11institutetext: Institute for Information and Transmission Problems, Moscow
22institutetext: Skolkovo Institute of Science and Technology, Moscow
33institutetext: Moscow Institute of Physics and Technology, Moscow

# Deep learning model solves change point detection for multiple change types

Alexander Stepikin 11 2 2 3 3 Evgenia Romanenkova 22 Alexey Zaytsev

22<EMAIL_ADDRESS>
###### Abstract

A change points detection aims to catch an abrupt disorder in data
distribution. Common approaches assume that there are only two fixed
distributions for data: one before and another after a change point. Real-
world data are richer than this assumption. There can be multiple different
distributions before and after a change. We propose an approach that works in
the multiple-distributions scenario. Our approach learn representations for
semi-structured data suitable for change point detection, while a common
classifiers-based approach fails. Moreover, our model is more robust, when
predicting change points. The datasets used for benchmarking are sequences of
images with and without change points in them.

###### Keywords:

change point detection, representation learning, sequential data

##  1 Introduction

The modern world is full of time series and sequential data: from server logs
and sensors’ reading while drilling to video analysis [1, 5, 6]. The change
point is an abrupt alteration of the distribution the data comes from. Its
occurrence means that the generating process has dramatically changed what may
be an emergency signal. Thus, there is an urgent need to detect those shifts
in a fast and accurate way.

The basic approaches mostly assume that there are only two unknown but fixed
data distributions: one before and one after a change point. On the contrary,
the real-world tasks require a method suitable for the case of multiple
possible distributions before and after change point, for example, in a video
monitoring system [5]: we observe numerous types of system behaviour and are
interested in their changes.

We adapt the existing method of change-point detection based on the
representation learning to the new setting with multiple possible
distributions.

We compare the performance of the recurrent neural network with principled
loss function to the one with basic binary cross-entropy loss in terms of the
key change point detection and classification metrics of quality. The results
show that our model can make precise and confident predictions in different
experimental setups with up to $10$ different types of distribution changes.
The considered changes are in Table 1.

Change number | New considered
---|---
| changes
$1$ | $[4,7]$
$2$ | $[7,4]$
$4$ |  $[1,9]$, $[9,1]$
$6$ |  $[2,5]$, $[5,2]$
$8$ |  $[0,8]$, $[8,0]$
$10$ |  $[3,6]$, $[6,3]$
Table 1: Considered types of changes for different number of considered
changes. $[x,y]$ designates one change with $x$ being a digit before a change
and $y$ being a digit after a change

##  2 Related work

The statistical approach includes the use of well-known statistics such as
cumulative sum, Shiryaev-Roberts statistics, posterior probability statistics,
or the use of smoothing and various filters, for example, the Kalman filter,
as well as various combinations of them [9, 2, 1]. Such statistics find
application in various fields because of their simplicity but strong
performance. However, they suggest the fixed distribution before and after a
change point. So, their performance may suffer in the case of multiple
distributions.

The seminal book [7] introduces an accurate overview of the mathematical
background behind these approaches and presents several examples of change
point detection applications in financial analysis. In the article [2], the
authors use ensembles of "weak"detectors (classical methods, but operating
under conditions of assumptions violation) to predict the degradation of
systems with extensive software in real-time. Paper [9] provides another
example of the usage of such approaches for medical issues. The authors
combine Kalman filters, cumulative sums, and an exponentially weighted moving
average to detect changes in the heart rate trend in real-time.

Statistical approaches were developed in the paper [5]. Based on criteria from
[7], the authors present differentiable loss functions for change point
detection via neural network. In this work, we consider this approach, so more
details are presented in the section 3.

Accurate reviews of state-of-the-art change point detection methods can be
found in [1, 8]. Authors describe different categories of methods such as
methods based on probability density ratio (both parametric and non-
parametric), subspace models, probabilistic models, kernel-based methods,
clustering, graph-based methods and classification methods.

##  3 Methods

In this section, we introduce the problem of change point detection (CPD), the
investigated approach based on the principled loss function presented in the
paper [5] and the design of the experiments.

###  3.1 Problem statement

The basic formulation of the change point detection problem is the following.
Consider a random process $X^{1:T}=\\{x_{1},\ldots,x_{T}\\}$ of length
$T\leq\infty$ consisting of observations $x_{t}\in\bbbr^{d}$,
$t\in\\{1,...,T\\}$. The process depends on a latent random variable
$\theta\in\\{0,1\ldots,+\infty\\}$ that comes from an unknown distribution
$G$. For each moment $t<\theta$, we assume the "normal" behaviour of the
process $\ x_{t}\sim f_{\infty}$, and for $t\geq\theta$, it is "abnormal" $\
x_{t}\sim f_{0}$. The distributions $f_{0}$ and $f_{\infty}$ are unknown as
well. The problem is to detect the true change point $\theta$ as quickly as
possible.

It should be emphasized that we work in an "online" setup which means that the
decision about a presence of a change point is based on the information
available at a current moment of time.

###  3.2 Principled loss function

The main idea of the method is to train a neural network parameterized by the
vector of weights $w$ with a special loss function [5]. In other words, the
model’s input is a dataset
$D=\\{(X_{1},\theta_{1}),\ldots,(X_{N},\theta_{N})\\}$ , where $X_{i}$ denotes
the $i$-th process and $\theta_{i}$ is the corresponding change point. The
model $f_{w}$ produces outputs $p_{i}=\\{p_{t,i}\\}_{t=0}^{T}$ where
$p_{t,i}=f_{w}(X_{i}^{1:t})$ is the probability of the change point in the
process $X_{i}$ at the particular time moment $t$.

The proposed loss function (1) is a sum of two terms that, as shown in [5],
approximate the expected detection delay (2) and the expected time to false
alarm if it appears (3) we aim to optimize, according to [7]:

$\mathcal{L}(f_{w},D)=\mathcal{L}_{delay}(f_{w},D)+c\cdot\mathcal{L}_{FA}(f_{w},D),$
(1)

where

$\mathcal{L}_{delay}(f_{w},D)=\frac{1}{N}\sum_{i=1}^{N}\left(\sum_{t=\theta_{i}}^{T}{(t-\theta_{i})p_{t,i}\prod_{k=\theta_{i}}^{t-1}(1-p_{k,i})+(T+1-\theta_{i})\prod_{k=\theta_{i}}^{T}(1-p_{k,i})}\right)$
(2)

$\mathcal{L}_{FA}(f_{w},D)=-\frac{1}{N}\sum_{i=1}^{N}\left(\sum_{t=0}^{\tilde{T}}tp_{t,i}\prod_{k=0}^{t-1}(1-p_{k,i})-(\tilde{T}+1)\prod_{k=0}^{\tilde{T}}(1-p_{k,i})\right),$
(3)

$\tilde{T}=\min{(\theta_{i},T)}$.

###  3.3 Final pipeline

As a result, the following end-to-end algorithm has been proposed summarized
in Figure 1. Training of a neural network for change point detection consists
of the following steps:

  1. 1.

Using a neural network $f_{w}$, get the change point probabilities
$p_{i}=f_{w}(X_{i})$ for each input $X_{i}$ in the dataset;

  2. 2.

Calculate the loss function given by (1);

  3. 3.

Train the neural network via SGD.

For making predictions, report about the change point if the probability
$p_{t,i}$ exceeds a pre-selected threshold $s$.

Figure 1: Teaser of investigated approach. For each moment, the model outputs
the probability of change point.

###  3.4 Classification approach

Alternatively, we can think of this problem as a binary classification: it is
necessary to classify all the process observations, whether they occurred
before or after the distribution change. In this case, it is possible to apply
a deep model with standard binary cross-entropy (BCE) loss.

###  3.5 Multiple types of change points

In this paper, we investigate a more complicated scenario when there are
multiple types of change points. In particular, it means that the observations
after the change point may come from several fixed distributions:
$f_{0,\infty}^{(1)},f_{0,\infty}^{(2)},\ldots f_{0,\infty}^{(K)}$.

We state that our embeddings would be universal enough to deal with the case
of multiple distributions. Our suggestion is that a proper selection of
hyperparameters can work in this case. So, a careful adaptation of existing
methods can make a neural network learn to detect the change itself even in
case of various scenarios for changes. However, at the same time, theoretical
guarantees exist only for the case of fixed distribution.

Thus, our paper focuses on the possibility to deal with the challenging
multiple distributions cases.

##  4 Results

In this section, we explain the conducted experiments and present their
results.

###  4.1 Datasets

Following [5], we used the datasets consisting of sequences of MNIST-like
images with smooth transitions from digit $x$ to digit $y$. The sequences were
generated by a Conditional Variational Autoencoder (CVAE). Each sequence has
the length of $64$ images and comprises of $3$ segments: $[x,x]$, $[x,y]$ and
$[y,y]$. The middle segment has a random length from $1$ to $10$ images. If
$x\neq y$, the sequence has a change point.

In all the experiments, datasets were balanced, i.e. they included equal
amounts of "normal" (without a change of digit) and "abnormal" (with a
transition from one digit to another) sequences. We constructed different
datasets: with $1$, $2$, $4$, $6$, $8$ and $10$ types of changes in data
distributions. More detailed, for different types of changes, we used
different starting and ending digits listed in Table 1. We generated $500$
sequences with each type of change point and $500$ corresponding sequences
without a digit transition. As a result, we got $6$ balanced datasets
consisting of $1000$, $2000$, $4000$, $6000$, $8000$ and $10000$ sequences
respectively.

The examples are shown in Figure 2.

Figure 2: Examples of generated sequences with (left) and without (right) a
transition of the digits.

###  4.2 Models architecture

We used models with the following architecture: recurrent neural network (RNN)
followed by $2$ fully-connected layers with ReLU activations and the output
layer with sigmoid activation. We chose LSTM [4] as a basic model because it
handles the sequential nature of the data. The code of data generation and the
experiments is available online111GitHub:
https://github.com/romanenkova95/PrincipledCPD.

###  4.3 Comparison details

In all the experiments, we compare the performance of two deep models with the
same architecture but different loss functions: custom CPD loss (1) with
balanced weights of the terms and BCE loss. All the other parameters were
identical. For the model with CPD loss, we tried different hyperparameters
$T$.

###  4.4 Main metrics

In order to evaluate the models‘ performance, we used different metrics:
standard classification metrics and specific metrics for change point
detection.

#### Classification metrics.

We interpret classification metrics for the change point detection problem in
the following way.

  1. 1.

True Positive (TP): the change point was correctly detected in the "abnormal"
sequence no earlier than it appears.

  2. 2.

False Positive (FP): a model predicts a change point in a "normal" sequence or
reports about it before it appears in an "abnormal" one.

  3. 3.

True Negative (TN): a model does not predict a change point in a "normal"
sequence.

  4. 4.

False Negative (FN): a model does not report about a change point in a
sequence where there is one.

Having these elements of the confusion matrix, we calculate accuracy, recall
and $F_{1}$-score based on the predictions of different models.

#### Covering.

As a specific metric for change point detection, we adopted the Covering
metric presented in [3] that comes from the image segmentation problem and
evaluates the similarity of two partitions of the sequences: the one based on
the model‘s predictions and the true one. The higher is the covering metric,
the better is the performance.

#### Mean Detection Delay and Mean Time to False Alarm.

These are the metrics commonly used for the evaluation of change point
detection algorithms. Detection Delay $(\tau-\theta)^{+}$ is a difference
between the change point time $\tau$ detected by a model and the real one
$\theta$. Time to false alarm (FA) is equal to $\tau$ – the difference between
the first positive prediction if there is one and the beginning of the
sequence. The custom loss function (1) is aimed to minimize the expected
Detection Delay while keeping the expected time to FA large enough.

#### Area under the detection curve (AUC).

The detection curve is a graph where the $x$-axis is the mean time to FA, and
the $y$-axis is the mean Detection Delay. The examples of the curves are in
Figure 5. The points on the graph were obtained by varying the detection
threshold $s$ in the range from $0$ to $1$. As the problem of the reasonable
threshold choice is not straightforward, the idea is to consider all the
possible variants and measure areas under the detection curves of different
models. The lower this area is, the better a model performs.

###  4.5 Experimental results

We evaluated the main quality metrics with a fixed set of alarm thresholds in
a range from $0.001$ to $0.999$. In addition, we calculated the areas under
the detection curves based on the models‘ predictions. The results are
presented in Table LABEL:table:metrcis and Figure 4. For clarity, we chose to
compare only those thresholds that optimize the considered metrics for
different models. A few typical predictions of the both models in different
experiments are depicted in the picture 3. The detection curves are depicted
in the figure 5.

Figure 3: Typical predictions of the models in experiments with multiple types of distribution changes. The CPD model is more confident in her predictions, while for BCE loss we observe outliers in change prediction probabilities. Table 2: Quality metrics for a selected set of thresholds. The "$\uparrow$" sign marks the metrics we would like to maximize, "$\downarrow$" – to minimize. Loss | Threshold | Time to FA $\uparrow$ | Detection Delay $\downarrow$ |  $F_{1}$-score $\uparrow$ | Covering $\uparrow$ | AUC $\downarrow$
---|---|---|---|---|---|---
|  |  | 1 type of change |  |  |
CPD | 0.01 | 39.84 | 0.03 | 1.00 | 0.998 | 294.98
| 0.1 | 41.64 | 0.03 | 1.00 | 0.999 |
| 0.2 | 41.64 | 0.03 | 1.00 | 0.999 |
| 0.5 | 41.64 | 0.03 | 1.00 | 0.999 |
| 0.7 | 41.64 | 0.05 | 1.00 | 0.999 |
BCE | 0.01 | 38.49 | 0.00 | 0.99 | 0.917 | 379.24
| 0.1 | 40.20 | 0.00 | 1.00 | 1.000 |
| 0.2 | 40.20 | 0.00 | 1.00 | 1.000 |
| 0.5 | 40.20 | 0.00 | 1.00 | 1.000 |
| 0.7 | 40.20 | 0.01 | 1.00 | 1.000 |
|  |  | 2 types of changes |  |  |
CPD | 0.001 | 39.63 | 0.05 | 0.89 | 0.844 | 297.93
| 0.01 | 39.74 | 0.05 | 0.91 | 0.949 |
| 0.5 | 40.19 | 0.09 | 0.93 | 0.985 |
| 0.99 | 41.50 | 0.19 | 0.97 | 0.988 |
| 0.9999 | 41.59 | 2.88 | 0.96 | 0.947 |
BCE | 0.001 | 8.59 | 0.00 | 0.16 | 0.459 | 414.69
| 0.01 | 39.15 | 0.01 | 0.89 | 0.847 |
| 0.5 | 40.56 | 0.02 | 0.96 | 0.998 |
| 0.99 | 39.36 | 0.88 | 1.00 | 0.980 |
| 0.9999 | 41.49 | 4.63 | 0.99 | 0.895 |
|  |  | 4 types of changes |  |  |
CPD | 0.001 | 41.76 | 0.70 | 0.92 | 0.974 | 284.68
| 0.2 | 41.02 | 0.89 | 0.93 | 0.976 |
| 0.5 | 41.12 | 0.95 | 0.94 | 0.975 |
| 0.99 | 41.09 | 1.16 | 0.93 | 0.974 |
| 0.9999 | 41.77 | 1.57 | 0.95 | 0.970 |
BCE | 0.001 | 36.15 | 0.06 | 0.70 | 0.732 | 309.90
| 0.2 | 39.96 | 0.11 | 0.94 | 0.996 |
| 0.5 | 40.05 | 0.20 | 0.96 | 0.995 |
| 0.99 | 41.09 | 0.62 | 1.00 | 0.984 |
| 0.9999 | 40.31 | 1.84 | 1.00 | 0.952 |
|  |  | 6 types of changes |  |  |
CPD | 0.001 | 38.98 | 0.83 | 0.81 | 0.931 | 317.41
| 0.7 | 40.13 | 1.13 | 0.86 | 0.958 |
| 0.9 | 40.20 | 1.14 | 0.87 | 0.959 |
| 0.99 | 40.42 | 1.25 | 0.88 | 0.959 |
| 0.9999 | 42.15 | 2.59 | 0.91 | 0.947 |
BCE | 0.001 | 12.29 | 0.00 | 0.16 | 0.382 | 356.14
| 0.7 | 40.23 | 0.40 | 0.95 | 0.984 |
| 0.9 | 40.53 | 0.70 | 0.97 | 0.982 |
| 0.999 | 41.42 | 2.98 | 0.99 | 0.931 |
| 0.9999 | 41.76 | 4.76 | 0.98 | 0.893 |
|  |  | 8 types of changes |  |  |
CPD | 0.001 | 37.20 | 1.07 | 0.78 | 0.903 | 327.02
| 0.5 | 38.37 | 1.48 | 0.83 | 0.929 |
| 0.9 | 39.41 | 1.71 | 0.84 | 0.929 |
| 0.999 | 41.89 | 2.69 | 0.88 | 0.937 |
| 0.9999 | 43.45 | 4.03 | 0.88 | 0.923 |
BCE | 0.001 | 16.41 | 0.00 | 0.17 | 0.576 | 350.65
| 0.5 | 39.68 | 0.27 | 0.88 | 0.974 |
| 0.9 | 41.00 | 1.31 | 0.95 | 0.973 |
| 0.999 | 43.53 | 5.78 | 0.94 | 0.891 |
| 0.9999 | 45.24 | 9.24 | 0.89 | 0.829 |
|  |  | 10 types of changes |  |  |
CPD | 0.001 | 39.07 | 0.99 | 0.81 | 0.940 | 296.34
| 0.2 | 39.71 | 1.27 | 0.83 | 0.943 |
| 0.7 | 40.14 | 1.45 | 0.83 | 0.944 |
| 0.999 | 40.68 | 1.68 | 0.86 | 0.950 |
| 0.9999 | 41.02 | 1.99 | 0.87 | 0.947 |
BCE | 0.001 | 21.21 | 0.00 | 0.25 | 0.660 | 338.22
| 0.2 | 40.27 | 0.22 | 0.92 | 0.986 |
| 0.7 | 41.39 | 0.87 | 0.97 | 0.981 |
| 0.999 | 42.96 | 6.78 | 0.93 | 0.871 |
| 0.9999 | 52.24 | 15.22 | 0.66 | 0.747 |

Figure 4: Dependence of the main metrics (AUC $\downarrow$ and Covering
$\uparrow$) on the number of change types. The quality of models‘ performance
deteriorates as the complexity of the problem increases.

1 type of change

2 types of changes

4 types of changes

6 types of changes

8 types of changes

10 types of changes

Figure 5: Detection curves of the models with CPD and BCE losses for the MNIST
experiments with multiple types of changes.

Based on the results obtained, the following conclusions can be drawn:

  * •

For the smallest thresholds, the model with BCE loss tends to make many false
alarms, resulting in low mean detection delay and low mean time to FA. At the
same time, the model with principled CPD loss does not make false alarms and
produces quality results according to these $2$ metrics and the Covering
metric as well.

  * •

On the other hand, for the highest thresholds, the model with BCE loss has a
large mean detection delay that is a sign of FN predictions. In this case, the
model with CPD loss outperforms it in terms of Covering and $F_{1}$-score.

  * •

These two peculiarities prove that the models with custom CPD loss are more
sure in their predictions than the ones with BCE loss: even a minor
fluctuation of change probabilities is often a sign of the change point.

  * •

In the middle cases, both models behave similarly. The area under the
detection curve is a metric of the model‘s overall performance considering
various alarm thresholds. The models with CPD loss have lower AUC metrics in
all the experiments we conducted. Thus, they make better predictions on
average.

  * •

Considering the main metrics, we can conclude that the more types of
distribution changes we have, the worse the models’ performances are. Both
approaches struggle when the problem becomes more complex.

##  5 Acknowledgements

The work of Alexey Zaytsev was supported by the Russian Foundation for Basic
Research grant 20-01-00203. The work of Evgenya Romanenkova was supported by
the Russian Science Foundation (project 20-71-10135).

##  6 Conclusion

We investigated the problem of change point detection in the case of multiple
data distributions for methods based on representation learning and estimated
the efficiency of the considered approach in terms of the main metrics of
change point detection quality.

We confirm that the models can work well for such data, but performance
worsens while the number of distribution types increases. We also show that
the principled model is more confident in its predictions for all cases. Thus,
it is easier to chose a threshold for predictions.

## References

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  * [3] Gerrit JJ Burg and Christopher KI Williams  “An evaluation of change point detection algorithms”  In _arXiv preprint arXiv:2003.06222_ , 2020
  * [4] Sepp Hochreiter and Jürgen Schmidhuber  “Long Short-term Memory”  In _Neural computation_ 9, 1997, pp. 1735–80  DOI: 10.1162/neco.1997.9.8.1735
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  * [6] Evgeniya Romanenkova et al.  “Real-time data-driven detection of the rock-type alteration during a directional drilling”  In _IEEE Geoscience and Remote Sensing Letters_ 17.11 IEEE, 2019, pp. 1861–1865
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# Normality from one family of meromorphic functions to another through
sharing of values

,  Kuldeep Singh Charak  Kuldeep Singh Charak Department of Mathematics
University of Jammu Jammu-180 006 India<EMAIL_ADDRESS>,  Manish
Kumar  Manish Kumar Department of Mathematics University of Jammu Jammu-180
006 India<EMAIL_ADDRESS>and  Rahul Kumar  Rahul Kumar Department of
Mathematics University of Jammu Jammu-180 006 India<EMAIL_ADDRESS>
###### Abstract.

Let $\mathcal{F}$ and $\mathcal{G}$ be two families of meromorphic functions
on a domain $D$, and let $a,\ b$ and $c$ be three distinct points in the
extended complex plane. Let $\mathcal{G}$ be a normal family in $D$ such that
all limit functions of $\mathcal{G}$ are non-constant. If for each
$f\in\mathcal{F}$, there exists $g\in\mathcal{G}$ such that $f$ and $g$ share
$a,\ b$ and $c$ partially, then $\mathcal{F}$ is normal in $D.$ This gives a
sharp improvement of a result due to X. J. Liu, S. H. Li and X. C. Pang. We
also prove some interesting related sharp results.

††footnotetext: 2010 Mathematics Subject Classification. 30D45,
30D30.††footnotetext: Keywords and phrases. Normal family, shared values,
meromorphic functions ††footnotetext: The work of the first author is
partially supported by Mathematical Research Impact Centric Support (MATRICS)
grant, File No. MTR/2018/000446, by the Science and Engineering Research Board
(SERB), Department of Science and Technology (DST), Government of India.

##  1\. Introduction and Main Results.

For the sake of brevity we shall use the following standard notations:

  * •

$\mathcal{H}(D):$ class of all holomorphic functions on a domain $D$ in
$\mathbb{C};$

  * •

$\mathcal{M}(D):$ class of all meromorphic functions on a domain $D$ in
$\mathbb{C};$

  * •

$E(f,a):$ the set of $a-$points of $f$ counted with multiplicity (CM);

  * •

$\bar{E}(f,a):$ the set of $a-$points of $f$ counted by ignoring multiplicity
(IM).

The present paper deals with passing of normality from one subfamily of
$\mathcal{M}(D)$ to another when the two subfamilies satisfy a certain
condition on sharing of values.

Let’s recall that a subfamily $\mathcal{F}$ of $\mathcal{M}(D)$ is said to be
normal in $D$ if from any given sequence in $\mathcal{F}$ we can extract a
subsequence that converges spherically locally uniformly in $D.$ The limit
function is either in $\mathcal{M}(D)$ or identically equal to $\infty$. For
complete introduction to normal families of meromorphic functions, the reader
may refer to [7] and [11].

Two functions $f,g\in\mathcal{M}(D)$ are said to share a given value
$a\in\mathbb{C}_{\infty}$ IM (CM) if $\bar{E}(f,a)=\bar{E}(g,a)$
($E(f,a)=E(g,a));$ further, $f$ and $g$ are said to share $a$ partially if
$\bar{E}(f,a)\subset\bar{E}(g,a).$ Maximum number of values shared by two non-
constant and distinct meromorphic functions is $4$ (see [4]). For detailed
account of sharing of values by meromorphic functions and their uniqueness,
one may refer to [10].

A study of normality of two families of meromorphic functions on a common
domain was initiated in $2013$, almost simultaneously, by Liu, Li and Pang [5]
and Yuan, Xiong and Lin [9]. Liu, Li and Pang [5] proved the following result:

Theorem A Let $\mathcal{F}$ and $\mathcal{G}$ be two subfamilies of
$\mathcal{M}(D)$ and let $a,\ b,\ c,\ d\in\mathbb{C}$ be four distinct values.
If $\mathcal{G}$ is normal in $D$, and for each $f\in\mathcal{F}$, there
exists $g\in\mathcal{G}$ such that $f$ and $g$ share the values $a,\ b,\ c$
and $d$, then $\mathcal{F}$ is normal in $D$.

Though the number $4$ of shared values in Theorem A cannot be reduced to $3$,
we can compensate the sharing of some of the values by a condition on limit
functions of $\mathcal{G}$ or on $a-$points of $\mathcal{F}$ . Precisely we
have obtained:

######  Theorem 1.1.

Let $\mathcal{F}$ and $\mathcal{G}$ be two subfamilies of $\mathcal{M}(D)$ and
let $a_{1},\ a_{2},\ a_{3}\in\mathbb{C}_{\infty}$ be three distinct values.
Let $\mathcal{G}$ be normal in $D$ such that all limit functions of
$\mathcal{G}$ are non-constant. If for each $f\in\mathcal{F}$, there exists
$g\in\mathcal{G}$ such that $f$ and $g$ share $a_{j},\ j=1,2,3,$ partially in
$D,$ then $\mathcal{F}$ is normal in $D$.

The condition that “all the limit functions of $\mathcal{G}$ are non-constant”
in Theorem 1.1 is essential as shown by the following example:

######  Example 1.2.

Let $\mathcal{F}=\left\\{f_{n}:n\in\mathbb{N}\right\\}$, where $f_{n}(z)=\tan
nz$, be the family of meromorphic functions on the open unit disk
$\mathbb{D}$. Let $z_{n,1},\ldots,z_{n,m_{n}}$ be the zeros of $\tan nz$ in
$\mathbb{D}$. Let $\mathcal{G}=\left\\{g_{n}\right\\}$, where

$g_{n}(z)=\frac{1}{n}\prod_{i=1}^{m_{n}}\left(\frac{z-z_{n,i}}{1-\bar{z}_{n,i}z}\right).$

Then $f_{n}$ omits $i,-i$ on $\mathbb{D}$ and each $f_{n}$ and $g_{n}$ share
$0$ partially. It is easy to see that all subsequences of
$\left\\{g_{n}\right\\}$ converge locally uniformly to 0. Since
$|g_{n}(z)|<1$, $\mathcal{G}$ is normal in $\mathbb{D}$. But $\mathcal{F}$ is
not normal at 0.

######  Theorem 1.3.

Let $a_{1},\ a_{2},\ a_{3},\ a_{4}\in\mathbb{C}_{\infty}$ be four distinct
values, $\mathcal{F}$ and $\mathcal{G}$ be two subfamilies of $\mathcal{M}(D)$
such that the multiplicity of $a_{1}-$points of each $f\in\mathcal{F}$ is at
least $2,$ and let $\mathcal{G}$ be normal in $D$. If for each
$f\in\mathcal{F}$, there exists $g\in\mathcal{G}$ such that $f$ and $g$ share
$a_{j},\ j=2,3,4,$ partially in $D$, then $\mathcal{F}$ is normal in $D$.

Example 1.2 also shows that in Theorem 1.3, the condition, “the multiplicity
of $a-points$ of $f\in\mathcal{F}$ is at least 2” is essential.

######  Theorem 1.4.

Let $a_{1},\ a_{2},\ a_{3},\ a_{4}\in\mathbb{C}_{\infty}$ be four distinct
values, $\mathcal{F}$ and $\mathcal{G}$ be two subfamilies of $\mathcal{M}(D)$
such that the multiplicities of $a_{1}-$points and $a_{2}-$points of each
$f\in\mathcal{F}$ are at least $2$ and $3$ respectively, and let $\mathcal{G}$
be normal in $D$. If for each $f\in\mathcal{F}$, there exists
$g\in\mathcal{G}$ such that $f$ and $g$ share $a_{3}$ and $a_{4}$ partially in
$D$, then $\mathcal{F}$ is normal in $D$.

To pass on the normality of $\mathcal{F}\subseteq\mathcal{M}(D)$ to a family
$\mathcal{R}$ of rational functions, we only require three values to be shared
partially:

######  Theorem 1.5.

Let $\mathcal{R}$ be a family of rational functions of degree at most
$m\in\mathbb{N}$ and let $\mathcal{G}\subseteq\mathcal{M}(D).$ Let $a_{1},\
a_{2},\ a_{3}\in\mathbb{C}_{\infty}$ be three distinct values. If
$\mathcal{G}$ is normal in $D$ and for each $R\in\mathcal{R}$, there exists
$g\in\mathcal{G}$ such that $R$ and $g$ share $a_{j},\ j=1,2,3,$ partially in
$D$, then $\mathcal{F}$ is normal in $D.$

######  Example 1.6.

Let
$\mathcal{F}=\\{f_{n}(z)=nz:n\in\mathbb{N}\\}~{}\mbox{and}~{}\mathcal{G}=\\{g_{n}(z)=z^{n}:n\in\mathbb{N}\\}$.
Then $\mathcal{F}$ is a family of rational map of degree $1.$ Also
$\mathcal{G}$ is normal in $\mathbb{D}$ and for each $f_{n}$ there exists
$g_{n}$ such that $f_{n}$ and $g_{n}$ share $0$ and $\infty$ partially. But
$\mathcal{F}$ is not normal in $\mathbb{D}.$ This shows that the number of
values cannot be reduced to two.

Liu, Li and Pang [5] also proved the following two results on the normality of
two families:

Theorem B Let $\mathcal{F},\ \mathcal{G}\subset\mathcal{H}(D)$, all of whose
zeros have multiplicity at least $k+1,\ k\in\mathbb{N}.$ Let $b$ be a non-zero
complex number. Assume that $\mathcal{G}$ is normal, and for any subsequence
$\left\\{g_{n}\right\\}\subseteq\mathcal{G}$ such that $g_{n}\to g$ locally
uniformly on $D$, we have $g\not\equiv\infty$ and $g^{(k)}\not\equiv b$ on
$D$. If for every $f\in\mathcal{F}$, there exists $g\in\mathcal{G}$ such that
$f$ and $g$ share $0$, and $f^{(k)}$ and $g^{(k)}$ share $b$, then
$\mathcal{F}$ is normal in $D$.

Theorem C Let $\mathcal{F},\ \mathcal{G}\subset\mathcal{M}(D)$, all of whose
zeros have multiplicity at least $k+1,\ k\in\mathbb{N},$ and let $b$ be a non-
zero complex number. Suppose that $\mathcal{G}$ is normal in $D$ and for any
sequence $\left\\{g_{n}\right\\}$ of $\mathcal{G}$ converging to a function
$g$ spherically locally uniformly in $D$, we have $g^{(k)}\not\equiv b$ and
$g\not\equiv\infty$. If for every $f\in\mathcal{F}$, there exists
$g\in\mathcal{G}$ such that $f$ and $g$ share $0$ and $\infty$ IM, and
$f^{(k)}$ and $g^{(k)}$ share $b$ CM, then $\mathcal{F}$ is normal in $D$.

We have obtained the following improvement of Theorem C:

######  Theorem 1.7.

Let $\mathcal{F},\ \mathcal{G}\subset\mathcal{M}(D)$, all of whose zeros have
multiplicity at least $k+1,\ k\in\mathbb{N},$ and let $a$ be a non-zero
complex number. Suppose that $\mathcal{G}$ is normal in $D$ such that all
limit functions of $\mathcal{G}$ are not identically equal to infinity. If for
each $f\in\mathcal{F}$, there exists $g\in\mathcal{G}$ such that $f$ and $g$
share $0$ and $\infty$ partially, and $f^{(k)}$ and $g^{(k)}$ share $a$
partially, then $\mathcal{F}$ is normal in $D$.

Note: When $\mathcal{F}$ and $\mathcal{G}$ in Theorem 1.7 are restricted to be
in $\mathcal{H}(D)$, we immediately obtain an improvement of Theorem B.

One can show that condition, “partial sharing of $0$ and $\infty$” in Theorem
1.7, is essential, for example one may refer to Examples 1.7 and 1.8 in [5].

######  Example 1.8.

Consider

$\mathcal{F}:=\left\\{f_{n}(z)=e^{nz},\ n\in\mathbb{N},\
z\in\mathbb{D}\right\\}$

and for $k\in\mathbb{N}$, consider

$\mathcal{G}:=\left\\{g_{n}(z)=(z-2)^{(k+1)},\
n\in\mathbb{N},z\in\mathbb{D}\right\\}.$

Then each $f\in\mathcal{F}$ omits 0 and $\infty$, $\mathcal{G}$ is normal in
$\mathbb{D}$ and all limit functions of $\mathcal{G}$ are not identically
equal to infinity. But $\mathcal{F}$ is not normal in $\mathbb{D}$. This shows
that the condition, “$f^{(k)}$ and $g^{(k)}$ share $a$, partially” in Theorem
1.7 is essential.

######  Example 1.9.

Consider the family

$\mathcal{F}:=\left\\{f_{n}(z)=z-\frac{e^{nz}}{n},\ n\in\mathbb{N},\
z\in\mathbb{D}\right\\}$

and let $\left\\{z_{n,i}\right\\}_{i=1}^{k_{n}}$ be the zeros of $f_{n}$ in
$\mathbb{D}$. Further consider

$\mathcal{G}:=\left\\{g_{n}(z)=\prod^{k_{n}}_{i=1}\left(\frac{z-z_{n,i}}{1-\overline{z_{n,i}}z}\right)^{2},n\in\mathbb{N},z\in\mathbb{D}\right\\}.$

Then $\mathcal{F},\mathcal{G}\subseteq\mathcal{H}(\mathbb{D})$ such that all
the zeros of $f\in\mathcal{F}$ are simple and all the zeros of
$g\in\mathcal{G}$ have multiplicity $2$. It is easy to see that for each
$f_{n}$, there exists $g_{n}\in\mathcal{G}$ such that $f_{n}$ and $g_{n}$
share $0$ partially and $f^{{}^{\prime}}_{n}$ and $g^{{}^{\prime}}_{n}$ share
$1$ partially. Since $|g_{n}(z)|<1$, $\mathcal{G}$ is normal in $\mathbb{D}$
and all limit functions of $\mathcal{G}$ are not identically equal to
infinity. But $\mathcal{F}$ is not normal in $\mathbb{D}$. This shows that the
condition, “all the zeros of $f\in\mathcal{F}$ have multiplicity at least
$k+1$” in Theorem 1.7, is essential.

######  Example 1.10.

Let $\mathcal{F}:=\left\\{f_{n}(z)=nz^{2k}\right\\}$ and
$\mathcal{G}:=\left\\{g_{n}(z)=z^{k}/k!\right\\}.$ Then
$\mathcal{F},\mathcal{G}\subseteq\mathcal{H}(\mathbb{D})$ with all zeros of
$f\in\mathcal{F}$ of multiplicity at least $k+1$ and all zeros of $g$ have
multiplicity exactly $k$. It is easy to see that for each $f\in\mathcal{F}$,
there exists $g\in\mathcal{G}$ such that $f$ and $g$ share $0$ partially, and
$f^{(k)}$ and $g^{(k)}$ share $1$ partially. Also $\mathcal{G}$ is normal in
$\mathbb{D}$ and all limit functions of $\mathcal{G}$ are not identically
equal to infinity. But $\mathcal{F}$ is not normal in $\mathbb{D}$. This shows
that in Theorem 1.7 the condition, “all the zeros of $g\in\mathcal{G}$ have
multiplicity at least $k+1$”, is essential.

Also, the condition, “all the limit functions of $\mathcal{G}$ are not
identically equally to infinity” in Theorem 1.7 is essential:

######  Example 1.11.

Let $\mathbb{D}$ be the open unit disk and let $k$ be a positive integer. Let

$\mathcal{F}:=\left\\{\frac{e^{nz}}{e^{nz}-1}:n\in\mathbb{N}\right\\}$

and

$\mathcal{G}:=\left\\{\frac{e^{nz}}{e^{nz}-1}+n:n\in\mathbb{N}\right\\}$

be two families of meromorphic functions on $\mathbb{D}$. Clearly, it
satisfies all the conditions of Theorem 1.7 except “all the limit functions of
$\mathcal{G}$ are not identically equal to infinity”. It is easy to see that
$\mathcal{F}$ is not normal at $0$.

Finally, we have the following related results:

######  Theorem 1.12.

Let $\mathcal{F}$ and $\mathcal{G}$ be two subfamilies of $\mathcal{M}(D)$ all
of whose zeros have multiplicity at least $k+1$. Let $a_{1}$ and $a_{2}$ be
two complex numbers, and let $\mathcal{G}$ be normal in $D$ such that all
limit functions of $\mathcal{G}$ are not identically equal to infinity. If,
for each $f\in\mathcal{F}$, there exists $g\in\mathcal{G}$ such that $f$ and
$g$ share $\infty$ partially, and $f^{(k)}$ and $g^{(k)}$ share $a_{j},\
j=1,2,$ partially, then $\mathcal{F}$ is normal in $D$.

######  Theorem 1.13.

Let $\mathcal{F}\subseteq\mathcal{M}(D)$ be such that all its zeros have
multiplicity at least $k+1,\ k\in\mathbb{N}$, and let $a_{1},a_{2}$ and
$a_{3}$ be three distinct complex numbers. Let
$\mathcal{G}\subset\mathcal{M}(D)$ be a normal family in $D$ such that all
limit functions of $\mathcal{G}$ are not identically equal to infinity. If for
each $f\in\mathcal{F}$, there exists $g\in\mathcal{G}$ such that $f^{(k)}$ and
$g^{(k)}$ share $a_{j},\ j=1,2,3,$ partially, then $\mathcal{F}$ is normal in
$D$.

Example 1.2 also shows that the condition, “all the limit functions of
$\mathcal{G}$ are not identically equal to infinity” in Theorem 1.12 and
Theorem 1.13, is essential.

######  Example 1.14.

Let

$\mathcal{F}:=\left\\{f_{n}(z)=nz^{k}:n\in\mathbb{N},\
z\in\mathbb{D}\right\\}$

and let

$\mathcal{G}:=\left\\{g_{n}(z)=z+1/n:n\in\mathbb{N},\
z\in\mathbb{D}\right\\}.$

Then $f_{n}^{(k)}$ and $g_{n}^{(k)}$ share any three given values partially,
outside the set $\\{k!n:n\in\mathbb{N}\\}$. Also, $\mathcal{G}$ is normal and
all the limit functions of $\mathcal{G}$ are not identically equal to
infinity. But $\mathcal{F}$ is not normal at $0$. This shows that in Theorem
1.13 the condition, “all the zeros of functions in $\mathcal{F}$ have
multiplicity at least $k+1$” is essential.

The condition, “$f$ and $g$ share $\infty$ partially” in Theorem 1.12 is
essential. Also, the number of shared values in Theorem 1.13 cannot be reduced
to two, for example one may refer to [5].

##  2\. Proofs of Main Theorems

Besides Zalcman’s Lemma [11] we shall use the following results-stated as
lemmas-in the proofs of our main theorems:

######  Lemma 2.1.

[1] Let $f$ be a transcendental meromorphic function of finite order on
$\mathbb{C}$ all of whose zeros have multiplicity at least $k+1$, where $k$ is
a positive integer. Then $f^{(k)}$ assumes every non-zero complex number
infinitely many times on $\mathbb{C}$.

######  Lemma 2.2.

[2] Let $a_{1},\ldots,a_{q}\in\mathbb{C}_{\infty}\mbox{ and
}m_{1},\ldots,m_{q}\in\mathbb{N}$, where $q$ is a positive integer. Suppose
$f\in\mathcal{M}(\mathbb{C})$ is non-constant such that all $a_{j}$-points of
$f$ have multiplicity at least $m_{j},(j=1,\ldots,q).$ Then

$\sum\limits_{j=1}^{q}\left(1-\frac{1}{m_{j}}\right)\leq 2.$

If $f$ does not assume the value $a_{j}$, then we take $m_{j}=\infty$.

######  Lemma 2.3.

[6] Let $\mathcal{F}$ be a family of meromorphic functions in the open unit
disk $\mathbb{D}$ with the property that for each $f\in\mathcal{F}$, all zeros
of $f$ are of multiplicity at least $k$. Suppose that there exists a number
$A\geq 1$ such that $|f^{k}(z)|\leq A$ whenever $f\in\mathcal{F}$ and
$f(z)=0$. If $\mathcal{F}$ is not normal in $\mathbb{D}$, then there exist,
for each $0\leq\alpha\leq k$, a number $0<r<1$, points $z_{n}$ with
$|z_{n}|<1$, functions $f_{n}\in\mathcal{F}$ and positive numbers
$\rho_{n}\rightarrow 0$ such that

$g_{n}(\zeta)=\rho_{n}^{-\alpha}f_{n}(z_{n}+\rho_{n}\zeta)\rightarrow
g(\zeta)$

spherically locally uniformly on $\mathbb{C}$, where $g$ is a non-constant
meromorphic function on $\mathbb{C}$.

######  Lemma 2.4.

[8] Let $f$ be a non-constant meromorphic function of finite order on
$\mathbb{C}$, all of whose zeros have multiplicity at least $k+1$. If
$f^{(k)}(z)\neq a$ on $\mathbb{C}$, where
$a\in\mathbb{C}\setminus\left\\{0\right\\}$, then

$f(z)=\frac{a\ (z-b)^{k+1}}{k!\ (z-c)},$

where $b$ and $c$ are two distinct complex numbers.

Proof of Theorem 1.1 Suppose that $\mathcal{F}$ is not normal at $z_{0}\in D$.
Then, by Zalcman Lemma, there exist $z_{n}\to z_{0}$, $f_{n}\in\mathcal{F}$
and $\rho_{n}\to 0^{+}$ such that $F_{n}(\zeta)=f_{n}(z_{n}+\rho_{n}\zeta)$
converges spherically locally uniformly to a non-constant meromorphic function
$F$ in $\mathbb{C}$. Now by hypothesis, there exists a sequence
$\\{g_{n}\\}\subset\mathcal{G}$ such that for each $n,$ $f_{n}$ and $g_{n}$
share $a_{j},\ j=1,2,3$, partially in $D$. We may assume that $\\{g_{n}\\}$
converges spherically locally uniformly to a non-constant meromorphic function
$g$ in $D$.

Claim. $F$ assumes at most one of the values $a_{j},\ j=1,2,3.$

Supposing on the contrary, we assume that there exist $\zeta_{j}\in\mathbb{C}$
such that $F(\zeta_{j})=a_{j}$ for $j=1,2.$ Then Hurwitz’s theorem ensures the
existence of sequences $\\{\zeta_{j,n}\\}:\zeta_{j,n}\to\zeta_{j}$ such that
for sufficiently large $n,$ $F_{n}(\zeta_{j,n})=a_{j},j=1,2.$ Thus, by
hypothesis, we find that $g_{n}(z_{n}+\rho_{n}\zeta_{j,n})=a_{j},j=1,2.$ That
is, $g(z_{0})=a_{j},\ j=1,2$ which is not allowed and hence the claim.

We assume that $F$ omits $a_{1}$ and $a_{2}$. Then $F$ is transcendental and
$F(\zeta_{0})=a_{3}$, for some $\zeta_{0}\in\mathbb{C}$. Using the preceding
argument, we arrive at $g(z_{0})=a_{3}$. Let $m\geq 1$ be the order of zero of
$g-a_{3}$ at $z_{0}$. Then again by Hurwitz’s theorem there are exactly $m$
zeros of $g_{n}-a_{3}$ in a neighborhood $N(z_{0})$ of $z_{0},$ for
sufficiently large $n.$ Since $F$ assumes $a_{3}$ infinitely often, let
$\zeta_{j},j=1,\ldots,m+1$, be the distinct complex numbers such that
$F(\zeta_{j})=a_{3}$. Once again by Hurwitz’s theorem, for each $j$ there
exist $\\{\zeta_{j,n}\\}$ such that $\zeta_{j,n}\to\zeta_{j}$ and
$F_{n}(\zeta_{j,n})=a_{3}$ for sufficiently large $n.$ That is, $f_{n}-a_{3}$
and hence $g_{n}-a_{3}$ has $m+1$ distinct zeros in $N(z_{0})$, namely, at
$z_{n}+\rho_{n}\zeta_{j,n},j=1,\ldots,m+1$. This is a contradiction. $\Box$

Proofs of Theorem 1.3, Theorem 1.4 and Theorem 1.5: Suppose that $\mathcal{F}$
is not normal in the domain $D$. By applying the arguments of the proof of the
Theorem 1.1 we find that limit function $F$ omits two values in Theorem 1.3,
one value in Theorem 1.4 and two values in Theorem 1.5. Then by Argument
Principle and Lemma 2.2 we arrive at a contradiction in Theorem 1.3 and
Theorem 1.4, whereas in Theorem 1.5 we arrive at a contradiction by using
Hurwitz’s theorem only.

Proof of the Theorem 1.7 Since normality is a local property, we assume $D$ to
be the open unit disk $\mathbb{D}.$ Suppose that $\mathcal{F}$ is not normal
at $z_{0}\in\mathbb{D}$. Then, by Lemma 2.3, there exist $z_{n}\to z_{0}$,
$f_{n}\in\mathcal{F}$ and $\rho_{n}\to 0^{+}$ such that
$F_{n}(\zeta)=\rho_{n}^{-k}f_{n}(z_{n}+\rho_{n}\zeta)$ converges spherically
locally uniformly to $F$ in $\mathbb{C}$, where $F$ is a non-constant
meromorphic function of finite order on $\mathbb{C}$ and all the zeros of $F$
have multiplicity at least $k+1$.
Without loss of generality, we may assume that the corresponding sequence
$\left\\{g_{n}\right\\}\subset\mathcal{G}$ converges spherically locally
uniformly to $g$ in $\mathbb{D}$, where $g\not\equiv\infty$ and all the zeros
of $g$ have multiplicity at least $k+1$.

Case-$1:$ When $F$ assumes 0.
Suppose that $F(\zeta_{0})=0$ for some $\zeta_{0}\in\mathbb{C}$. Then, by
Hurwitz’s theorem, there exists $\zeta_{n}\to\zeta_{0}$ such that for
sufficiently large $n,$ $F_{n}(\zeta_{n})=0.$ That is,
$f_{n}(z_{n}+\rho_{n}\zeta_{n})=0$, and hence
$g_{n}(z_{n}+\rho_{n}\zeta_{n})=0,$ for sufficiently large $n.$ That is,
$g(z_{0})=0.$ Since zeros of $g$ have multiplicity at least $k+1$,
$g^{(k)}(z_{0})=0.$
Claim:  $F^{(k)}(\zeta)\neq a$ on $\mathbb{C}$.
Suppose that there exists $\zeta_{1}\in\mathbb{C}$ such that
$F^{(k)}(\zeta_{1})=a.$ If $F^{(k)}\equiv a$, then $F$ is a polynomial of
degree $k$, a contradiction to the fact that all the zeros of $F$ are of
multiplicity at least $k+1$. Therefore, $F^{(k)}\not\equiv a$, and so by
Hurwitz’s theorem, there exists $\zeta_{1,n}:\zeta_{1,n}\to\zeta_{1}$ such
that for sufficiently large $n$, $F^{(k)}_{n}(\zeta_{1,n})=a.$ This by our
hypothesis implies that $g^{(k)}_{n}(z_{n}+\rho_{n}\zeta_{1,n})=a$, and so
$g^{(k)}(z_{0})=a\ (\neq 0)$, a contradiction. This proves the claim.
By Lemma 2.4, $F$ is a rational function and therefore, there exists
$\zeta_{2}\in\mathbb{C}$ such that $F(\zeta_{2})=\infty$. Again by Hurwitz’s
theorem, there exists $\\{\zeta_{2,n}\\}:\zeta_{2,n}\to\zeta_{2}$ such that
for sufficiently large $n$, $F_{n}(\zeta_{2,n})=\infty.$ This by our
hypothesis implies that for sufficiently large $n,$
$g_{n}(z_{n}+\rho_{n}\zeta_{2,n})=\infty,$ and hence $g(z_{0})=\infty$, a
contradiction.

Case-$2:$ When $F$ omits zero.
In this case we first prove that $F(\zeta)\neq\infty$ on $\mathbb{C}.$ For,
suppose that $F(\zeta_{3})=\infty$, for some $\zeta_{3}\in\mathbb{C}.$ Then
there exists $\\{\zeta_{3,n}\\}:\zeta_{3,n}\to\zeta_{3}$ such that
$F_{n}(\zeta_{3,n})=\infty$, for sufficiently large $n$. This by our
hypothesis implies that $g_{n}(z_{n}+\rho_{n}\zeta_{3,n})=\infty$, and so
$g(z_{0})=\infty$. Since $F\neq 0$ on $\mathbb{C}$, by Hayman’s
Alternative[3], $F^{(k)}(\zeta_{4})=a$, for some $\zeta_{4}\in\mathbb{C}$. By
Hurwitz’s theorem, there exists a sequence $\\{\zeta_{4,n}\\}$ converging to
$\zeta_{4}$ such that for sufficiently large $n,$
$F_{n}^{(k)}(\zeta_{4,n})=a,$ and hence by hypothesis we have
$g_{n}^{(k)}(z_{n}+\rho_{n}\zeta_{4,n})=a$, and so $g^{(k)}(z_{0})=a$, a
contradiction. Therefore, $F$ is transcendental and again by Hayman’s
Alternative, $F^{(k)}$ assumes $a$. As in Case-$1$, we get $g^{(k)}(z_{0})=a$.
If $g^{(k)}\equiv a$, then $g$ is a polynomial of degree $k$, a contradiction
to the fact that zeros of $g$ are of multiplicity at least $k+1$. Assume that
$m\ (\geq 1)$ be the order of zero of $g^{(k)}-a$ at $z_{0}$. By Hurwitz’s
theorem, for sufficiently large $n$, $g_{n}^{(k)}-a$ has exactly $m$ zeros in
a neighborhood $N(z_{0})$ of $z_{0}$. Now by Lemma 2.1, $F^{(k)}$ assumes $a$
infinitely many times and let
$\zeta^{{}^{\prime}}_{1},\ldots,\zeta^{{}^{\prime}}_{m+1}$ be the distinct
zeros of $F^{(k)}-a$. By Hurwitz’s theorem, there exists
$\\{\zeta^{{}^{\prime}}_{j,n}\\}$ converging to $\zeta^{{}^{\prime}}_{j}$ such
that for sufficiently large $n,$ $F_{n}^{(k)}(\zeta^{{}^{\prime}}_{j,n})=b,\
j=1,\ldots,m+1.$ This implies that $f^{(k)}_{n}-a$ has $m+1$ distinct zeros in
$N(z_{0})$, and hence $g^{(k)}_{n}-a$ has $m+1$ distinct zeros in $N(z_{0}),$
a contradiction. $\Box$

Proof of the Theorem 1.12 Normality being a local property allows to replace
the domain $D$ by the open unit disk $\mathbb{D}.$ By supposing on the
contrary that $\mathcal{F}$ is not normal at $z_{0}\in\mathbb{D},$ Lemma 2.3
ensures the existence of sequences $\\{z_{n}\\}$ in $\mathbb{D},$
$\\{f_{n}\\}$ in $\mathcal{F}$ and $\\{\rho_{n}\\}$ in $(0,1)$ such that
$z_{n}\to z_{0},\ \rho_{n}\to 0^{+}$ and

$F_{n}(\zeta):=\frac{f_{n}(z_{n}+\rho_{n}\zeta)}{\rho_{n}^{k}}\to F(\zeta),$

spherically locally uniformly on $\mathbb{C}$, where $F$ is a non-constant
meromorphic function on $\mathbb{C}$ all of whose zeros are of multiplicity at
least $k+1$.

We may assume that the corresponding sequence $\\{g_{n}\\}$ converges
spherically locally uniformly to $g\not\equiv\infty$ on $\mathbb{D}$. For the
sake of convenience we denote the cardinality of a set $A$ by $\\#(A).$

Claim:
$\\#\left(F^{(k)}(\mathbb{C})\cap\left\\{a_{1},a_{2}\right\\}\right)\leq 1$.
Suppose that there exist $\zeta_{j}\in\mathbb{C}$ such that
$F^{(k)}(\zeta_{j})=a_{j},\ j=1,2.$ If $F^{(k)}$ is a constant function, then
$F$ is a polynomial of degree at most $k$ which is not the case as all the
zeros of $F$ are of multiplicity at least $k+1$. Now Hurwitz’s theorem implies
the existence of sequences $\\{\zeta_{j,n}\\}$ such that
$F_{n}^{(k)}(\zeta_{j,n})=a_{j},\ j=1,2,$ which in turn implies that
$g^{(k)}_{n}(z_{n}+\rho_{n}\zeta_{j,n})=a_{j}$ and hence
$g^{(k)}(z_{0})=a_{j},\ j=1,2;$ which is absurd. This proves the claim.

We may assume that $F^{(k)}$ omits $a_{1}$. Then $F^{(k)}$ assumes $a_{2},$
otherwise $F^{(k)}$ would reduce to a constant enforcing $F$ to be a
polynomial of degree at most $k$, a contradiction since all zeros of $F$ are
of multiplicity at least $k+1$. Thus there exists $\zeta_{0}\in\mathbb{C}$
such that $F^{(k)}(\zeta_{0})=a_{2}$ which implies that
$g^{(k)}(z_{0})=a_{2}.$ Now we have two cases: First, when $F$ assumes
$\infty.$ In this case, since $f$ and $g$ share $\infty$ partially,
$g(z_{0})=\infty$ which is not true as $g^{(k)}(z_{0})=a_{2}$ and
$g\not\equiv\infty.$ The second case when $F$ omits $\infty.$ Then $F^{(k)}$
assume $a_{2}$ infinitely many times, and one can arrive at a contradiction by
the arguments used in the proof of Theorem 1.7. $\Box$

Proof of the Theorem 1.13 As in the last proof, we obtain a non-constant
meromorphic function $F$ on $\mathbb{C}$ all of whose zeros are of
multiplicity at least $k+1$. Precisely $F$ is the spherical local uniform
limit of the scaled sequence
$F_{n}(\zeta):=\rho_{n}^{-k}f_{n}(z_{n}+\rho_{n}\zeta)$ in $\mathbb{D}.$
Further, one can establish that

$\\#\left(F^{(k)}(\mathbb{C})\cap\left\\{a_{j}:j=1,2,3\right\\}\right)\leq 1.$

Thus $F^{(k)}$ omits two values in $\mathbb{C}$ and hence $F^{(k)}$ reduces to
a constant. This implies that $F$ is a polynomial of degree at most $k$, a
contradiction as all zeros of $F$ are of multiplicity at least $k+1$. $\Box$

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# Ideals in the convolution algebra of periodic distributions

Amol Sasane  Department of Mathematics
London School of Economics
Houghton Street
London WC2A 2AE
United Kingdom<EMAIL_ADDRESS>
###### Abstract.

The ring of periodic distributions on $\mathbb{R}^{\tt d}$ with usual addition
and with convolution is considered. Via Fourier series expansions, this ring
is isomorphic to the ring $\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ of all
maps $f:\mathbb{Z}^{\tt d}\rightarrow\mathbb{C}$ of at most polynomial growth
(i.e., there exist a real $M>0$ and a nonnegative integer ${\tt m}$ such that
for all $\bm{n}=({\tt n}_{1},\cdots,{\tt n}_{\tt d})\in\mathbb{Z}^{\tt d}$,
$|f(\bm{n})|\leq M(1+|{\tt n}_{1}|+\cdots+|{\tt n}_{\tt d}|)^{\tt m}$), with
pointwise operations. It is shown that finitely generated ideals in
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ are principal, and ideal membership
is characterised analytically. Calling an ideal in
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ fixed if there is a common index
$\bm{n}\in\mathbb{Z}^{\tt d}$ where each member vanishes, the fixed maximal
ideals are described, and it is shown that not all maximal ideals are fixed.
It is shown that finitely generated proper prime ideals in
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ are fixed maximal ideals. The Krull
dimension of $\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ is proved to be
infinite, while the weak Krull dimension is shown to be equal to $1$.

###### Key words and phrases:

Krull dimension, prime ideals, maximal ideals, ring of periodic distributions,
convolution, sequences of at most polynomial growth

###### 2010 Mathematics Subject Classification:

Primary 54C40; Secondary 13A15, 15A24, 13J99

##  1\. Introduction

The aim of this article is to study ideals in a naturally arising ring in
harmonic analysis and distribution theory, namely the ring
$\mathcal{D}^{\prime}_{\mathbf{V}}(\mathbb{R}^{\tt d})$ of periodic
distributions with the usual addition $+$ distributions, and with convolution
$\ast$ taken as multiplication. Via a Fourier series expansion, the ring
$(\mathcal{D}^{\prime}_{\mathbf{V}}(\mathbb{R}^{\tt d}),+,\ast)$ is isomorphic
to the ring $\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ consisting of all
$f:\mathbb{Z}^{\tt d}\rightarrow\mathbb{C}$ of at most polynomial growth, with
pointwise operations, and we recall this below.

###  1.1. The ring of periodic distributions.

For background on periodic distributions and its Fourier series theory, we
refer the reader to [2, Chapter 16] and [7, pp.527-529].

Let $\mathbb{N}=\\{1,2,3,\cdots\\}$ be the set of natural numbers and
$\mathbb{Z}$ be the set of integers. Consider the space
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ of all complex valued maps on
$\mathbb{Z}^{\tt d}$ of at most polynomial growth, that is,

$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d}):=\Big{\\{}f:\mathbb{Z}^{\tt
d}\rightarrow\mathbb{C}\;\Big{|}\;\begin{array}[]{ll}\exists\textrm{ a real
}M>0\;\exists\,{\tt m}\in\mathbb{N}\cup\\{0\\}\;\textrm{ such that}\\\
\forall\bm{n}\in\mathbb{Z}^{\tt d},\;|f(\bm{n})|\leq
M(1+\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,\bm{n}\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,)^{\tt
m}\end{array}\Big{\\}},$

where
$\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,\bm{n}\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,:=|{\tt
n}_{1}|+\cdots+|{\tt n}_{\tt d}|$ for all $\bm{n}=({\tt n}_{1},\cdots,{\tt
n}_{\;\\!\tt d})\in\mathbb{Z}^{\tt d}$. Then
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ is a unital commutative ring with
pointwise operations, and the multiplicative unit element $1_{\mathbb{Z}^{\tt
d}}$ is the constant function $\mathbb{Z}^{\tt d}\owns\bm{n}\mapsto 1$. The
set $\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ equipped with pointwise
operations, is a commutative, unital ring. Moreover,
$(\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d}),+,\cdot)$ is isomorphic as a ring,
to the ring $(\mathcal{D}^{\prime}_{\mathbf{V}}(\mathbb{R}^{\tt d}),+,\ast)$,
where $\mathcal{D}^{\prime}_{\mathbf{V}}(\mathbb{R}^{\tt d})$ is the set of
all periodic distributions (with periods described by ${\mathbf{V}}$, see the
definition below), with the usual pointwise addition of distributions, and
multiplication taken as convolution of distributions.

Let $\mathcal{D}(\mathbb{R}^{d})$ denote the space of compactly supported
infinitely many times differentiable complex valued functions on
$\mathbb{R}^{\tt d}$, and $\mathcal{D}^{\prime}(\mathbb{R}^{\tt d})$ the space
of distributions on $\mathbb{R}^{\tt d}$. For $\bm{v}\in{{\mathbb{R}}}^{\tt
d}$, the translation operator
${\mathbf{S}}_{{\bm{v}}}:\mathcal{D}^{\prime}(\mathbb{R}^{\tt
d})\rightarrow\mathcal{D}^{\prime}(\mathbb{R}^{\tt d})$, is defined by
$\langle{\mathbf{S}}_{\bm{v}}(T),\varphi\rangle=\langle
T,\varphi(\cdot+\bm{v})\rangle$ for all
$\varphi\in{\mathcal{D}}({\mathbb{R}}^{\tt d})$. A distribution
$T\in{\mathcal{D}}^{\prime}({\mathbb{R}}^{\tt d})$ is called periodic with a
period $\bm{v}\in{\mathbb{R}}^{\tt d}\setminus\\{\mathbf{0}\\}$ if
$T={\mathbf{S}}_{\bm{v}}(T)$. Let
${\mathbf{V}}:=\\{\bm{v}_{1},\cdots,\bm{v}_{\tt d}\\}$ be a linearly
independent set of ${\tt d}$ vectors in ${\mathbb{R}}^{\tt d}$. Let
${\mathcal{D}}^{\prime}_{{\mathbf{V}}}({\mathbb{R}}^{\tt d})$ denote the set
of all distributions $T$ that satisfy ${\mathbf{S}}_{\bm{v}_{\tt k}}(T)=T$ for
all ${\tt k}\in\\{1,\cdots,{\tt d}\\}$. From [1, §34], $T$ is a tempered
distribution, and from the above it follows by taking Fourier transforms that
$(1-e^{2\pi i\bm{v}_{\tt k}\cdot\bm{y}})\widehat{T}=0$, for ${\tt
k}\in\\{1,\cdots,{\tt d}\\}$, $\bm{y}\in\mathbb{R}^{d}$. Then
$\widehat{T}=\sum_{\bm{v}\in V^{-1}{\mathbb{Z}}^{\tt
d}}\alpha_{\bm{v}}(T)\delta_{\bm{v}},$ for some scalars
$\alpha_{\bm{v}}(T)\in{\mathbb{C}}$, and where $V$ is the matrix with its rows
equal to the transposes of the column vectors ${\bm{v}_{1},\cdots,\bm{v}_{\tt
d}}$: $V^{\textrm{t}}:=\left[\begin{smallmatrix}\bm{v}_{1}&\cdots&\bm{v}_{\tt
d}\end{smallmatrix}\right],$ with $V^{\textrm{t}}$ denoting the transpose of
the matrix $V$. Also, in the above, $\delta_{\bm{v}}$ denotes the usual Dirac
measure with support in $\bm{v}$, i.e.,
$\langle\delta_{\bm{v}},\varphi\rangle=\varphi(\bm{v})$ for all
$\varphi\in{\mathcal{D}}({\mathbb{R}}^{\tt d})$. Then the Fourier coefficients
$\alpha_{\bm{v}}(T)$ give rise to an element in
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$, and vice versa, every element in
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ is the set of Fourier coefficients
of some periodic distribution. In fact, the ring
$(\mathcal{D}^{\prime}_{\mathbf{V}}(\mathbb{R}^{\tt d}),+,\ast)$ of periodic
distributions on $\mathbb{R}^{\tt d}$ is isomorphic to the ring
$(\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d}),+,\cdot).$

In [5], some algebraic-analytical properties of
$(\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d}),+,\cdot)$ were established; see
also [4]. In this article, the structure of ideals in this ring is studied,
akin to an analogous investigation in [8] for a ring of entire functions.

###  1.2. Main results and organisation of the article

  * •

In §2, we show that finitely generated ideals in
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ are principal, and ideal membership
is characterised analytically.

  * •

In §3, we describe fixed maximal ideals in
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$, and it is shown that not all
maximal ideals are fixed.

  * •

In §4, we show that finitely generated proper prime ideals in
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ are fixed maximal ideals. Also, the
Krull dimension of $\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ is proved to be
infinite, while the weak Krull dimension is shown to be equal to $1$.

##  2\. Finitely generated ideals

######  Proposition 2.1.

$g$ is a divisor of $f$ in $\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ if and
only if there exist a real number $M>0$ and a nonnegative integer ${\tt m}$
such that for all $\bm{n}\in\mathbb{Z}^{\tt d}$, $|f(\bm{n})|\leq
M(1+\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,\bm{n}\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,)^{\tt
m}|g(\bm{n})|$.

###### Proof.

(‘If’ part:) Define $d:\mathbb{Z}^{\tt d}\rightarrow\mathbb{C}$ by

$d(\bm{n})=\left\\{\begin{array}[]{cl}\frac{f(\bm{n})}{g(\bm{n})}&\textrm{if
}g(\bm{n})\neq 0,\\\ 0&\textrm{if }g(\bm{n})=0.\end{array}\right.$

Thus for $g(\bm{n})\neq 0$, we have $|d(\bm{n})|\leq
M(1+\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,\bm{n}\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,)^{\tt
m}$, and this also holds trivially when $g(\bm{n})=0$, since the left-hand
side is $0$. Thus $d\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$. Moreover,
for $g(\bm{n})\neq 0$, we have $d(\bm{n})g(\bm{n})=f(\bm{n})$, and when
$g(\bm{n})=0$, the inequality $|f(\bm{n})|\leq
M(1+\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,\bm{n}\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,)^{\tt
m}|g(\bm{n})|$ yields $f(\bm{n})=0$ too, showing that
$d(\bm{n})g(\bm{n})=d(\bm{n})0=0=f(\bm{n})$. Hence $dg=f$, as wanted.

(‘Only if’ part:) Suppose that $d\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$
is such that $dg=f$. Since $d\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$,
there exist $M>0$ and a nonnegative integer ${\tt m}$ such that
$|d(\bm{n})|\leq
M(1+\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,\bm{n}\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,)^{\tt
m}$. So
$|f(\bm{n})|\\!\leq\\!|d(\bm{n})||g(\bm{n})|\\!\leq\\!M(1\\!+\\!\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,\bm{n}\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,)^{\tt
m}|g(\bm{n})|$ for all $\bm{n}\in\mathbb{Z}^{\tt d}$. ∎

In particular, $f$ is invertible in $\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$
if and only if there exists a real number $\delta>0$ and a nonnegative integer
${\tt m}$ such that for all $\bm{n}\in\mathbb{Z}^{\tt d}$,
$|f(\bm{n})|\geq\delta(1+|\bm{n}|)^{-{\tt m}}$.

######  Proposition 2.2.

Every finite number of elements $f_{1},\cdots,f_{\tt
K}\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ $({\tt K}\in\mathbb{N})$ have a
greatest common divisor $d$. The element $d$ is given $($up to invertible
elements$)$ by $d(\bm{n})=\max\\{|f_{1}(\bm{n})|,\cdots,|f_{\tt
K}(\bm{n})|\\}$ $(\bm{n}\in\mathbb{Z}^{\tt d})$.

###### Proof.

Let $d(\bm{n})=\max\\{|f_{1}(\bm{n})|,\cdots,|f_{\tt K}(\bm{n})|\\}$ for all
$\bm{n}\in\mathbb{Z}^{\tt d}$. Clearly
$d\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$. As $|f_{\tt
k}(\bm{n})|\leq|d(\bm{n})|$ for all $\bm{n}\in\mathbb{Z}^{\tt d}$ and all
${\tt k}\in\\{1,\cdots,{\tt K}\\}$, Proposition 2.1 implies that $d$ is a
common divisor of $f_{1},\cdots,f_{\tt K}$.

If $\widetilde{d}\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ is a common
divisor of $f_{1},\cdots,f_{\tt K}$, then by Proposition 2.1 again, there
exist real $M_{\tt k}>0$ and positive integers ${\tt m}_{\tt k}$, for each
${\tt k}\in\\{1,\cdots,{\tt K}\\}$, such that $|f_{\tt k}(\bm{n})|\leq M_{\tt
k}(1+\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,\bm{n}\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,)^{{\tt
m}_{\tt k}}|\widetilde{d}(\bm{n})|$ for all ${\bm{n}}\in\mathbb{Z}^{\tt d}$.
Setting $M:=\max\\{M_{1},\cdots,M_{\tt K}\\}$ and $m:=\max\\{{\tt
m}_{1},\cdots,{\tt m}_{\tt K}\\}$, we get $|d(\bm{n})|\leq
M(1+\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,\bm{n}\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,)^{\tt
m}|\widetilde{d}(\bm{n})|$ for all $\bm{n}\in\mathbb{Z}^{\tt d}$. By
Proposition 2.1, $\widetilde{d}$ divides $d$ in
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$. ∎

######  Proposition 2.3.

Let $\langle f_{1},\cdots,f_{\tt K}\rangle$ denote the ideal generated ${\tt
K}\in\mathbb{N}$ elements $f_{1},\cdots,f_{\tt
K}\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$. Then $f\in\langle
f_{1},\cdots,f_{\tt K}\rangle$ if and only if there exists an $M>0$ and a
nonnegative integer ${\tt m}$ such that

$|f(\bm{n})|\leq
M(1+\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,\bm{n}\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,)^{\tt
m}\sum_{{\tt k}=1}^{\tt K}|f_{\tt k}(\bm{n})|\textrm{ for all
}\bm{n}\in\mathbb{Z}^{\tt d}.$

###### Proof.

(‘If’ part:) For ${\tt k}\in\\{1,\cdots,{\tt N}\\}$, define $g_{\tt
k}:\mathbb{Z}^{\tt d}\rightarrow\mathbb{C}$ by

$g_{\tt k}(\bm{n})=\left\\{\begin{array}[]{cl}\frac{\overline{f_{\tt
k}(\bm{n})}}{\sum\limits_{\tt j=1}^{\tt K}|f_{\tt
j}(\bm{n})|^{2}}f(\bm{n})&\textrm{if }{\scaleobj{0.81}{\sum\limits_{\tt
j=1}^{\tt K}}}|f_{\tt j}(\bm{n})|^{2}\neq 0,\\\ 0&\textrm{if
}{\scaleobj{0.81}{\sum\limits_{\tt j=1}^{\tt K}}}|f_{\tt
j}(\bm{n})|^{2}=0.\end{array}\right.$

If $Q(\bm{n}):={\scaleobj{0.81}{\sum\limits_{\tt j=1}^{\tt K}}}|f_{\tt
j}(\bm{n})|^{2}\neq 0$, then (by Cauchy-Schwarz in the last step),

$\begin{array}[]{rcl}|g_{\tt k}(\bm{n})|=\frac{|\overline{f_{\tt
k}(\bm{n})}|}{\sum\limits_{\tt j=1}^{\tt K}|f_{\tt
j}(\bm{n})|^{2}}|f(\bm{n})|\\!\\!\\!&\leq&\\!\\!\\!\frac{\sum\limits_{\tt
j=1}^{\tt K}|f_{\tt j}(\bm{n})|}{\sum\limits_{\tt j=1}^{\tt K}|f_{\tt
j}(\bm{n})|^{2}}M(1\\!+\\!\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,\bm{n}\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,)^{\tt
m}{\scaleobj{0.81}{\sum\limits_{{\tt k}=1}^{\tt K}}}|f_{\tt k}(\bm{n})|\\\
\\!\\!\\!&\leq&\\!\\!\\!\frac{(\sum\limits_{\tt j=1}^{\tt K}|f_{\tt
j}(\bm{n})|)^{2}}{\sum\limits_{\tt j=1}^{\tt K}|f_{\tt
j}(\bm{n})|^{2}}M(1\\!+\\!\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,\bm{n}\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,)^{\tt
m}\leq
KM(1\\!+\\!\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,\bm{n}\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,)^{\tt
m}.\end{array}$

So $g_{1},\cdots,g_{\tt K}\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$. We
claim that $f_{1}g_{1}+\cdots+f_{\tt K}g_{\tt K}=f$. The evaluation of the
left-hand side at an $\bm{n}\in\mathbb{Z}^{\tt d}$ such that $Q(\bm{n})\neq 0$
is easily seen to be $f(\bm{n})$ by the definition of $g_{1},\cdots,g_{\tt
K}$. On the other hand, if $Q(\bm{n})=0$, then each $f_{\tt k}(\bm{n})=0$, and
by the given inequality in the statement of the proposition, so is
$f(\bm{n})=0$. Thus in this case the evaluations at $\bm{n}$ of both sides of
$f_{1}g_{1}+\cdots+f_{\tt K}g_{\tt K}=f$ are zeroes.

(‘Only if’ part:) If $f\in\langle f_{1},\cdots,f_{\tt K}\rangle$, then there
exist $g_{1},\cdots,g_{\tt K}\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ such
that $f=f_{1}g_{1}+\cdots+f_{\tt K}g_{\tt K}$. Let $M_{\tt k}>0$ and ${\tt
m}_{\tt k}\in\mathbb{N}\cup\\{0\\}$, ${\tt k}\in\\{1,\cdots,{\tt K}\\}$, be
such that $|g_{\tt k}(\bm{n})|\leq M_{\tt
k}(1+\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,\bm{n}\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,)^{{\tt
m}_{\tt k}}$ ($\bm{n}\in\mathbb{Z}^{\tt d}$). Then with
$M:=\max\\{M_{1},\cdots,M_{\tt K}\\}$ and ${\tt m}:=\max\\{{\tt
m}_{1},\cdots,{\tt m}_{\tt K}\\}$, we get

$|f(\bm{n})|\leq{\scaleobj{0.81}{\sum_{{\tt k}=1}^{\tt K}}}|f_{\tt
k}(\bm{n})||g_{\tt k}(\bm{n})|\leq
M(1+\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,\bm{n}\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,)^{\tt
m}{\scaleobj{0.81}{\sum_{{\tt k}=1}^{\tt K}}}|f_{\tt k}(\bm{n})|.$ ∎

It follows from Propositions 2.2 and 2.3 that every finite generated ideal is
principal. (Indeed, $\langle f_{1},\cdots,f_{\tt K}\rangle=\langle d\rangle$:
That $f_{\tt k}\subset\langle d\rangle$ for each ${\tt k}$ is obvious as $d$
divides $f_{\tt k}$, in turn showing $\langle f_{1},\cdots,f_{\tt
K}\rangle\subset\langle d\rangle$. For the reverse inclusion,
$|d(\bm{n})|\\!=\\!\max\\{|f_{1}(\bm{n})|,\cdots,|f_{\tt
K}(\bm{n})|\\}\\!\leq\\!\sum_{{\tt k}=1}^{\tt K}|f_{\tt k}(\bm{n})|$
($\bm{n}\\!\in\\!\mathbb{Z}^{\tt d}$), and so by Proposition 2.3, $d\in\langle
f_{1},\cdots,f_{\tt K}\rangle$. Thus we get $\langle d\rangle\subset\langle
f_{1},\cdots,f_{\tt K}\rangle$.)

##  3\. Maximal ideals

######  Definition 3.1.

An ideal $\mathfrak{i}$ of $\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ is fixed
if there exists an $\bm{k}\in\mathbb{Z}^{\tt d}$ such that for all
$f\in\mathfrak{i}$, $f(\bm{k})=0$.

######  Theorem 3.2.

For $\bm{k}\in\mathbb{Z}^{\tt d}$, let
$\mathfrak{m}_{\bm{k}}:=\\{f\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt
d}):f(\bm{k})=0\\}$. Then $\mathfrak{m}_{\bm{k}}$ is a fixed maximal ideal of
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$. Every fixed maximal ideal of
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ is equal to $\mathfrak{m}_{\bm{k}}$
for some $\bm{k}\in\mathbb{Z}^{\tt d}$.

###### Proof.

The fixedness of $\mathfrak{m}_{\bm{k}}$ is clear. We now show that
maximality. As $1_{\mathbb{Z}^{\tt d}}\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt
d})\setminus\mathfrak{m}_{\bm{k}}$,
$\mathfrak{m}_{\bm{k}}\subsetneq\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$. Let
$\mathfrak{i}$ be an ideal such that
$\mathfrak{m}_{\bm{k}}\subsetneq\mathfrak{i}$. Suppose that
$f\in\mathfrak{i}\setminus\mathfrak{m}_{\bm{k}}$. Then $f(\bm{k})\neq 0$.
Define $g\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ by $g=1_{\mathbb{Z}^{\tt
d}}-\frac{f}{f(\bm{k})}$. As $g(\bm{k})=0$, we have
$g\in\mathfrak{m}_{\bm{k}}\subset\mathfrak{i}$. Also,
$\frac{f}{f(\bm{k})}\in\mathfrak{i}$. Thus $1_{\mathbb{Z}^{\tt
d}}=g+\frac{f}{f(\bm{k})}\in\mathfrak{i}$, i.e.,
$\mathfrak{i}=\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$.

Next, let $\mathfrak{m}$ be a fixed maximal ideal of
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$. Since $\mathfrak{m}$ is fixed,
there exists a $\bm{k}\in\mathbb{Z}^{\tt d}$ such that
$\mathfrak{m}\subset\mathfrak{m}_{\bm{k}}\subsetneq\mathcal{S}^{\prime}(\mathbb{Z}^{\tt
d})$. By the maximality of $\mathfrak{m}$, we conclude that
$\mathfrak{m}=\mathfrak{m}_{\bm{k}}$. ∎

######  Example 3.3 (Non-fixed maximal ideals).

Let $({\tt k}_{\tt j})_{{\tt j}\in\mathbb{N}}$ be any subsequence of the
sequence of natural numbers. Set $\bm{k}_{\tt j}=({\tt k}_{\tt j},\cdots,{\tt
k}_{\tt j})\in\mathbb{Z}^{\tt d}$. Define
$\mathfrak{i}:=\\{f\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d}):\lim_{{\tt
j}\rightarrow\infty}e^{{\tt k}_{\tt j}}f(\bm{k}_{\tt j})=0\\}.$ Then
$\mathfrak{i}$ is an ideal of $\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$. (It
is clear that if $f,g\in\mathfrak{i}$, then $f+g\in\mathfrak{i}$. If
$f\in\mathfrak{i}$ and $g\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$, then
there exist a real $M>0$ and an ${\tt m}\in\mathbb{N}\cup\\{0\\}$ such that
$|g(\bm{n})|\leq
M(1+\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,\bm{n}\,\rule[-2.58334pt]{1.30005pt}{9.90276pt}\,)^{\tt
m}$ for all $\bm{n}\in\mathbb{Z}^{\tt d}$, and so

$|(fg)(\bm{k}_{\tt j})|\leq|f(\bm{k}_{\tt j})|M(1+{\tt d}{\tt k}_{\tt j})^{\tt
m}=e^{{\tt k}_{\tt j}}|f(\bm{k}_{\tt j})|e^{-{\tt k}_{\tt j}}M(1+{\tt d}{\tt
k}_{\tt j})^{\tt m}\stackrel{{\scriptstyle{\tt
j}\rightarrow\infty}}{{\longrightarrow}}0,$

showing that $fg\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$.) Moreover,
$\mathfrak{i}\neq\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ since
$1_{\mathbb{Z}^{\tt d}}\not\in\mathfrak{i}$: $e^{{\tt k}_{\tt
j}}|1_{\mathbb{Z}^{\tt d}}(\bm{k}_{\tt j})|=e^{{\tt k}_{\tt j}}1>1$ for all
$n\in\mathbb{N}$. Hence there exists a maximal ideal $\mathfrak{m}$ in
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ such that
$\mathfrak{i}\subset\mathfrak{m}$. We note that for each
$\bm{k}\in\mathbb{Z}^{\tt d}$, $\mathfrak{m}\neq\mathfrak{m}_{\bm{k}}$: Define
$1_{\\{\bm{k}\\}}:\mathbb{Z}^{\tt d}\rightarrow\mathbb{C}$ by
$1_{\\{\bm{k}\\}}(\bm{n})=0$ for all $\bm{n}\neq\bm{k}$ and
$1_{\\{\bm{k}\\}}(\bm{k})=1$. Then
$1_{\\{\bm{k}\\}}\in\mathfrak{i}\subset\mathfrak{m}$, but
$1_{\\{\bm{k}\\}}\not\in\mathfrak{m}_{\bm{k}}$ as
$1_{\\{\bm{k}\\}}(\bm{k})=1\neq 0$. $\Diamond$

##  4\. Prime ideals

###  4.1. Finitely generated proper prime ideals

######  Theorem 4.1.

Let $\mathfrak{p}$ be a finitely generated proper prime ideal of
${\mathcal{S}}^{\prime}(\mathbb{Z}^{\tt d})$. Then there exists an
$\bm{n}_{*}\in\mathbb{Z}^{\tt d}$ such that

$\mathfrak{p}=\\{f\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt
d}):f(\bm{n}_{*})=0\\}.$

###### Proof.

We carry out the proof in several steps.

Step 1. $\mathfrak{p}$ is principal. If $\mathfrak{p}\\!=\\!\langle d\rangle$,
then $d$ has at most one zero. As $\mathfrak{p}$ is finitely generated, it is
principal. Let $d\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ be such that
$\mathfrak{p}=\langle d\rangle$. Let $\bm{m},\bm{n}\in\mathbb{Z}^{\tt d}$ be
distinct and $d(\bm{n})=0=d(\bm{m})$. Define $a:\mathbb{Z}^{\tt
d}\rightarrow\mathbb{C}$ by $a(\bm{k})=1$ for all $\bm{k}\neq\bm{n}$ and
$a(\bm{n})=0$. Then $a\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$. Also, let
$b:\mathbb{Z}^{\tt d}\rightarrow\mathbb{C}$ be defined by
$b(\bm{k})=d(\bm{k})$ for all $\bm{k}\not\in\\{\bm{m},\bm{n}\\}$,
$b(\bm{m})=0$ and $b(\bm{n})=1$. Then clearly
$b\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ too (since it matches with $d$
everywhere except at the single index $\bm{n}$). Now $(ab)(\bm{k})=d(\bm{k})$
for all $\bm{k}\in\mathbb{Z}^{\tt d}$: If $\bm{k}\not\in\\{\bm{m},\bm{n}\\}$
this is clear from the definitions since the left-hand side is $1\cdot
d(\bm{k})$, and if $\bm{k}=\bm{m}$ or $\bm{n}$, then both sides are $0$. So
$ab=d\in\langle d\rangle=\mathfrak{p}$. But $a\not\in\langle d\rangle$ since
otherwise $a=d\tilde{a}$ for some
$\tilde{a}\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ and then
$1=a(\bm{m})=d(\bm{m})\tilde{a}(\bm{m})=0\tilde{a}(\bm{m})=0$, a
contradiction. Also, $b\not\in\langle d\rangle$ since otherwise $b=d\tilde{b}$
for some $\tilde{b}\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ and then
$1=b(\bm{n})=d(\bm{n})\tilde{b}(\bm{n})=0\tilde{b}(\bm{m})=0$, a
contradiction. So neither $a$ nor $b$ belong to $\langle
d\rangle=\mathfrak{p}$, contradicting the primality of $\mathfrak{p}$.

Step 2. Let ${\scaleobj{0.96}{\mathfrak{p}\\!=\\!\langle d\rangle}}$ (as in
Step 1). For each ${\scaleobj{0.96}{\bm{n}\\!\in\\!\mathbb{Z}^{\tt d}}}$, let
${\scaleobj{0.96}{d(\bm{n})\\!=\\!|d(\bm{n})|e^{i\theta(\bm{n})}}}$ for some
$\theta(\bm{n})\in(-\pi,\pi]$. Define
$h\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ by
${\scaleobj{0.96}{h(\bm{n})=\sqrt{|d(\bm{n})|}e^{i\theta(\bm{n})/2}}}$ for all
$\bm{n}\in\mathbb{Z}^{\tt d}$. Then $h^{2}=d\in\mathfrak{p}$, and as
$\mathfrak{p}$ is prime, $h\in\mathfrak{p}$.

Step 3. $d$ has exactly one zero. We will now show that there exists an
$\bm{n}_{*}\in\mathbb{Z}^{\tt d}$ such that $d(\bm{n}_{*})=0$. Suppose this is
not true. Then by Step 1, $d(\bm{n})\neq 0$ for all $\bm{n}\in\mathbb{Z}^{\tt
d}$. If $h\in\mathfrak{p}$ is as in Step 2, then there exists a
$k\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ such that $h=kd$, i.e.,
$\sqrt{|d(\bm{n})|}e^{i\theta(\bm{n})/2}=|d(\bm{n})|e^{i\theta(\bm{n})}k(\bm{n})$,
which yields $1=d(\bm{n})(k(\bm{n}))^{2}$ ($\bm{n}\in\mathbb{Z}^{\tt d}$).
Thus $dk^{2}=1_{\mathbb{Z}^{\tt d}}$, showing that $d$ is invertible in
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$, contradicting the properness of
the ideal $\mathfrak{p}$.

Step 4. We now show that
$\mathfrak{p}=\\{f\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt
d}):f(\bm{n}_{*})=0\\}$. That
$\mathfrak{p}\subset\\{f\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt
d}):f(\bm{n}_{*})=0\\}$ is clear. Let
$f\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ be such that $f(\bm{n}_{*})=0$.
Define $g:\mathbb{Z}^{\tt d}\rightarrow\mathbb{C}$ by

$g(\bm{n})=\left\\{\begin{array}[]{cl}\frac{f(\bm{n})}{d(\bm{n})}&\textrm{if
}\bm{n}\neq\bm{n}_{*},\\\ 0&\textrm{if }\bm{n}=\bm{n}_{*}.\end{array}\right.$

Then $f=dg$ (note that $f(\bm{n})=d(\bm{n})g(\bm{n})$ for
$\bm{n}\neq\bm{n}_{*}$ follows from the definition of $g$, and
$f(\bm{n}_{*})=d(\bm{n}_{*})g(\bm{n}_{*})$ too since both sides are $0$). As
the $h$ from Step 2 is in $\mathfrak{p}$, there exists a
$k\in\mathcal{S}^{\prime}(Z^{\tt d})$ such that $h=kd$, and so for all
$\bm{n}\in\mathbb{Z}^{\tt d}$, we get
$\sqrt{|d(\bm{n})|}e^{i\theta(\bm{n})/2}=|d(\bm{n})|e^{i\theta(\bm{n})}k(\bm{n})$,
giving $1=|d(\bm{n})||k(\bm{n})|^{2}$. Hence for $\bm{n}\neq\bm{n}_{*}$,
$\frac{1}{|d(\bm{n})|}=|k(\bm{n})|^{2}\leq M(1+|\bm{n}|)^{\tt m}$ for some
$M>0$ and a nonnegative integer ${\tt m}$. This estimate shows that
$g\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$, and hence $f=gd\in
d\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})=\langle d\rangle=\mathfrak{p}$. ∎

###  4.2. Krull dimension

######  Definition 4.2.

The Krull dimension of a commutative ring $R$ is the supremum of the lengths
of chains of distinct proper prime ideals of $R$.

Recall that the Hardy algebra $H^{\infty}$ is the Banach algebra of bounded
and holomorphic functions on the unit disc
$\mathbb{D}:=\\{z\in\mathbb{C}:|z|<1\\}$, with pointwise operations and the
supremum norm $\|\cdot\|_{\infty}$. In [9], von Renteln showed that the Krull
dimension of $H^{\infty}$ is infinite. We adapt the idea given in [9], to show
that the Krull dimension of $\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ is
infinite too. A key ingredient of the proof in [9] was the use of a canonical
factorisation of $H^{\infty}$ elements used to create ideals with zeroes at
prescribed locations with prescribed multiplicities. Instead, we will look at
the zero set in $\mathbb{Z}^{\tt d}$ for
$f\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$, and use the notion of ‘zero-
order’ introduced below.

If $f\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ and $\bm{n}=({\tt
n}_{1},\cdots,{\tt n}_{\tt d})\in\mathbb{Z}^{\tt d}$ is such that
$f(\bm{n})=0$, then we define the zero-order $m(f,\bm{n})$ by

$m(f,\bm{n})\\!=\\!\min_{1\leq{\tt k}\leq{\tt
d}}\max{\scaleobj{1.1}{\Big{\\{}}}{\tt
i}\\!\in\\!\mathbb{N}:\\!\\!{\scaleobj{0.9}{\begin{array}[]{ll}f({\tt
n}_{1},\cdots,{\tt n}_{{\tt k}-1},{\tt n}_{\tt k}\\!+\\!{\tt j},{\tt n}_{{\tt
k}+1},\cdots,{\tt n}_{\tt d})=0\\\ \textrm{ whenever }0\leq{\tt j}\leq{\tt
i}-1\end{array}}}\\!\\!{\scaleobj{1.1}{\Big{\\}}}}.$

If $f({\tt n}_{1},\cdots,{\tt n}_{{\tt k}-1},{\tt n}_{\tt k}\\!+\\!{\tt
j},{\tt n}_{{\tt k}+1},\cdots,{\tt n}_{\tt d})=0$ for all ${\tt
j}\in\mathbb{N}\cup\\{0\\}$, and all ${\tt k}\in\\{1,\cdots,{\tt d}\\}$, then
we set $m(f,\bm{n})=\infty$. If $f(\bm{n})\neq 0$, then we set
$m(f,\bm{n})=0$. Analogous to the multiplicity of a zero of a (not identically
vanishing) holomorphic function, the zero-order satisfies the following
property.

$\begin{array}[]{ll}\textrm{(P1): If
}f,g\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})\textrm{ and
}\bm{n}\in\mathbb{Z}^{\tt d},\\\ \phantom{(P1):}\textrm{then
}m(f+g,\bm{n})\geq\min\\{m(f,\bm{n}),m(g,\bm{n})\\}.\end{array}$

The multiplicity of a zero $\zeta$ of the pointwise product of two holomorphic
functions is the sum of the multiplicities of $\zeta$ as a zero of each of the
two holomorphic functions. For the zero-order, we have the following instead:

$\begin{array}[]{ll}\textrm{(P2): If
}f,g\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})\textrm{ and
}\bm{n}\in\mathbb{Z}^{\tt d},\\\ \phantom{(P1):}\textrm{then
}m(fg,\bm{n})\geq\max\\{m(f,\bm{n}),m(g,\bm{n})\\}.\end{array}$

We will use the following known result; see [3, Theorem, §0.16, p.6].

######  Proposition 4.3.

If $\mathfrak{i}$ is an ideal in a ring $R,$ $M\subset R$ is a set that is
closed under multiplication, and $M\cap\mathfrak{i}=\emptyset$, then there
exists an ideal $\mathfrak{p}$ such that $\mathfrak{i}\subset\mathfrak{p}$ and
$\mathfrak{p}\cap M=\emptyset,$ and $\mathfrak{p}$ maximal with respect to
these properties. Moreover, such an ideal $\mathfrak{p}$ is necessarily prime.

######  Theorem 4.4.

The Krull dimension of ${\mathcal{S}}^{\prime}(\mathbb{Z}^{\tt d})$ is
infinite.

###### Proof.

For ${\tt i}\in\\{1,\cdots,{\tt d}\\}$, let $\bm{e}_{\tt i}\in\mathbb{Z}^{\tt
d}$ be the vector all of whose components are zeroes except for the ${\tt
i}^{\textrm{th}}$ one, which is defined as $1$. For ${\tt n}\in\mathbb{N}$,
define $f_{\tt n}\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ by

$\displaystyle\\!\\!\\!f_{\tt n}(2^{\tt k}\bm{e}_{1}\\!+\\!{\tt j}\bm{e}_{\tt
i})=0\textrm{ if }{\tt k}\in\mathbb{N}\cup\\{0\\},\,1\\!\leq\\!{\tt
i}\\!\leq\\!{\tt d},\,0\\!\leq\\!{\tt j}\\!\leq\\!{\tt k}^{{\tt n}+1},$
$\displaystyle\\!\\!\\!f_{\tt n}(\bm{m})=1\textrm{ if }\bm{m}\not\in\\{2^{\tt
k}\bm{e}_{1}\\!+\\!{\tt j}\bm{e}_{\tt i}:{\tt
k}\in\mathbb{N}\cup\\{0\\},\,1\\!\leq\\!{\tt i}\\!\leq\\!{\tt
d},\,0\\!\leq\\!{\tt j}\\!\leq\\!{\tt k}^{{\tt n}+1}\\}.$

Note that $m(f_{\tt n},2^{\tt k}\bm{e}_{1})\geq{\tt k}^{{\tt n}+1}$, but for
each fixed ${\tt n}\in\mathbb{N}$, there exists a ${\tt K}_{\tt
n}\in\mathbb{N}\cup\\{0\\}$ such that the gap between the indices,

$2^{{\tt k}+1}-2^{\tt k}=2^{\tt k}>{\tt k}^{{\tt n}+1}\textrm{ for all }{\tt
k}>{\tt K}_{\tt n},$

and so $m(f_{\tt n},2^{\tt k}\bm{e}_{1})={\tt k}^{{\tt n}+1}$ for all ${\tt
k}>{\tt K}_{\tt n}$. Hence

$\lim\limits_{{\tt k}\rightarrow\infty}\frac{m(f_{\tt n},2^{\tt
k}\bm{e}_{1})}{{\tt k}^{\tt n}}=\infty\quad\textrm{ and
}\quad\lim\limits_{{\tt k}\rightarrow\infty}\frac{m(f_{\tt n},2^{\tt
k}\bm{e}_{1})}{{\tt k}^{{\tt n}+1}}=1<\infty.$ (1)

Let

$\mathfrak{i}_{*}:=\\{f\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt
d}):\exists\,{\tt k}_{0}(f)\in\mathbb{N}_{0}\textrm{ such that }\forall\,{\tt
k}>{\tt k}_{0}(f),\;f(2^{\tt k}\bm{e}_{1})=0\\}.$

The set $\mathfrak{i}_{*}$ is nonempty since $0\in\mathfrak{i}_{*}$. Clearly
$\mathfrak{i}_{*}$ is closed under addition, and $fg\in\mathfrak{i}_{*}$
whenever $f\in\mathfrak{i}_{*}$ and $g\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt
d})$. So $\mathfrak{i}_{*}$ is an ideal of
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$. For ${\tt n}\in\mathbb{N}$, define

$\displaystyle\mathfrak{i}_{\tt n}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\Big{\\{}f\in\mathfrak{i}_{*}:\lim\limits_{{\tt
k}\rightarrow\infty}{\scaleobj{0.87}{\frac{m(f,2^{\tt k}\bm{e}_{1})}{{\tt
k}^{\tt n}}}}=\infty\Big{\\}},$ $\displaystyle M_{\tt n}\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\Big{\\{}f\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt
d}):\sup\limits_{{\tt k}\in\mathbb{N}}{\scaleobj{0.87}{\frac{m(f,2^{\tt
k}\bm{e}_{1})}{{\tt k}^{\tt n}}}}<\infty\Big{\\}}.$

Clearly $f_{\tt n}\in\mathfrak{i}_{\tt n}$, and so $\mathfrak{i}_{\tt n}$ is
not empty. Using (P1), we see that if $f,g\in\mathfrak{i}_{\tt n}$, then
$f+g\in\mathfrak{i}_{\tt n}$. If $g\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt
d})$ and $f\in\mathfrak{i}_{\tt n}$, then (P2) implies that
$fg\in\mathfrak{i}_{\tt n}$. Hence $\mathfrak{i}_{\tt n}$ is an ideal of
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$.

The identity element $1_{\mathbb{Z}^{\tt d}}\in M_{\tt n}$ for all ${\tt
n}\in\mathbb{N}$. If $f,g\in M_{\tt n}$, then it follows from (P2) that $fg\in
M_{\tt n}$. Thus $M_{\tt n}$ is a nonempty multiplicatively closed subset of
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$.

It is easy to check that for all ${\tt n}\in\mathbb{N}$, $\mathfrak{i}_{{\tt
n}+1}\subset\mathfrak{i}_{\tt n}$ and $M_{\tt n}\subset M_{{\tt n}+1}$. We now
prove that the inclusions are strict for each ${\tt n}\in\mathbb{N}$. From
(1), it follows that $f_{\tt n}\in\mathfrak{i}_{\tt n}$ but $f_{\tt
n}\not\in\mathfrak{i}_{{\tt n}+1}$. Also $f_{\tt n}\in M_{{\tt n}+1}$ and
$f_{\tt n}\not\in M_{\tt n}$.

Next we show that $\mathfrak{i}_{\tt n}\cap M_{\tt n}=\emptyset$. Indeed, if
$f\in\mathfrak{i}_{\tt n}\cap M_{\tt n}$, then

$\infty=\lim\limits_{{\tt
k}\rightarrow\infty}{\scaleobj{0.87}{\frac{m(f,2^{\tt k}\bm{e}_{1})}{{\tt
k}^{\tt n}}}}=\limsup\limits_{{\tt
k}\rightarrow\infty}{\scaleobj{0.87}{\frac{m(f,2^{\tt k}\bm{e}_{1})}{{\tt
k}^{\tt n}}}}\leq\sup\limits_{{\tt
k}\in\mathbb{N}}{\scaleobj{0.87}{\frac{m(f,2^{\tt k}\bm{e}_{1})}{{\tt k}^{\tt
n}}}}<\infty,$

a contradiction. But $\mathfrak{i}_{\tt n}\cap M_{{\tt n}+1}\neq\emptyset$,
since $f_{\tt n}\in\mathfrak{i}_{\tt n}$ and $f_{\tt n}\in M_{{\tt n}+1}$.

We will now show that the Krull dimension of
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ is infinite by showing that for all
${\tt N}\in\mathbb{N}$, we can construct a chain of strictly decreasing prime
ideals $\mathfrak{p}_{{\tt N}+1}\subsetneq\mathfrak{p}_{\tt
N}\subset\cdots\subsetneq\mathfrak{p}_{2}\subsetneq\mathfrak{p}_{1}$ in
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$.

Fix an ${\tt N}\in\mathbb{N}$. Applying Proposition 4.3 by taking
$\mathfrak{i}=\mathfrak{i}_{{\tt N}+1}$ and $M=M_{{\tt N}+1}$, we obtain the
existence of a prime ideal $\mathfrak{p}=\mathfrak{p}_{{\tt N}+1}$ in
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$, which satisfies
$\mathfrak{i}_{{\tt N}+1}\subset\mathfrak{p}_{{\tt N}+1}$ and
$\mathfrak{p}_{{\tt N}+1}\cap M_{{\tt N}+1}=\emptyset$.

We claim the ideal $\mathfrak{i}_{\tt N}+\mathfrak{p}_{{\tt N}+1}$ of
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ satisfies
${\scaleobj{0.999}{(\mathfrak{i}_{\tt N}+\mathfrak{p}_{{\tt N}+1})\cap M_{\tt
N}=\emptyset}}$. Let $h=f+g\in\mathfrak{i}_{\tt N}+\mathfrak{p}_{{\tt N}+1}$,
where $f\in\mathfrak{i}_{\tt N}$ and $g\in\mathfrak{p}_{{\tt N}+1}$. Since
$g\in\mathfrak{p}_{{\tt N}+1}$, by the construction of ${\mathfrak{p}}_{{\tt
N}+1}$ it follows that $g\not\in M_{{\tt N}+1}$. But $M_{\tt N}\subset M_{{\tt
N}+1}$, and so $g\not\in M_{\tt N}$ as well. Thus there exists a subsequence
$({\tt k}_{\tt j})_{\tt j\in\mathbb{N}}$ of $({\tt k})_{{\tt k}\in\mathbb{N}}$
such that

$\lim_{{\tt j}\rightarrow\infty}{\scaleobj{0.87}{\frac{m(g,2^{{\tt k}_{\tt
j}}\bm{e}_{1})}{{\tt k}_{\tt j}^{\tt N}}}}=\infty.$

From (P1), we obtain

${\scaleobj{0.87}{\frac{m(h,2^{{\tt k}_{\tt j}}\bm{e}_{1})}{{\tt k}_{\tt
j}^{\tt N}}}}\geq\min\Big{\\{}{\scaleobj{0.87}{\frac{m(f,2^{{\tt k}_{\tt
j}}\bm{e}_{1})}{{\tt k}_{\tt j}^{\tt N}}}},{\scaleobj{0.87}{\frac{m(g,2^{{\tt
k}_{\tt j}}\bm{e}_{1})}{{\tt k}_{\tt j}^{\tt N}}}}\Big{\\}}.$

As $f\in\mathfrak{i}_{\tt N}$, it follows that

$\sup\limits_{{\tt j}\in\mathbb{N}}{\scaleobj{0.87}{\frac{m(h,2^{{\tt k}_{\tt
j}}\bm{e}_{1})}{{\tt k}_{\tt j}^{\tt
N}}}}\geq\min\Big{\\{}\limsup\limits_{{\tt
j}\rightarrow\infty}{\scaleobj{0.87}{\frac{m(f,2^{{\tt k}_{\tt
j}}\bm{e}_{1})}{{\tt k}_{\tt j}^{\tt N}}}},\limsup\limits_{{\tt
j}\rightarrow\infty}{\scaleobj{0.87}{\frac{m(g,2^{{\tt k}_{\tt
j}}\bm{e}_{1})}{{\tt k}_{\tt j}^{\tt N}}}}\Big{\\}}\geq\infty.$

Thus $h\not\in M_{\tt N}$. Consequently, $(\mathfrak{i}_{\tt
N}+\mathfrak{p}_{{\tt N}+1})\cap M_{\tt N}=\emptyset$.

Clearly $\mathfrak{i}_{\tt N}\subset\mathfrak{i}_{\tt N}+\mathfrak{p}_{{\tt
N}+1}$. Applying Proposition 4.3 again, now taking
$\mathfrak{i}=\mathfrak{i}_{\tt N}+\mathfrak{p}_{{\tt N}+1}$ and $M=M_{\tt
N}$, we obtain the existence of a prime ideal $\mathfrak{p}=\mathfrak{p}_{\tt
N}$ in $\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ such that $\mathfrak{i}_{\tt
N}+\mathfrak{p}_{{\tt N}+1}\subset\mathfrak{p}_{\tt N}$ and $\mathfrak{p}_{\tt
N}\cap M_{\tt N}=\emptyset$. Thus $\mathfrak{p}_{{\tt
N}+1}\subset\mathfrak{i}_{\tt N}+\mathfrak{p}_{{\tt
N}+1}\subset\mathfrak{p}_{\tt N}$. The first inclusion is strict as $f_{\tt
N}\in\mathfrak{i}_{\tt N}\subset\mathfrak{i}_{\tt N}+\mathfrak{p}_{{\tt
N}+1}$. But $f_{\tt N}\not\in\mathfrak{p}_{{\tt N}+1}$ (since $f_{\tt N}\in
M_{{\tt N}+1}$ and $\mathfrak{p}_{{\tt N}+1}\cap M_{{\tt N}+1}=\emptyset$ by
the construction of $\mathfrak{p}_{{\tt N}+1}$). Thus $\mathfrak{p}_{{\tt
N}+1}\subsetneq\mathfrak{p}_{\tt N}$.

Now consider the ideal $\mathfrak{i}:=\mathfrak{i}_{{\tt
N}-1}+\mathfrak{p}_{\tt N}\supset\mathfrak{i}_{{\tt N}-1}$ of
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ and the multiplicatively closed set
$M:=M_{{\tt N}-1}$ of $\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$. Similar to
the argument given above, we show below that $\mathfrak{i}\cap
M=(\mathfrak{i}_{{\tt N}-1}+\mathfrak{p}_{\tt N})\cap M_{{\tt
N}-1}=\emptyset$.

Let $h=f+g\in\mathfrak{i}_{{\tt N}-1}+\mathfrak{p}_{{\tt N}}$, where
$f\in\mathfrak{i}_{{\tt N}-1}$ and $g\in\mathfrak{p}_{\tt N}$. Since
$g\in\mathfrak{p}_{\tt N}$, by the construction of $\mathfrak{p}_{\tt N}$,
$g\not\in M_{\tt N}\supset M_{{\tt N}-1}$, and so $g\not\in M_{{\tt N}-1}$.
Thus there exists a subsequence $({\tt k}_{\tt j})_{{\tt j}\in\mathbb{N}}$ of
$({\tt k})_{{\tt k}\in\mathbb{N}}$ such that

$\lim\limits_{{\tt j}\rightarrow\infty}{\scaleobj{0.87}{\frac{m(g,2^{{\tt
k}_{\tt j}}\bm{e}_{1})}{{\tt k}_{\tt j}^{{\tt N}-1}}}}=\infty.$

As $f\in\mathfrak{i}_{{\tt N}-1}$,

$\sup\limits_{{\tt j}\in\mathbb{N}}{\scaleobj{0.87}{\frac{m(h,2^{{\tt k}_{\tt
j}}\bm{e}_{1})}{{\tt k}_{\tt j}^{{\tt
N}-1}}}}\geq\min\Big{\\{}\limsup\limits_{{\tt
j}\rightarrow\infty}{\scaleobj{0.87}{\frac{m(f,2^{{\tt k}_{\tt
j}}\bm{e}_{1})}{{\tt k}_{\tt j}^{{\tt N}-1}}}},\limsup\limits_{{\tt
j}\rightarrow\infty}{\scaleobj{0.87}{\frac{m(g,2^{{\tt k}_{\tt
j}}\bm{e}_{1})}{{\tt k}_{\tt j}^{{\tt N}-1}}}}\Big{\\}}\geq\infty.$

Thus $h\not\in M_{{\tt N}-1}$. So $(\mathfrak{i}_{{\tt N}-1}+\mathfrak{p}_{\tt
N})\cap M_{{\tt N}-1}=\emptyset$.

By Proposition 4.3, taking $\mathfrak{i}=\mathfrak{i}_{{\tt
N}-1}+\mathfrak{p}_{\tt N}\supset\mathfrak{i}_{{\tt N}-1}$ and $M=M_{{\tt
N}-1}$, there exists a prime ideal $\mathfrak{p}=\mathfrak{p}_{{\tt N}-1}$ in
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ such that $\mathfrak{i}_{{\tt
N}-1}+\mathfrak{p}_{\tt N}\subset\mathfrak{p}_{{\tt N}-1}$ and
$\mathfrak{p}_{{\tt N}-1}\cap M_{N-1}=\emptyset$. Thus $\mathfrak{p}_{\tt
N}\subset\mathfrak{i}_{{\tt N}-1}+\mathfrak{p}_{\tt
N}\subset\mathfrak{p}_{{\tt N}-1}$, and again the first inclusion is strict
(because $f_{{\tt N}-1}\in\mathfrak{i}_{{\tt N}-1}\subset\mathfrak{i}_{{\tt
N}-1}+\mathfrak{p}_{\tt N}$, $f_{{\tt N}-1}\in M_{\tt N}$ and $M_{\tt
N}\cap\mathfrak{p}_{\tt N}=\emptyset$).

Proceeding in this manner, we obtain the chain of distinct prime ideals
$\mathfrak{p}_{{\tt N}+1}\subsetneq\mathfrak{p}_{\tt
N}\subsetneq\mathfrak{p}_{{\tt
N}-1}\subsetneq\cdots\subsetneq\mathfrak{p}_{1}.$ in
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$. As ${\tt N}\in\mathbb{N}$ was
arbitrary, it follows that the Krull dimension of
$\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ is infinite. ∎

###  4.3. Weak Krull dimension

Recall the following definition from [6]:

######  Definition 4.5.

The weak Krull dimension of a commutative ring $R$ is the supremum of the
lengths of chains of distinct proper finitely generated prime ideals of $R$.

######  Theorem 4.6.

The weak Krull dimension of ${\mathcal{S}}^{\prime}(\mathbb{Z}^{\tt d})$ is
$1$.

###### Proof.

Let $\mathfrak{p}_{1}$ and $\mathfrak{p}_{2}$ be finitely generated proper
prime ideals in ${\mathcal{S}}^{\prime}(\mathbb{Z}^{\tt d})$ such that
$\mathfrak{p}_{1}\subset\mathfrak{p}_{2}$. For each ${\tt i}\in\\{1,2\\}$, by
Proposition 2.2, there exists an $\bm{n}_{\tt i}\in\mathbb{Z}^{\tt d}$ such
that $\mathfrak{p}_{\tt i}=\\{f\in\mathcal{S}^{\prime}(\mathbb{Z}^{\tt
d}):f(\bm{n}_{\tt i})=0\\}$. But as $\mathfrak{p}_{1}\subset\mathfrak{p}_{2}$,
it follows that $\bm{n}_{1}=\bm{n}_{2}$ (by considering the function which is
zero at all $\bm{n}\in\mathbb{Z}^{\tt d}\setminus\\{\bm{n}_{2}\\}$ and equal
to $1$ at $\bm{n}_{2}$), and so $\mathfrak{p}_{1}=\mathfrak{p}_{2}$. So the
weak Krull dimension of $\mathcal{S}^{\prime}(\mathbb{Z}^{\tt d})$ is $1$. ∎

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