}}{2}y^{2n-2k-1+j}x^{k-1-j}\,.$

Integrating with respect to $x$, there is a constant $a^{2n-k-1}_{0}\in F$
such that

$\varphi^{2n-k-1}=a\binom{n}{k+1}y^{2n-2k-2}x^{k+1}+\sum_{j=0}^{k-1}\frac{(2n-2k+1+j)a^{2n-k}_{k-1-j}}{2(k-1-j)}y^{2n-2k-1+j}x^{k-j}+a^{2n-k-1}_{0}y^{2n-k-1}\,.$

Defining $a^{2n-k-1}_{k-j}=\frac{(2n-2k+1+j)a^{2n-k}_{k-1-j}}{2(k-1-j)}$, we
get

$\varphi^{2n-k-1}=a\binom{n}{k+1}y^{2n-2k-2}x^{k+1}+\sum_{j=0}^{k}a^{2n-k-1}_{k-j}y^{2n-2k-1+j}x^{k-j}$

as wished, and this completes the recurrence.

Now, replacing the expressions obtained in (10) and (12) in the equation (9),
we have

$\displaystyle\varphi$
$\displaystyle=\sum_{k=0}^{n}\varphi^{2n-k}z^{k}+\sum_{k=n+1}^{2n}\varphi^{2n-k}z^{k}$
$\displaystyle=\varphi^{2n}+\sum_{k=1}^{n}\varphi^{2n-k}z^{k}+\sum_{k=n+1}^{2n}\varphi^{2n-k}z^{k}$
$\displaystyle=ay^{2n}+\sum_{k=1}^{n}\left(a\binom{n}{k}y^{2n-2k}x^{k}+\sum_{j=0}^{k-1}a^{2n-k}_{k-1-j}y^{2n-2k+1+j}x^{k-1-j}\right)z^{k}+\sum_{k=n+1}^{2n}\varphi^{2n-k}z^{k}$
$\displaystyle=a\sum_{k=0}^{n}\binom{n}{k}y^{2n-2k}x^{k}z^{k}+\sum_{k=1}^{n}\left(\sum_{j=0}^{k-1}a^{2n-k}_{k-1-j}y^{2n-2k+1+j}x^{k-1-j}\right)z^{k}+\sum_{k=n+1}^{2n}\varphi^{2n-k}z^{k}\,.$

From the last equality, we deduce that

$\varphi=a(xz+y^{2})^{n}+z\psi,$

where $\psi\in F[x,y,z]$ is a homogeneous polynomial of degree equal to
$2n-1$. By the linearity of the solutions of the equation (5), we obtain that
$z\psi$ is a solution, hence $\psi$ is also a solution.

($b$) Let us suppose that:

(13)
$\varphi=\varphi^{2n+1}+\varphi^{2n}+\cdots+\varphi^{1}z^{2n}+\varphi^{0}z^{2n+1}\,,$

where $\varphi^{2n-k}$ is a homogeneous polynomial in $F[x,y]$ of degree equal
to $2n+1-k$, for all $0\leq k\leq 2n+1$. By Lemma 4, we get the following
equalities:

(14) $\displaystyle\varphi^{2n+1}$ $\displaystyle=ay^{2n+1}$ (15)
$\displaystyle 2y\varphi^{2n-k}_{x}$
$\displaystyle=\varphi^{2n+1-k}_{y}\,\,\,\textnormal{para
todo}\,\,\,k=0,1,\dots,2n-1\,.$

As in the even case, we claim that there are constants $a,a_{j}^{2n-k}$, with
$0\leq k\leq n-1$ and $0\leq j\leq k$, such that:

(16)
$\varphi^{2n-k}=\frac{(2n-2k+1)\cdots(2n-1)(2n+1)}{2^{k}\cdot(k+1)!}ay^{2n-2k-1}x^{k+1}+\sum_{j=0}^{k}b^{2n-k}_{k-j}y^{2n-2k+j}x^{k-j}\,.$

Replacing $k=0$ in the equation (14) and joining with (15), we have

$2y\varphi^{2n}_{x}=\varphi^{2n+1}_{y}=(2n+1)ay^{2n}\,,$

hence $\varphi_{x}^{2n}=(2n+1)ay^{2n-1}$. Integrating with respect to $x$, we
obtain

$\varphi^{2n}=(2n+1)ay^{2n-1}x+b^{2n}_{0}y^{2n}$

thus (16) is true for $k=1$. Suppose we have (16). Taking the derivative with
respect to $y$, we have

$\varphi^{2n-k}_{y}=\frac{(2n-2k-1)\cdots(2n-1)(2n+1)}{2^{k}\cdot(k+1)!}ay^{2n-2k-2}x^{k+1}+\sum_{j=0}^{k}(2n-2k+j)b^{2n-k}_{k-j}y^{2n-2k+j-1}x^{k-j}\,.$

From the equality (15), we get

$\varphi^{2n-k-1}_{x}=\frac{(2n-2k-1)\cdots(2n-1)(2n+1)}{2^{k+1}\cdot(k+1)!}ay^{2n-2k-3}x^{k+1}+\sum_{j=0}^{k}\frac{(2n-2k+j)b^{2n-k}_{k-j}}{2}y^{2n-2k+j-2}x^{k-j}\,.$

Integrating with respect to $x$, there is a constant $b^{2n-k-1}_{0}\in F$
such that $\varphi^{2n-k-1}$ is equal to

$\frac{(2n-2k-1)\cdots(2n-1)(2n+1)}{2^{k+1}\cdot(k+2)!}ay^{2n-2k-3}x^{k+2}+\sum_{j=0}^{k}\frac{(2n-2k+j)b^{2n-k}_{k-j}}{2(k-j+1)}y^{2n-2k+j-2}x^{k-j+1}+b^{2n-k-1}_{0}y^{2n-k-1}\,.$

Defining $b^{2n-k-1}_{k-j}=\frac{(2n-2k+j)b^{2n-k}_{k-j}}{2(k-j+1)}$, we get

$\varphi^{2n-k-1}=\frac{(2n-2k+-1)\cdots(2n-1)(2n+1)}{2^{k+1}\cdot(k+2)!}ay^{2n-2k-3}x^{k+2}+\sum_{j=0}^{k+1}\frac{(2n-2k+j)b^{2n-k}_{k-j}}{2(k-j+1)}y^{2n-2k+j-2}x^{k-j+1}$

as wished, and this completes the recurrence. In particular, replacing $k=n-1$
in (16), we obtain:

$\varphi^{n+1}=\frac{1\cdot 3\cdots(2n+1)}{2^{n-1}\cdot
n!}ayx^{n}+\sum_{j=0}^{n-1}b^{n+1}_{n-1-j}y^{j+2}x^{n-1-j}\,.$

We solve the equation (15) for $k=n$ and we get:

$\displaystyle-2y\varphi^{n}_{x}$ $\displaystyle=\varphi^{n+1}_{y}\,,$ (17)
$\displaystyle-2y\varphi^{n}_{x}$ $\displaystyle=\frac{1\cdot
3\cdots(2n+1)}{2^{n-1}\cdot
n!}ax^{n}+\sum_{j=0}^{n-1}(j+2)b^{n+1}_{n-1-j}y^{j+1}x^{n-1-j}\,.$

From the equality (17), we deduce that $a=0$ and as consequence from this we
obtain that $\varphi^{2n+1}=0$. Therefore,

$\displaystyle\varphi=z\varphi^{2n}+z^{2}\varphi^{2n-1}+\cdots+z^{2n}\varphi^{1}+\varphi^{0}z^{2n+1}=z\psi\,,$

where $\psi\in F[x,y,z]$ is a homogeneous polynomial of degree $2n$, and as in
the previous case, we obtain that $\psi$ is also a solution.

∎

##  4\. Proof of the main results

###### Proof of Theorem 1.

Suppose that $\varphi=p(xz+y^{2},z)$, where $p$ is a polynomial in
$F[t_{1},t_{2}]$. Replacing the partial derivatives

$\displaystyle\varphi_{x}$ $\displaystyle=z\cdot p_{s}(xz+y^{2},z)\,,$
$\displaystyle\varphi_{y}$ $\displaystyle=2y\cdot p_{s}(xz+y^{2},z)\,.$

in the linear PDE (5), one verifies that $\varphi=p(xz+y^{2},z)$ is a
solution. Reciprocally, suppose that $\varphi\in F[x,y,z]$ is a solution of
the equation (5). We shall prove that $\varphi$ has the form
$\varphi=p(xz+y^{2},z)$, where $p$ is a polynomial in $F[t_{1},t_{2}]$. By
Lemma 3, we can assume that $\varphi\in F[x,y,z]$ is a homogeneous polynomial.
More precisely, we shall prove by recurrence the following:

(18)
$\varphi=\left\\{\begin{array}[]{rcl}\sum_{k=0}^{n}a_{2k}z^{2k}(xz+y^{2})^{2n-2k}&\mbox{if}&\deg\varphi=2n\,,\\\
&&\\\
\sum_{k=1}^{n}a_{2k-1}z^{2k-1}(xz+y^{2})^{2n-2k}&\mbox{if}&\deg\varphi=2n-1\,.\end{array}\right.$

Indeed, applying Proposition 4 for $n=1$, we obtain:

$\varphi=\left\\{\begin{array}[]{ccl}a(xz+y^{2})+z\psi&\mbox{if}&\deg\varphi=2\,,\\\
&&\\\ z\psi_{1}&\mbox{if}&\deg\varphi=1\,,\end{array}\right.$

where $\psi$ and $\psi_{1}$ are homogeneous polynomials with $\deg\psi=1$ and
$\deg\psi_{1}=0$. Then there are constants $b,c\in F$ such that $\psi=cz$ and
$\psi_{1}=b\in F$, so (18) is true for $n=1$.
Let us suppose that (18) holds for homogeneous polynomial solutions of degree
$2n$ or $2n-1$. If $\deg\varphi=2n+2$, then by Proposition 4 ($a$), we obtain
that $\varphi=a(xz+y^{2})^{2n+2}+z\psi$, where $\psi$ is a solution of the
equation (5). By the inductive hypothesis, we have

$\psi=\sum_{k=1}^{n}a_{2k+1}z^{2k-1}(xz+y^{2})^{2n-2k}\,.$

Hence,

$\displaystyle\varphi$ $\displaystyle=a(xz+y^{2})^{2n+2}+z\psi$
$\displaystyle=a(xz+y^{2})^{2n+2}+z\left(\sum_{k=1}^{n}a_{2k+1}z^{2k-1}(xz+y^{2})^{2n-2k}\right)\,.$

If $\deg\varphi=2n+1$, then by Proposition 4 ($b$), we obtain that
$\varphi=z\psi$, where $\psi$ is a solution of the equation (5) with
$\deg\psi=2n$. By the inductive hypothesis, one has

$\psi=\sum_{k=0}^{n}a_{2k}z^{2k}(xz+y^{2})^{2n-2k}\,.$

Hence,

$\displaystyle\varphi$ $\displaystyle=z\psi$
$\displaystyle=z\left(\sum_{k=0}^{n}a_{2k}z^{2k}(xz+y^{2})^{2n-2k}\right)$
$\displaystyle=\sum_{k=0}^{n}a_{2k}z^{2k+1}(xz+y^{2})^{2n-2k}$
$\displaystyle=\sum_{k=1}^{n+1}a_{2k-2}z^{2k-1}(xz+y^{2})^{2(n+1)-2k}\,,$

which completes the proof. ∎

###### Proof of the Theorem 2.

One implication is straightforward. Suppose that $\det J(f,g,h)$ is a nonzero
constant. By Proposition 1, $\varphi$ satisfies the linear PDE (5). By Theorem
1, the solution of the equation (5) has the form $\varphi=p(xz+y^{2},z)$ where
$p$ is a polynomial in $F[t_{1},t_{2}]$. Finally, by Proposition 2, $(f,g,h)$
is an automorphism and its inverse is given in (2). ∎

##  5\. Conclusions

Theorem 2 says that the automorphisms of the form (2) satisfy the well know
Jacobian conjecture via the approach of solving first order partial
differential equations. A further problem is whether those automorphisms might
be naturally extended on polynomial rings with more than three variables
satisfying the aforementioned result.

## References

  * [1] Nagata, M., (1972) On the automorphism group of $k[x,y]$. Lectures in Math., Kyoto Univ., Kinokuniya, Tokyo.
  * [2] Shestakov, I. P., Umirbaev, U. U. (2004). The Tame and the Wild Automorphisms of Polynomial Rings in Three Variables. Journal of the American Mathematical Society, 17(1), 197–227. http://www.jstor.org/stable/20161190
  * [3] Shestakov, I. P., Umirbaev,U. U. (2004) Poisson brackets and two-generated subalgebras of rings of polynomials., J. Am. Math. Soc. 17 no. 1, p. 181-196.


# Evidence of random spin-singlet state in a three-dimensional quantum spin
liquid candidate Sr3CuNb2O9

S. M. Hossain  Department of Physics, Shiv Nadar Institution of Eminence,
Gautam Buddha Nagar, Uttar Pradesh 201314, India  S. S. Rahaman  S.N. Bose
National Centre for Basic Sciences, Salt Lake, Kolkata 700106, India  H.
Gujrati  Department of Physics, Shiv Nadar Institution of Eminence, Gautam
Buddha Nagar, Uttar Pradesh 201314, India  Dilip Bhoi  The Institute for Solid
State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan.  A. Matsuo
The Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba
277-8581, Japan.  K. Kindo  The Institute for Solid State Physics, University
of Tokyo, Kashiwa, Chiba 277-8581, Japan.  M. Kumar
<EMAIL_ADDRESS>S.N. Bose National Centre for Basic Sciences,
Salt Lake, Kolkata 700106, India  M. Majumder<EMAIL_ADDRESS>Department
of Physics, Shiv Nadar Institution of Eminence, Gautam Buddha Nagar, Uttar
Pradesh 201314, India

###### Abstract

Disorder is ubiquitous in any quantum many-body system and is usually
considered to be an obstacle to the elucidation of the underlying physics of
complex systems, but its presence can often introduce exotic phases of matter
that cannot generally be realized in a clean system. We report here a detailed
experimental and theoretical study of magnetic properties of highly disordered
Sr3CuNb2O9 material which exhibits random site mixing between Cu and Nb. The
magnetic moments (Cu2+) are arranged in a quasi-cubic (three-dimensional)
manner, leading to a high degree of frustration with a Curie-Weiss temperature
($\theta_{CW}$) of about -60 K without any long-range magnetic ordering down
to 466 mK. These observations suggest that Sr3CuNb2O9 is a candidate for a
quantum spin liquid. More interestingly, the susceptibility
($\chi=M/\mu_{0}H$) and the $C_{m}/T$ ($C_{m}$ is the magnetic part of the
heat capacity) follow a power-law behavior with decreasing temperature. In
addition, $M(T,\mu_{0}H)$ and $C_{m}(T,\mu_{0}H)/T$ show scaling relationships
over a wide temperature and field range. This unusual behavior with respect to
the conventional behavior of a QSL can be discussed qualitatively as the
coexistence of a disorder-induced random spin singlet (RSS) state and a QSL
state. A quantitative description has been given by numerical calculations
considering a power-law probability distribution $P(J)\propto J^{-\gamma}$
($J$ is the exchange interaction) of random spin singlets. The parameters
extracted from the numerical calculations are in excellent agreement with the
experimental data. Furthermore, the analytical results are also consistent
with the power-law and scaling behavior of $\chi$ and $C_{m}(T,\mu_{0}H)/T$ as
a whole. Thus, our comprehensive experimental and theoretical analysis
provides evidence for the stabilization of the RSS state in a three-
dimensional lattice.

††preprint: APS/123-QED

Introduction: The presence of disorder is inevitable in any real-life quantum
many-body system and it hinders us from elucidating the actual physics of a
system. More specifically, the effects of disorder on the phases or phase
transitions in quantum many-body systems is reflected by the disappearance of
spontaneous symmetry breaking or smearing out the singularities associated
with phase transitions and critical phases [1]. On the contrary, there are
also examples of exotic ground states and new phenomena driven by disorder,
which can not be realized in a clean or disorder-free system. Thus,
discovering and characterizing those ground states in a quantum many-body
system where there exists the interplay of disorder and quantum fluctuations
is a current field of study in condensed matter physics [1, 2]. These systems
are also crucial for several application purposes, e.g. they can be the key
elements for memories and state transfer channels in quantum computing [3, 4,
5, 6]. One of the first prominent examples of disorder-driven states is
Anderson localization, where the wavefunction of non-interacting quantum
particles is highly localized near a point in space due to a strong random
potential [7, 8]. Recently, disorder-driven topological Anderson localization
has also been observed [9, 10]. Quasi-particle interference (QPI) is an
powerful experimental tool and it originates because of the presence of
disorder and helps to explore the momentum space informations in a 2D
system[11]. Disorder-driven phenomena are also often observed in correlated
many-body systems, e.g. Kondo disorder state, quantum Griffiths phase, which
shows exotic non-Fermi liquid behaviors [1, 12, 13, 14]. Experimentally and
theoretically, it was also observed that the presence of a strong disorder can
turn the first-order magnetic phase transition to a second-order phase
transition near the quantum critical point, and thus disorder-induced quantum
critical point can be obtained [15, 2].

The effects of disorder in frustrated magnets is nowadays a focus area.
Frustration may lead to a huge accidental degeneracy of different spin
configurations and the ground state is the superposition of the degenerate
states, known as the quantum spin liquid (QSL) state. In the QSL state, due to
the absence of spontaneous symmetry breaking, the spins are dynamic even at T
= 0 K and one can expect exotic fractionalized excitations.[16, 17, 18]. The
presence of disorder in a frustrated system gives rise to a glassy state, the
spin glass (SG) state [19]. Super-lattice structure produced by site disorder
may also give rise to a frustrated triangular lattice and further may lead to
the observation of a QSL state[20]. Very recently, it has been pointed out by
Kimchi et al. that in some disordered frustrated systems, peculiar features in
heat capacity and magnetization can be qualitatively explained as the presence
of a $"\text{random spin-singlet}"$ (RSS) state admixed with QSL state [21].
The RSS state have been first discussed for doped semiconductors in which the
antiferromagnetic exchange energies follow a power-law probability
distribution due to the presence of disorder [22, 23, 24]. Further, this idea
of RSS state has also been extended to two-dimensional (2D) and three-
dimensions (3D) [25, 26, 27, 28] for doped semiconductors. Whereas, only a
handful of the 2D frustrated systems show such exotic states where RSS state
and QSL coexist [29, 30, 31, 32] and so far the presence of such a state has
not been observed in any 3D frustrated system experimentally. Note that in
some 3D pyrochlore systems, the signature of QSL-like state was interpreted to
be due to a RSS state but the probability distribution of exchange energies
was not a power-law probability distribution [33]. Thus, discovering new
three-dimensional systems with strong frustration along with disorder and
elucidating the complex ground state qualitatively and quantitatively is a
primary goal in this field. Needless to say that stabilizing such a state in a
3D lattice is inherently more challenging compared to low-dimensional systems,
primarily due to the enhanced dimensionality reduces quantum fluctuations and
hence more tendency to order magnetically.

In this letter, we have reported a detailed study of a three-dimensional
quasi-cubic system Sr3CuNb2O9. Magnetization and heat capacity measurements
down to 466 mK indicate the absence of long-range magnetic ordering (LRO)
despite a strong exchange interactions (of the order of -60 K) between
Cu2+-moments and thus established Sr3CuNb2O9 as a 3D QSL candidate.
Furthermore, the power-law divergence of susceptibility and heat capacity,
along with the scaling behaviors of magnetization and heat capacity, indicate
the presence of RSS state admixed with QSL due to the presence of strong Cu/Nb
site disorder. Furthermore, numerical and analytical calculations described
the experimental data quantitatively.

Figure 1: (a) Magnetic sites forming an octahedral environment with the oxygen
atoms (red) and the green spheres showing strontium atoms intercalated in
between. (b) Arrangement of the magnetic Cu2+ (blue) sharing the same site
with the non-magnetic Nb5+ (magenta).(c) A clean XRD pattern shows the absence
of any impurity and has been analysed with the Rietveld refinement method.

Crystal structure: Here Sr3CuNb2O9 has been synthesized via a conventional
solid-state reaction technique (details have been discussed in the
supplementary materials[34]). Sr3CuNb2O9 belongs to a family of triple
perovskite crystal structures with a general formula A3B3O9 shown in Fig.1(a).
The compound stabilises in a tetragonal crystal structure of space group
P4/mmm (space group no. 123), with lattice parameters ($a=b\approx$ 3.953 Å),
$c$ = 4.089 Å, and $\alpha$ = $\beta$ = $\gamma$ = 90∘ obtained from PXRD
rietveld refinement Fig.1(c). No extra peaks represent the existence of
single-phase formation. Most importantly, no super lattice peaks have been
detected, indicating a random distribution of Cu and Nb atoms in contrast to
Sr3CuSb2O9 [20]. Thus, the magnetic skeleton is formed by Cu atoms randomly
distributed over Nb sites, with occupancy ratios of 1/3 and 2/3, respectively
and the in-plane distance between magnetic ions $\approx$ 3.95 Å, whereas out-
of-plane distance is $\approx$ 4.07 Å, makes it almost a cubic magnetic
lattice structure shown in Fig.1(b). Such a quasi-cubic 3D structure, if one
considers the next nearest-neighbour interactions along with nearest-
neighbour, may give rise to a 3D frustrated lattice. Thus, the present
compound on the one hand, may provide a 3D frustrated lattice and, on the
other hand, provide strong disorder (random site-disorder of Cu and Nb), which
may then be favourable to realize a RSS state. In order to verify our
expectation from the structural analysis, we have explored the ground state by
performing detailed magnetization and heat capacity measurements in a broad
temperature-field region.

Figure 2: (a) $\chi$ versus temperature at an applied magnetic field of 1 T
(ZFC) and extrapolation of the Curie-Wiess fit (ranging from 100-300 K) in the
inverse susceptibility data. (b) ZFC and FC data at 50 Oe applied magnetic
field, and the inset shows the power-law behavior with the exponent
$\gamma=0.63$. (c) Scaling behavior of the magnetization data with respect to
field at different temperatures with an exponent of $\gamma-1=0.35$. (d)
Scaling of the magnetization with respect to the temperature is showing
excellent data collapse in a wide applied magnetic field range from 50 Oe to 8
T.

Magnetization: Magnetic susceptibility ($\chi=M/\mu_{0}H$) has been measured
as a function of temperature at different applied magnetic fields. No anomaly
associated with long-range magnetic ordering (LRO) has been observed down to
1.85 K as seen from Fig.2(a). The susceptibility data between 100-300 K,
measurement at an applied magnetic field of 1 T, has been fitted with Curie-
Weiss law ($\chi=\frac{C}{T-\theta_{CW}}+\chi_{0}$) shown as a solid line in
Fig.2(a). Curie-Weiss temperature ($\theta_{CW}$) obtained $\approx$ -60 K
with a curie constant $C=0.49$ cm3Kmol-1. The negative sign indicates the
nature of exchanges is antiferromagnetic with a high degree of frustration
parameter, defined as $f=\theta_{CW}/T_{N}$. For any magnetic system,
$\theta_{CW}$ indicates the strength of exchange interactions between magnetic
moments, and $T_{N}$ is the ordering temperature. In the present systems, even
though the moments are highly correlated with high $\theta_{CW}$, the system
cannot go to an LRO state at least down to 1.85 K due to strong frustration.
The effective moment has been calculated using the formula
$\mu_{eff}=\sqrt{8C}$ gives 1.98 $\mu_{B}$, where $C$ is the Curie constant
obtained from $1/{\chi}$ vs $T$ fitting shown in Fig. 2(a). This indicates
that the magnetic ions are in Cu2+ (S=1/2) state. Furthermore, the absence of
branching in $\chi$ between the zero-field cooling and field cooling measured
at 50 Oe rules out the possibility of spin freezing state (e.g. spin glass) at
least down to 1.85 K (Fig. 2(b)). Thus, magnetization measurements ensure that
Sr3CuNb2O9 can be considered as a highly frustrated magnetic system promising
to have a QSL ground state.

More interestingly, if one closely observes the temperature dependence of
$\chi$, then one can see that $\chi$ follows a power-law behavior
($T^{-\gamma}$) between 20 K to 1.85 K with $\gamma$ to be 0.63. Such a power-
law behavior was predicted to be a signature of RSS state in a frustrated
system[21]. Along with the power-law in $\chi$, the $M$ versus $\mu_{0}H$ data
at different temperatures also collapses in one curve following a scaling
behavior as shown in Fig.2(c) as $MT^{\gamma-1}$ against $\mu_{0}H/T$. Even a
scaling has also been observed in $(dM/dT)\times(\mu_{0}H)^{\gamma}$ versus
$T/\mu_{0}H$. All these power-law and scaling behaviors support the presence
of the predicted RSS state admixed with a QSL state.

Heat capacity: Heat capacity ($C_{p}(T,\mu_{0}H)$) measurements have been
performed from 300 K down to 466 mK at various applied magnetic fields to
explore the ground state of Sr3CuNb2O9. Fig.3(a) shows $C_{p}$ versus $T$ at
different applied magnetic fields. The absence of any lambda-like anomaly
indicates that Sr3CuNb2O9 is avoided of any LRO down to 466 mK even though
there is a strong correlation between magnetic moments and thus established
that Sr3CuNb2O9 is a 3D QSL candidate with a high degree of frustration
($f=\theta_{CW}/T_{min}\simeq 120$, where $T_{min}$ represents the lowest
temperature heat capacity has been measured). In general, $C_{p}/T$ follows
either exponential behaviour or power-law ($T^{\gamma}$) for a gapped and
gapless QSL, respectively[35, 36]. In contrast to this expectation, $C_{p}/T$
follows a power-law behavior of $T^{-\gamma}$ in the present system at zero
applied magnetic field. To get an qualitative description of the $C_{p}$ data,
we modeled it by considering three contributions as

$C_{p}(T,\mu_{0}H)=C_{ph}+C_{Sch}+C_{power-law}$ (1)

where $C_{power-law}=A_{power-law}T^{1-\gamma}$ [29]. The other two terms
$C_{ph}$, $C_{Sch}$ represent phonon and two-level Schottky contribution
(representing the hump), respectively. At sufficiently low temperatures, the
phonon contribution behaves as $C_{ph}=\beta_{ph}T^{3}$, where $\beta_{ph}$ is
the Debye coefficient. And

$C_{Sch}=A_{Sch}R\left[\frac{\Delta(\mu_{0}H)}{k_{B}T}\right]^{2}\frac{e^{\left(\frac{\Delta(\mu_{0}H)}{k_{B}T}\right)}}{\left[1+e^{\left(\frac{\Delta(\mu_{0}H)}{k_{B}T}\right)}\right]^{2}}$
(2)

where $\frac{\Delta}{k_{B}}$ represents the energy gap.

Figure 3: (a) Temperature dependence of $C(T,\mu_{0}H)$ at different applied
magnetic fields. Solid lines show the fit considering three different
contributions described in the text. (b) Power-law fit of the $C_{m}/T$ (at 0
T) as a function of temperature with the an exponent of 0.58. (c) Scaling of
the $C_{m}(T,\mu_{0}H)/T$.

$C_{m}(\mu_{0}H=0)/T$ at zero field has been plotted as a function of
temperature in Fig.3(b), which was estimated by subtracting the phonon (Cph)
and Schottky contributions (due to 1.8% of weakly coupled impurities; for
details see [34]) from the total $C_{p}$. $C_{m}(\mu_{0}H=0)/T$ shows a power-
law divergence with an exponent $\gamma=0.58$ with lowering temperature. Also,
note that the exponent $\gamma$ is in agreement with the obtained $\gamma$
from magnetization measurements.

From the qualitative analysis of the $C_{p}$ it is now clear the spins are
correlated (as it is seen from high $\theta_{CW}$) and may form random
singlets by a power-law distribution of exchange interactions due to the
presence of strong site-disorder. Because of the power-law distribution of the
exchange interactions (responsible for forming singlets), the excitations are
gapless overall, giving rise to the power-law behavior in magnetization or
heat capacity and eventually the scaling behaviors. These correlated spins
(which form the singlets) may also form resonating state due to the presence
of strong frustration, which may give rise to a QSL behavior.

Kimchi et al.[21] pointed out that the magnetic contribution of the total heat
capacity at finite fields (represented by $C_{m}(\mu_{0}H)=C_{p}-C_{ph}$; see
[34] for details) for this disorder-induced state, collapses into a single
curve of the form

$\frac{{C_{m}[\mu_{0}H,T]}}{T}\sim\frac{1}{{(\mu_{0}H)^{\gamma}}}F_{q}\left[{\frac{T}{\mu_{0}H}}\right],$
(3)

where $Fq(X)$ is a scaling function which is determined by the energy
distribution of the singlets

$F_{q}[X]\sim\left\\{{\begin{array}[]{*{20}{c}}{X^{q}}&{X\ll 1}\\\
{X^{-\gamma}\left({1+\frac{{c_{0}}}{{X^{2}}}}\right)}&{X\gg
1}\end{array}}\right.$ (4)

where $q=1$ and $q=0$ represent the situation with and without Dzyaloshinskii-
Moriya (DM) interactions in the effective low-energy theory coupling the
orphan spins, respectively. Fig.3(b) represents $(\mu_{0}H)^{0.5}\times
C_{m}/T$ as a function of $T/\mu_{0}H$ in which all the $C_{m}$ data points
fall on top of each other and thus validates the scaling relationship as
proposed by the theory. In the range where $X<<1$, the data saturates,
indicating the absence of DM interactions, whereas in the limit of $X>>1$, the
data agrees reasonably well with the predicted scaling function Fig.3(c). It
should be pointed out that, the scaling behavior still holds for the total
heat capacity (see Fig.6 of the supplementary material[34]) except for the
high-temperature region where the lattice contribution dominates.

Thus, the detailed magnetization and heat capacity measurements in a broad
temperature and field range indicate a coexistence of RSS state and QSL state
in a three-dimensional Sr3CuNb2O9. Even though the experimental data
represented here qualitatively matches the theoretical prediction, more
quantitative description will be crucial to elucidate the underlying complex
physics.

Theory: We further carried out numerical calculations to have a quantitative
description of the temperature and field dependence of magnetization and heat
capacity to confirm the possibility of the presence of RSS state in
Sr3CuNb2O9. To understand the magnetic properties of the material we first
analyze the percentage distribution ($\%\rho$) of Cu in various clusters of
sizes up to 30x30x30 cubic lattice of the compound Sr3CuNb2O9 with only $33\%$
of Cu. The distance between the two Cu atoms sitting in two nearest octahedra
of the present compound is approximately 4 Å, therefore, a spin is a part of
cluster if the distance of a spin is less than equal to 4 Å from any of spin
in the cluster and with this criterion the percentage distribution ($\%\rho$)
of the Cu atoms decays exponentially with cluster size (L) as shown in the
supplementary material with the details of the calculation[34]. The most
dominating spin clusters are free spins and part of these spins may be forming
weakly interacting Cu structures with random exchange as the distances among
the spins are larger than 4 Å and random. These spins can have comparable
nearest, next-nearest exchanges due to their distribution in three-dimensional
space. Therefore, these weakly interacting frustrated spins can be treated as
random dimer singlet with a broad distribution of the exchanges. Other spin
cluster bigger than dimer are very small in numbers and may have negligible
effect on the magnetic properties of the material. Hence, we adopt an
isotropic dimer Heisenberg spin-1/2 model Hamiltonian, incorporating a
distribution of exchanges. This idea is inspired by the spin-1/2 model
proposed by Dasgupta, Ma, and Fisher [22] for highly disordered one-
dimensional systems, as well as the model presented by Bhatt and Lee [25] for
highly disordered two and three-dimensional systems. The dimer model can be
written as

$H=\sum_{i}J_{i}S_{i,1}.S_{i,2}=\sum_{i}H_{i}$ (5)

where $H_{i}$ is $i^{th}$ isolated dimer Hamiltonian for spin 1 and 2, and
$J_{i}$ is isotropic exchange interaction of the $i^{th}$ dimer. We assume
$J_{i}$ can take any continuous values (in between 0 and 1 in the reduced
unit) and it follows the distribution $P(J)=J^{-\gamma}$, i.e. the smaller J
has a higher probability distribution. A detailed description of the
calculation is given in the Supplementary file[34].

Figure 4: Upper panel: Temperature dependence of $\chi$ (obtained
experimentally) represented by the scattered points and the solid lines
indicate the data obtained numerically. Lower panel: The solid lines and the
scattered points represents experimentally and numerically obtained data of
$C_{m}/T$.

We use $g=2.22$, the average $J=<J_{i}>\approx 72K$ and $\gamma=0.6$ to fit
experimental curve of $C_{m}(T,\mu_{0}H)/T$ and $\chi(T)$. The $\frac{M}{H}$
for four different fields are shown in the upper panel of Fig. 4, and
$\chi(=\frac{M}{H})$ curves show a power-law decay as a function of
temperature. The experimental curve of $C_{m}(T,\mu_{0}H)/T$ is shown as
points and the theoretical curve as lines for six different magnetic fields,
all the experimental curves can be fitted well with the theoretical curves as
shown in lower panel of Fig. 4. In the lower panel of Fig. 4, the linear
behavior of the $C_{m}/T$ plot clearly shows the power-law behavior for the
zero-field but finite field data shows criticality as a function of $T$ and
$H$. This behavior is well known as data-collapse in $\frac{T}{H}$ for the
resonating limit $|J-H|<k_{B}T$. It is noticed that both $C_{m}(T,\mu_{0}H)/T$
and $\chi$ can be fitted well with the experimental data with the same
exponent which is $0.6$ in this case. We analyse these results for different
limits of temperature and field in the Supplementary file[34].

Thus, our numerical calculations quantitatively established that the
temperature and field dependence of magnetization and heat capacity are due to
the presence of disorder-driven RSS state.

Conclusion: The present manuscript has studied a novel disorder-induced state
in great detail. We report here a comprehensive experimental and theoretical
investigation of the compound Sr3CuNb2O9. The disorder introduced by site-
mixing between Cu and Nb and frustration due to competing nearest-neighbour
and next nearest-neighbour of the three-dimensional quasi-cubic lattice gives
rise to an exotic state in which RSS state coexists with QSL. Numerical
calculations further gave a quantitative description of the experimental
results and provided evidence of the realization of the RSS state in a three-
dimensional lattice. Analytical calculations again confirm the experimentally
observed behaviors of $\chi$ and $C_{m}(T,\mu_{0}H)/T$. These observations
established Sr3CuNb2O9 as one of the rare compounds where such a RSS occurs in
a 3D-lattice.

Acknowledgement: SMH and MM would like to acknowledge UGC-DAE Collaborative
Research Scheme (ref. CRS/2021-22/01/393) for funding. MK thanks DST-SERB for
funding through projects CRG/2020/000754.

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