# A Federated Learning Approach to Privacy Preserving
Offensive Language Identification

###### Abstract

The spread of various forms of offensive speech online is an important concern
in social media. While platforms have been investing heavily in ways of coping
with this problem, the question of privacy remains largely unaddressed. Models
trained to detect offensive language on social media are trained and/or fine-
tuned using large amounts of data often stored in centralized servers. Since
most social media data originates from end users, we propose a privacy
preserving decentralized architecture for identifying offensive language
online by introducing Federated Learning (FL) in the context of offensive
language identification. FL is a decentralized architecture that allows
multiple models to be trained locally without the need for data sharing hence
preserving users’ privacy. We propose a model fusion approach to perform FL.
We trained multiple deep learning models on four publicly available English
benchmark datasets (AHSD, HASOC, HateXplain, OLID) and evaluated their
performance in detail. We also present initial cross-lingual experiments in
English and Spanish. We show that the proposed model fusion approach
outperforms baselines in all the datasets while preserving privacy.

Keywords: federated learning, offensive language identification, privacy

A Federated Learning Approach to Privacy Preserving
Offensive Language Identification

Marcos Zampieri1, Damith Premasiri2, Tharindu Ranasinghe3,
---
1George Mason University, USA, 2Lancaster University, UK, 3Aston University,
UK
<EMAIL_ADDRESS>

Abstract content

## 1\. Introduction

NLP systems relying on modern deep learning paradigms are trained on very
large amounts of data. In several applications and domains (e.g., social
media), most data used to train machine learning models comes from end users.
Such confidential data often cannot be shared without compromising users’
privacy. This is an important concern for organizations that handle large
amounts of confidential data, such as financial institutions, healthcare
facilities, law firms, and many others. With the widespread use of personal
computing devices (e.g., PCs, smartphones, and virtual assistants), data
privacy also became a great concern to individuals, which motivated several
countries to pass legislation aiming to protect users’ privacy such as the
European Union General Data Protection Regulation (GDPR)111https://gdpr.eu/
and the Swiss Datenschutzgesetz
(DSG).222https://www.edoeb.admin.ch/edoeb/de/home/datenschutz/ueberblick/datenschutz.html

The need for privacy-preserving machine learning models that can handle
confidential data while protecting organizations’ and users’ privacy emerges
from this situation. To address this important challenge, Federated Learning
(FL) has become an increasingly popular machine learning paradigm McMahan et
al. (2017) as it allows us to train robust machine learning models across
multiple devices or servers without exchanging data. In FL, multiple clients
work together under the coordination of a central server. Each client’s data
is stored locally and not exchanged among clients or with the central server.
FL, therefore, offers the possibility of training robust machine learning
models on large numbers of decentralized local data repositories without
compromising privacy. FL models have been successfully applied in a wide range
of applications in computer networks Lim et al. (2020), computer vision Yan et
al. (2021), information retrieval Wang et al. (2021), NLP Chen et al. (2019),
and many others.

| Training | Testing |
---|---|---|---
Dataset | Inst. | OFF % | Inst. | OFF % | Data Sources
AHSD Davidson et al. (2017) | 19,822 | 0.83 | 4,956 | 0.82 | Twitter
HASOC Mandl et al. (2020) | 5,604 | 0.36 | 1,401 | 0.35 | Twitter, Facebook
HateXplain Mathew et al. (2021) | 11,535 | 0.59 | 3,844 | 0.58 | Twitter, Gab
OLID Zampieri et al. (2019a) | 13,240 | 0.33 | 860 | 0.27 | Twitter

Table 1: The four datasets, including the number of instances (Inst.) in the
training and testing sets, the OFF % in each set and the data source.

In this paper, we explore the use of FL in offensive language identification
online through a model fusion technique Choshen et al. (2022). Datasets
containing the various forms of offensive speech (e.g., hate speech,
cyberbullying, etc.) are sensitive in nature, which creates an interesting use
case for FL. The use of FL and other privacy-preserving paradigms allows
social media platforms to work together to solve this important issue without
the need to exchange confidential information, thus preserving users’ privacy.
While FL has recently started to be explored in NLP Chen et al. (2019); Lin et
al. (2022b), including the workshop on Federated Learning for NLP (FL4NLP) at
ACL-2022 Lin et al. (2022a), to the best of our knowledge, no studies have yet
explored the use of FL in the context of offensive language identification.
Our work fills this gap by introducing FL in the context of offensive language
identification online and by providing the community with an evaluation of FL
methods using four publicly available English offensive language benchmark
datasets presented in Section 3.

One recent study Gala et al. (2023) proposed FL in offensive language
identification, but it lacks the consideration of combining different data.
Their architecture solely focuses on distributed training on the same dataset
with multiple clients and evaluating fedopt Reddi et al. (2021), fedprox Sahu
et al. (2019) algorithms to optimise the global model. Our main focus in this
study is on combining multiple models using FL, which could identify offensive
content in different data.

## 2\. Related Work

Offensive Language Identification The task of automatically identifying
offensive language online has been substantially explored in the literature
MacAvaney et al. (2019); Melton et al. (2020); Zia et al. (2022); Weerasooriya
et al. (2023). Multiple types of offensive content have been addressed, such
as aggression, cyberbulling, and hate speech using classical machine learning
classifiers (e.g., Support Vector Machines) Malmasi and Zampieri (2017, 2018),
neural networks Gambäck and Sikdar (2017); Djuric et al. (2015); Hettiarachchi
and Ranasinghe (2019), pre-trained general-purpose transformer-based language
models Ranasinghe and Zampieri (2020, 2021), and fine-tuned language models on
offensive language datasets Caselli et al. (2020); Sarkar et al. (2021). The
vast majority of studies addressed offensive content in English and other
widely-spoken resource-rich languages such as Arabic Mubarak et al. (2021),
Portuguese Fortuna et al. (2019) and Turkish Çöltekin (2020) while a few
studies dealt with low-resource languages Fišer et al. (2017); Gaikwad et al.
(2021); Raihan et al. (2023). Multiple competitions on this topic have been
organized creating important benchmark datasets such as OffensEval Zampieri et
al. (2019b, 2020), HASOC Mandl et al. (2020); Modha et al. (2021); Satapara et
al. (2022), TRAC Kumar et al. (2018, 2020), and HatEval Basile et al. (2019).
While substantial progress has been made in the past few years, to the best of
our knowledge, none of the aforementioned studies or competitions has
addressed the question of data privacy.

Federated Learning in NLP With the goal of preserving users’ data privacy, FL
architectures have been extensively studied in a variety of domains Wang et
al. (2021) in the past several years. Only more recently, however, FL has been
explored for text and speech processing Lin et al. (2022b); Silva et al.
(2023); Zhang et al. (2023); Che et al. (2023). Recent workshops co-located
with top-tier conferences confirm this growing interest in FL and privacy in
general. The workshop on Privacy in Natural Language Processing (PrivateNLP)
Feyisetan et al. (2022), which is currently in its fourth edition, addressed
the interplay between NLP and data privacy while the aforementioned FL4NLP
workshop Lin et al. (2022a) co-located with ACL-2022 was the first workshop
organized focusing exclusively on FL for NLP. Most papers presented in the
workshop, however, dealt with language modelling and learning representation
rather than with downstream tasks and applications such as offensive language
identification. As we mentioned before, a recent study applied different FL
strategies in offensive language identification Gala et al. (2023). However,
their study focuses on distributed training on the same dataset Sahu et al.
(2019).

## 3\. Data

We use four popular publicly available datasets containing English data
summarized in Table 1. As the datasets were annotated using different
guidelines and labels, following the methodology described in previous work
Ranasinghe and Zampieri (2020), we map all labels to OLID level A Zampieri et
al. (2019a), which contains the labels offensive (OFF) vs. not offensive
(NOT). We choose OLID due to the flexibility provided by its general three-
level hierarchical taxonomy below, where the OFF class contains all types of
offensive content, from general profanity to hate speech, while the NOT class
contains non-offensive examples. The OLID taxonomy is presented next:

Figure 1: The three stages of the FL pipeline in the proposed fused model.

  * •

Level A: Offensive (OFF) vs. Non-offensive (NOT).

  * •

Level B: Classification of the type of offensive (OFF) tweet - Targeted (TIN)
vs. Untargeted (UNT).

  * •

Level C: Classification of the target of a targeted (TIN) tweet - Individual
(IND) vs. Group (GRP) vs. Other (OTH).

In the OLID taxonomy, offensive (OFF) posts targeted (TIN) at an individual
are often cyberbulling whereas offensive (OFF) posts targeted (TIN) at a group
is often hate speech.

AHSD Davidson et al. (2017) is one of the most popular hate speech datasets
available. The dataset contains data retrieved from Twitter and it was
annotated using crowdsourcing. The annotation taxonomy contains three classes;
Offensive, Hate, and Neither. We conflate Offensive and Hate under a class OFF
while neither class corresponds to OLID’s NOT class.

OLID Zampieri et al. (2019a) is the official dataset of the SemEval-2019 Task
6 (OffensEval) Zampieri et al. (2019b). It contains data from Twitter
annotated with a three-level hierarchical annotation in which level A
classifies posts into offensive and not offensive; level B differentiates
between targeted pots (insults and threats) and untargeted posts (general
profanity); and level C classifies them into three targets: individual, group,
or other. We adopt the labels in OLID level A as our classification labels.

HASOC Mandl et al. (2020) is the dataset used in the HASOC shared task 2020.
It contains posts retrieved from Twitter and Facebook. The upper level of the
annotation taxonomy used in HASOC is the same as OLID’s level A, which allows
us to directly use the same labels in our models.

HateXplain Mathew et al. (2021) is a recent dataset collected for the
explainability of hate speech. It contains both token- and post-level
annotation of Twitter and Gab posts. The annotation taxonomy contains three
classes; hate speech, offensive speech, and normal. Following the annotation
guidelines of OLID Zampieri et al. (2019a), we mapped the hate speech and
offensive speech classes to offensive (OFF) and normal class to not offensive
(NOT).

## 4\. Methodology

The proposed FL pipeline contains three steps depicted in Figure 1. We
describe these steps below.

Initial Model Training Transformer models have achieved state-of-the-art
performance in many NLP tasks Devlin et al. (2019), including offensive
language identification Ranasinghe et al. (2019); Sarkar et al. (2021).
Therefore, our methodology in this paper builds around pre-trained
transformers. For the text classification tasks such as offensive language
identification, we use the pre-trained transformer models by utilizing the
hidden representation of the classification token (CLS) as shown in Figure 2.
For this task, we implemented a softmax layer on top of the CLS token, i.e.,
the predicted probabilities are
$\bm{y}^{(B)}=\operatorname*{softmax}(W\bm{h}+b)$, where
$W\in\mathbb{R}^{k\times d}$ is the softmax weight matrix, and $k$ is the
number of labels. which in our case is always equal to two.

Figure 2: A sample transformer model for offensive language identification
Ranasinghe and Zampieri (2020) predicting offensive and not offensive labels.

We used this text classification architecture to build separate models for
each dataset that we introduced in the previous section. We trained the model
using the training sets of each dataset. We employed a batch-size of 16, Adam
optimiser with learning rate $4\mathrm{e}{-5}$, and a linear learning rate
warm-up over 10% of the training data. During the training process, the
parameters of the transformer model and the parameters of the subsequent
layers were updated. The models were evaluated while training using an
evaluation set that had one-fifth of the rows in training data. We performed
early stopping if the evaluation loss did not improve over three evaluation
steps. All the models were trained for three epochs.

Dataset | Approach |  | Models |  | Macro F1
---|---|---|---|---|---
AHSD | non-fused | AHSD |  | - | - | 0.931 ±0.01
fusion with FT | AHSD | OLID | - | - | 0.921 ±0.00
fusion without FT | AHSD | OLID | - | - | 0.866 ±0.00
ensemble | AHSD | OLID | - | - | 0.845 ±0.01
OLID | non-fused | - | OLID | - | - | 0.854 ±0.00
fusion with FT | AHSD | OLID | - | - | 0.837 ±0.03
fusion without FT | AHSD | OLID | - | - | 0.836 ±0.00
ensemble |  | OLID | - | HateX | 0.785 ±0.04
HASOC | non-fused | - | - | HASOC | - | 0.798 ±0.01
fusion without FT | AHSD | OLID | HASOC | - | 0.770 ±0.01
fusion with FT | AHSD | OLID | HASOC | - | 0.754 ±0.07
ensemble | AHSD |  | HASOC | - | 0.647±0.02
HateX | non-fused | - |  | - | HateX | 0.795 ±0.01
fusion with FT | AHSD | - | - | HateX | 0.777 ±0.00
fusion without FT | AHSD | OLID | - | HateX | 0.772 ±0.01
ensemble | - | - | HASOC | HateX | 0.654 ±0.01

Table 2: The best result for each dataset for each approach; non-fused models,
fused models with fine-tuning (FT), fused models without finetuning and
ensemble. We only report the results with fBERT. The results are ordered from
Macro F1.

We repeated this process with two popular pre-trained transformer models;
bert-large-cased Devlin et al. (2019) and fBERT Sarkar et al. (2021). The
bert-large-cased is a general purpose pre-trained transformer model while
fBERT is a domain-specific pre-trained transformer model for offensive
language identification that has been trained on over $1.4$ million offensive
tweets in SOLID dataset Rosenthal et al. (2021) and has shown state-of-the-art
results in several offensive language identification benchmarks Sarkar et al.
(2021).

Model Fusion In order to combine the different models created using different
datasets, we followed a recent approach named model fusion Choshen et al.
(2022). Model Fusion is the process of taking several fine-tuned models and
creating a new base model. Formally, given an initialization base model $P$
and $n$ models fine-tuned on it, let $W_{1},W_{2}\ldots
W_{n}\in\mathbb{R}^{d}$ be the weights fine-tuned by the models over $P$.
Fusing is a function

$\displaystyle W_{fuse}$ $\displaystyle=f(W_{1},W_{2},\ldots,W_{n})$
$\displaystyle\mathbb{R}^{d}\times\mathbb{R}^{d}\times\ldots\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$
(1)

In this work, we propose the simplest form of fusion. For each weight shared
by all models, assign the average weight to the model.

$\displaystyle W_{fuse}$
$\displaystyle=f\left(W_{1},W_{2},\ldots,W_{n}\right)$
$\displaystyle=\frac{W_{1}+W_{2}+\ldots+W_{n}}{n}$ (2)

In order to empirically evaluate model fusion in offensive language
identification, we consider all possible seven combinations. These include
different combinations of two models, such as $AHSD+OLID$ and $HASOC+HateX$,
different combinations of three models, such as $AHSD+OLID+HASOC$ and
$AHSD+OLID+HASOC$ and finally, the combination of all four models.

Further Finetunning The weights of the fused model resulting from step 2 can
be anomalous as we followed a naive averaging method. Therefore, we performed
a further finetuning step on the fused model. In this step, we fine-tuned the
fused model using only one available dataset in a particular environment. We
followed the same classification objective described in step 1 and used the
same configurations. However, to avoid the model being biased toward the
finetuning dataset, we only used $20\%$ of the available training data in the
finetuning step.

The whole pipeline described above simulates a real-life scenario where the
data can not be shared. The machine learning models are trained in separate
environments using their own data, as in the first step. In the second step,
with model fusion, we combined the models. In the final step. We further fine-
tuned the fused model on a particular dataset where we repeated the process
for all four datasets. Therefore with this pipeline, the datasets are not
shared, and privacy is preserved among the different environments.

### 4.1. Baseline Models

We compared our fusion-based approach to two baseline models.

Non-fused Baseline We train a transformer-based baseline using the training
set of one of the datasets and evaluate it on the test set of that particular
dataset as well as on the test sets of other datasets. We repeated the process
for all four datasets with two transformer models; bert-large-cased Devlin et
al. (2019) and fBERT Sarkar et al. (2021). This baseline reflects the most
common approach in offensive language detection, where a model is trained on a
dataset available for a particular environment, but evaluated on other
datasets in different environments as well.

Ensemble Baseline We also used an ensemble baseline; where we trained four
separate transformer models on each dataset. For each test instance, we
predicted values from all four models, and the final label is the label
predicted with the highest probability from all four models. Similar to our
previous experiments we repeated the process for bert-large-cased Devlin et
al. (2019) and fBERT Sarkar et al. (2021).

## 5\. Results and Discussion

In Table 2, we present the best results from each approach for each dataset.
We show the results for fBERT as it provided better overall results. For the
AHSD test set, the best result, $0.921$ Macro F1 score, is obtained when fBERT
models are trained on AHSD and OLID and fused, then further fine-tuned on
AHSD. For OLID the best result, $0.839$ Macro F1 score was provided when BERT-
large-cased models trained on AHSD and OLID were fused and further fine-tuned
on AHSD. Similarly, for HateX the best result, $0.777$ was provided when the
fBERT models trained on AHSD and HateX were fused and further fine-tuned on
HateX. However, HASOC follows a different pattern, and the best result was
produced when fBERT models trained on AHSD, OLID and HASOC were fused, and
further fine-tuned on AHSD. Overall, fBERT models provided slightly better
results than BERT-large-cased models in most experiments. This is mainly
because the fBERT model was trained on domain-specific data on offensive
language identification. Finally, we present all results of the fused models
and the non-fused model baseline in Table 3 in terms of Macro F1 score.

### 5.1. Discussion

Fine-tuned Dataset |  | Fused Models |  | BERT-large-cased | fBERT
---|---|---|---|---|---
|  |  |  | AHSD | OLID | HASOC | HATEX | AHSD | OLID | HASOC | HATEX
AHSD | AHSD | OLID | - | - | 0.900±0.00 | 0.830±0.07 | 0.610±0.00 | 0.554±0.06 | 0.921±0.00 | 0.836±0.09 | 0.627±0.00 | 0.628±0.00
AHSD | - | HASOC | - | 0.778±0.14 | 0.627±0.00 | 0.637±0.00 | 0.607±0.02 | 0.776±0.04 | 0.722±0.00 | 0.632±0.00 | 0.677±0.05
AHSD | - | - | HATEX | 0.727±0.03 | 0.697±0.00 | 0.660±0.04 | 0.594±0.00 | 0.781±0.03 | 0.707±0.00 | 0.673±0.03 | 0.648±0.00
AHSD | OLID | HASOC | - | 0.919±0.00 | 0.837±0.08 | 0.766±0.02 | 0.636±0.00 | 0.915±0.00 | 0.835±0.08 | 0.770±0.01 | 0.623±0.00
AHSD | - | HASOC | HATEX | 0.705±0.06 | 0.674±0.00 | 0.595±0.03 | 0.565±0.00 | 0.734±0.03 | 0.704±0.00 | 0.643±0.00 | 0.643±0.00
AHSD | OLID | - | HATEX | 0.905±0.00 | 0.813±0.09 | 0.628±0.00 | 0.719±0.05 | 0.914±0.00 | 0.834±0.08 | 0.627±0.00 | 0.772±0.01
AHSD | OLID | HASOC | HATEX | 0.716±0.03 | 0.708±0.00 | 0.646±0.05 | 0.652±0.06 | 0.730±0.01 | 0.724±0.00 | 0.668±0.04 | 0.684±0.04
Non-fused Baseline |  | 0.926±0.01 | 0.699±0.03 | 0.630±0.05 | 0.586±0.06 | 0.931±0.01 | 0.743±0.03 | 0.682±0.04 | 0.606±0.06
OLID | AHSD | OLID | - | - | 0.893±0.00 | 0.839±0.05 | 0.647±0.00 | 0.621±0.03 | 0.866±0.00 | 0.837±0.03 | 0.601±0.00 | 0.598±0.00
- | OLID | HASOC | - | 0.715±0.00 | 0.405±0.01 | 0.392±0.00 | 0.651±0.06 | 0.718±0.00 | 0.725±0.07 | 0.655±0.00 | 0.667±0.05
- | OLID | - | HATEX | 0.696±0.00 | 0.692±0.08 | 0.656±0.04 | 0.616±0.00 | 0.679±0.07 | 0.723±0.07 | 0.611±0.00 | 0.650±0.00
AHSD | OLID | HASOC | - | 0.868±0.00 | 0.826±0.04 | 0.756±0.00 | 0.608±0.00 | 0.840±0.00 | 0.819±0.02 | 0.759±0.09 | 0.606±0.00
- | OLID | HASOC | HATEX | 0.687±0.00 | 0.649±0.09 | 0.586±0.01 | 0.596±0.00 | 0.729±0.00 | 0.694±0.08 | 0.637±0.01 | 0.630±0.00
AHSD | OLID | - | HATEX | 0.847±0.00 | 0.812±0.04 | 0.642±0.00 | 0.751±0.09 | 0.861±0.00 | 0.831±0.03 | 0.615±0.00 | 0.752±0.01
AHSD | OLID | HASOC | HATEX | 0.713±0.00 | 0.777±0.00 | 0.672±0.07 | 0.699±0.08 | 0.708±0.08 | 0.793±0.00 | 0.682±0.08 | 0.707±0.09
Non-fused Baseline |  | 0.685±0.02 | 0.845±0.00 | 0.636±0.05 | 0.620±0.06 | 0.702±0.01 | 0.851±0.00 | 0.653±0.05 | 0.645±0.08
HASOC | AHSD | - | HASOC | - | 0.777±0.13 | 0.419±0.00 | 0.652±0.00 | 0.356±0.06 | 0.792±0.11 | 0.785±0.05 | 0.680±0.00 | 0.708±0.08
- | OLID | HASOC | - | 0.147±0.00 | 0.707±0.05 | 0.656±0.00 | 0.220±0.07 | 0.717±0.00 | 0.734±0.05 | 0.683±0.00 | 0.673±0.04
- | - | HASOC | HATEX | 0.530±0.05 | 0.480±0.00 | 0.695±0.04 | 0.738±0.00 | 0.761±0.03 | 0.791±0.00 | 0.689±0.00 | 0.690±0.00
AHSD | OLID | HASOC | - | 0.864±0.00 | 0.812±0.05 | 0.763±0.08 | 0.624±0.00 | 0.805±0.00 | 0.801±0.00 | 0.754±0.07 | 0.635±0.00
AHSD | - | HASOC | HATEX | 0.754±0.01 | 0.419±0.00 | 0.686±0.01 | 0.698±0.00 | 0.734±0.09 | 0.780±0.00 | 0.668±0.01 | 0.661±0.00
- | OLID | HASOC | HATEX | 0.732±0.00 | 0.700±0.04 | 0.675±0.01 | 0.686±0.00 | 0.736±0.00 | 0.712±0.06 | 0.671±0.00 | 0.676±0.00
AHSD | OLID | HASOC | HATEX | 0.703±0.09 | 0.647±0.00 | 0.651±0.00 | 0.651±0.00 | 0.719±0.06 | 0.781±0.00 | 0.702±0.06 | 0.718±0.06
Non-fused Baseline |  | 0.620±0.03 | 0.492±0.01 | 0.788±0.01 | 0.555±0.06 | 0.645±0.02 | 0.532±0.01 | 0.798±0.01 | 0.575±0.05
HATEX | AHSD | - | - | HATEX | 0.758±0.01 | 0.449±0.00 | 0.531±0.08 | 0.744±0.00 | 0.671±0.01 | 0.591±0.00 | 0.587±0.00 | 0.777±0.00
- | OLID | - | HATEX | 0.650±0.00 | 0.689±0.06 | 0.557±0.09 | 0.749±0.00 | 0.584±0.02 | 0.668±0.01 | 0.599±0.00 | 0.775±0.00
- | - | HASOC | HATEX | 0.538±0.01 | 0.545±0.0 | 0.710±0.05 | 0.756±0.00 | 0.527±0.05 | 0.573±0.00 | 0.707±0.07 | 0.772±0.00
AHSD | - | HASOC | HATEX | 0.692±0.04 | 0.529±0.00 | 0.693±0.05 | 0.741±0.00 | 0.636±0.10 | 0.588±0.00 | 0.688±0.08 | 0.767±0.00
- | OLID | HASOC | HATEX | 0.561±0.00 | 0.640±0.09 | 0.690±0.06 | 0.755±0.00 | 0.526±0.00 | 0.664±0.08 | 0.689±0.08 | 0.772±0.00
AHSD | OLID | - | HATEX | 0.522±0.00 | 0.597±0.08 | 0.607±0.00 | 0.645±0.09 | 0.532±0.00 | 0.563±0.03 | 0.613±0.00 | 0.633±0.10
AHSD | OLID | HASOC | HATEX | 0.627±0.08 | 0.532±0.00 | 0.635±0.09 | 0.642±0.11 | 0.631±0.09 | 0.565±0.00 | 0.652±0.09 | 0.671±0.11
Non-fused Baseline |  | 0.569±0.03 | 0.504±0.01 | 0.604±0.02 | 0.782±0.02 | 0.581±0.01 | 0.523±0.01 | 0.612±0.01 | 0.795±0.01

Table 3: Macro F1 score results for the fuse models (BERT-large-cased and
fBERT) compared to the baseline systems fine-tuned on the four datasets.
Results are reported on 10 runs along with standard deviation. The best
results from the fused approach for each model are in bold. Results for the
non-fused baseline model evaluated on the same dataset are underlined.

We discuss the following four main findings from our results;

(1) The fused model performs better when evaluated on the same dataset used in
further finetunning. All the datasets except for HASOC, the best result was
produced when the fused model was further fine-tuned on that particular
dataset. For HASOC too, when the fBERT model trained on AHSD, OLID and HASOC
were fused and further fine-tuned on HASOC provided 0.754 Macro F1 score,
which is very close to the best result (0.770). With the results, we can
conclude that the fused model performs better when evaluated on the same
dataset used in further finetunning. This observation reflects an ideal
scenario in real-world applications where we want an ML model to perform
excellently in data specific to our environment/ platform. This objective can
be achieved successfully with model fusion and finetunning as we see in the
results.

(2) The fused model generalizes well across datasets even when it is not used
in finetunning. One drawback of fused models is that the result slightly
decreases compared to the non-fused models trained only using a particular
dataset. In the results, this is clear as there is a decrease in the Macro F1
score between underlined values and bolded values. Furthermore as you can see
in Table 2, the best result in all the datasets were produced with the non-
fused baseline. However, after further investigating this, it is clear that
non-fused models do not often generalise well across other datasets. For
example in Table 3, the non-fused model trained on AHSD only provides 0.699
Macro F1 score for OLID. However, AHSD and OLID fused model further fine-tuned
on AHSD provides 0.830 Macro F1 score. This is similar to the majority of the
experiments, and fused models provide better results than non-fused models in
other datasets. This observation again reflects an ideal scenario in real-
world applications where we want an ML model to perform well across data not
specific to our environment/ platform. As we see in the results, this
objective can be achieved successfully with model fusion.

(3) The Fused model outperforms the ensemble baseline in all the datasets. As
shown in Table 2, model fusion approaches with and without finetunning on a
particular dataset outperform the best ensemble model. For HASOC, there is a
large gap between the ensemble model and fused models as the ensemble model
produces only 0.670 Macro F1 score while the fused model provides 0.770 Macro
F1 score. The other datasets also follow a similar pattern. This is a key
observation because we have presented a fusion based approach for FL that can
surpass an ensemble based model preserving privacy across different datasets.
The platforms/ environments that are interested in developing a FL approach
should focus on model fusion based strategies that outperform ensemble based
models as we showed in the results.

(4) The Fused model performance heavily depends on the datasets it was trained
on. Our final observation is that the fused model performance depends on the
datasets that it was trained on. For example, when the model fusion was
performed between AHSD and OLID, the final model provided excellent results on
both datasets. This is due to the general nature of these two datasets
covering multiple types of offensive content rather than focusing on a
particular type of offensive content. On the other hand, results are not the
same when the model fusion was performed between AHSD and HASOC where the
final model did not provide good results for both datasets. This can be
explained by the demography of the dataset as HASOC data is collected on
Twitter users based in India. It is clear that model fusion would thrive in
similar kinds of datasets, but would not perform well with different kinds of
data.

Overall, model fusion produces excellent results on the dataset that it was
fine-tuned on, and it generalizes well across other datasets. Fused models
outperform both of our baselines in all the datasets. Therefore, model fusion
provides a successful approach to FL.

### 5.2. Multilingual Experiments

We conducted initial multilingual experiments with the same FL setting. We
used OffendES Plaza-del Arco et al. (2021), a Spanish offensive language
identification dataset. For English we used the OLID dataset described before.
Each instance in OffendES is labelled as belonging to one of the five classes;
Offensive and targeted to a person (OFP), Offensive and targeted to a group
(OFG), Offensive and not targeted to a person or a group (OFO), Non-offensive,
but with expletive language (NOE), and Non-offensive (NO). We map the
instances belonging to the OFP, OFG, OFO, and NOE to OLID OFF, and the NO
class as NOT. Even though, the label NOE is considered non-offensive in
OffendES, it contains profanity so we map it to OLID label OFF to conform with
the OLID guidelines.

Instead of the monolingual BERT models we used in the previous experiments, we
use cross-lingual models, specifically XLM-R Conneau et al. (2019). We used
the same FL settings and compared it with ensemble baseline. The results are
shown in Table 4.

Dataset | Approach | Macro F1
---|---|---
English | non-fused | 0.845 ±0.01
fusion with FT | 0.829 ±0.03
fusion without FT | 0.831 ±0.00
ensemble | 0.776 ±0.02
Spanish | non-fused | 0.812 ±0.04
fusion with FT | 0.809 ±0.02
fusion without FT | 0.792 ±0.01
ensemble | 0.761 ±0.02

Table 4: The results for multilingual experiments on English and Spanish; non-
fused models, fused models with fine-tuning (FT), fused models without
finetuning and ensemble. We report the results with xlm-roberta. The results
are ordered from Macro F1.

The results show that fusion based FL outperforms ensemble baseline in
multilingual settings too. This opens new avenues for privacy preserving
models for languages other than English and more specifically, low-resource
languages.

## 6\. Conclusion and Future Work

This paper introduced FL in the context of combining different offensive
language identification models. While a recent study Gala et al. (2023) uses
FL learning in offensive language identification, their work is limited to
distributed training on the same dataset with multiple clients. As far as we
know, our research is the first study to use FL in combining multiple
offensive language identification models. We evaluated a fusion-based FL
architecture using a general BERT model and a fine-tuned fBERT model on four
publicly available English benchmark datasets. We also presented initial
cross-lingual experiments in English and Spanish. Our results show that the
fusion model performances outperform the performance of an ensemble baseline
model. We also show that the fused model generalizes well across all datasets
tested. As the FL architecture does not require data sharing, we believe that
FL is a promising research direction in offensive language identification due
to its privacy preserving nature.

In future work, we would like to explore other FL architectures and compare
their performance to the fused model proposed in this paper. Finally. we would
like to evaluate the performance of recently proposed large language models
(LLMs) (e.g., GPT-4, LLama 2) for this task in FL settings.

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# Periodic multivariate formal power series

Xue Zhang<EMAIL_ADDRESS>
###### Abstract

A system of multivariate formal power series $\varphi$ with a homogeneous
decomposition $\varphi=\sum_{k=0}^{\infty}\varphi_{k}$ is invertible under
composition if $\varphi_{0}=0$ and $\mathrm{det}(\varphi_{1})\neq 0.$ All
invertible series over a field $K$ form a formal transformation group
$G_{\infty}(n,K).$ We prove that every periodic series $\varphi\in
G_{\infty}(n,K)$ with $\varphi_{1}$ diagonalizable is conjugate to
$\varphi_{1}.$ This classifies all periodic series in
$G_{\infty}(n,\mathbb{C}).$ A constraint for a periodic series is obtained
when its first term is a multiple of identity.

Mathematical subject classification: 13F25, 16W60, 47J07, 65Q30

Keywords formal power series, period, conjugacy, recursive relation

##  1 Introduction

The study of formal power series involves many areas of mathematics, for
example analysis [4, 20], combinatorics [11, 15], dynamical systems [16],
differential equations [24, 19] and the famous Jacobian conjecture [1, 6, 23].
There are two kinds of multiplication of formal power series, usual
multiplication and composition. This note deals with periodic series in the
group of formal power series in multiple variables with the composition law.

Given a commutative ring $R$ and a system of formal power series
$\varphi=(f_{1},\cdots,f_{n})$ in $R[[x_{1},\cdots,x_{n}]]^{n}$ of the form
$\varphi=\sum_{k=0}^{\infty}\varphi_{k}$ with $\varphi_{k}$ consisting of
terms of degree $k.$ If $\varphi_{0}=0$ and
$\varphi_{1}=(x_{1},\cdots,x_{n}),$ then there is a unique series
$\phi=(g_{1},\cdots,g_{n})\in R[[x_{1},\cdots,x_{n}]]^{n}$ such that
$\varphi\circ\phi=\phi\circ\varphi=id,$ i.e.

$(f_{1}(\phi),\cdots,f_{n}(\phi))=(g_{1}(\varphi),\cdots,g_{n}(\varphi))=(x_{1},\cdots,x_{n}).$

All invertible series in $R[[x_{1},\cdots,x_{n}]]^{n}$ form a non-abelian
group, called _formal transformation group_. The unit of this group is the
identity $id=(x_{1},\cdots,x_{n}).$ The inverse of a invertible series is
given by the Lagrange inversion formula, a family of integer polynomials that
is independent of $R.$ There are many proofs of it, including Raney [20],
Labelle [15], Gessel [11], Haiman-Schmitt [12] and Johnston-Prochno [14]. In
addition, see [10] for the non-commutative case.

Let $R=K$ be a field and let $GL_{n}(K)$ denote the general linear group over
$K.$ We consider the formal transformation group $G_{\infty}(H)$ of formal
power series over $K$ with respect to a subgroup $H$ of $GL_{n}(K),$ which is
given by

$G_{\infty}(H):=\left\\{\varphi=\sum_{i=0}^{\infty}\varphi_{i}\in
K[[x_{1},\cdots,x_{n}]]^{n}|\ \varphi_{0}=0,\varphi_{1}\in H\right\\}.$

Here $\sum_{i=0}^{\infty}\varphi_{i}$ is the unique homogeneous decomposition
of $\varphi.$ When $H=GL_{n}(K),$ we write $G_{\infty}(n,K)$ for
$G_{\infty}(GL_{n}(K)).$

A series in $G_{\infty}(n,K)$ is said to be _periodic_ if its order in
$G_{\infty}(n,K)$ is finite and is said to be _$m$ -periodic_ if its order
equals to $m.$ For $n=1,$ it is well known that
$\varphi=\omega_{m}x+\sum_{k=2}^{\infty}a_{k}x^{k}$ is $m$-periodic if and
only if $\varphi$ is conjugate to $\omega_{m}x,$ where $\omega_{m}$ is a
primitive $m$-th root of unit in $K.$ See, for example, [2],[21], [5] and
[18]. Brewer [3] generalized this result to high dimension and proved that
every $m$-periodic series $\varphi$ with $\varphi_{1}=\omega_{m}\cdot id$ is
conjugate to $\varphi_{1},$ where $\omega\cdot id$ is a multiple of the
identity.

This note generalizes Brewer’s result to a more general case where
$\varphi_{1}$ is a diagonalizable matrix. Let $\mathrm{char}\,K$ denote the
characteristic of $K.$ In general, two elements $g_{1},g_{2}$ in a group $G$
are said to be _conjugate_ to each other if $g_{1}h=hg_{2}$ for some $h\in G.$
Now we state our main results.

######  Theorem 1.

If $\mathrm{char}\,K=0$ and $\varphi\in G_{\infty}(n,K)$ is periodic, then
$\mathrm{ord}(\varphi)=\mathrm{ord}(\varphi_{1}).$

######  Theorem 2.

Suppose $m\geq 2$ and $m\notin\mathrm{char}\,K\cdot\mathbb{Z}.$ Let
$\varphi\in G_{\infty}(n,K)$ be a $m$-periodic series with $\varphi_{1}$
diagonalizable. Then $\varphi$ is conjugate to $\varphi_{1},$ i.e. there
exists $\phi\in G_{\infty}(n,K)$ such that $\varphi=\phi^{-1}\varphi_{1}\phi.$

Next result says that if the first term of $\varphi$ is a multiple of the
identity, then its $m$-periodicity is completely determined by a constraint.

######  Theorem 3.

Suppose $m\geq 2$ and $m\notin\mathrm{char}\,K\cdot\mathbb{Z}.$ Let $\omega$
be a primitive $m$-th root of unit in $K.$ Let $\varphi=\omega\cdot
id+\sum_{i=2}^{\infty}\varphi_{i}\in G_{\infty}(n,K).$ Then $\varphi$ is
$m$-periodic if and only if $\varphi_{mk+1}=-\frac{\omega}{m}((\varphi_{\leq
mk})^{m})_{mk+1}$ for any $k\geq 1,$ where $\varphi_{\leq
mk}=\sum_{i=1}^{mk}\varphi_{i}.$

######  Remark 4.

By reducing to $SL_{n}(\mathbb{Z}),$ it is not difficult to see that
$G_{\infty}(n,\mathbb{Z})$ has no $m$-periodic series for $m>2.$ As an
interesting consequence of Theorem 3, there are $2$-periodic series in
$G_{\infty}(n,\mathbb{Z})$ since the coefficients in $((\varphi_{\leq
2k})^{2})_{2k+1}$ are all even when the constraint holds.

We get an immediate consequence of Theorem 3.

######  Corollary 5.

If $\varphi\in G_{\infty}(n,\mathbb{C})$ is periodic, then $\varphi$ is
conjugate to $\varphi_{1}.$

This note is arranged as follows. In section 2 we review background material
from [7, 22, 17, 3], including formal transformation group and its basic
properties. In section 3, we give a recursive description (Theorem 10) of the
Lagrange inversion formula and some lemmas. In section 4, we introduce a kind
of subgroup $G_{p,q}(n,K)$ of $G_{\infty}(n,K)$ depending on a pair of
integers $p$ and $q.$ In section 5, we classify periodic series with an
initial condition and prove our main results.

##  2 Notation and conventions

We begin by recalling the basic properties of formal transformation groups.
Let $K$ be a field and let $K\\{x_{1},\cdots,x_{n}\\}$ (or simply denoted by
$K[[X]]$) be the formal power series ring in $n$ variables
$x_{1},\cdots,x_{n}$ over $K$ whose element has the form

$\sum_{\alpha}a_{\alpha}x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}},$

where $a_{\alpha}\in K$ and the index $\alpha=(\alpha_{1},\cdots,\alpha_{n})$
runs over $\mathbb{N}^{n}.$ For each $k\in\mathbb{N},$ we consider a set
$K[X]_{k}^{n}$ that consists of all polynomial vectors
$(f_{1},\cdots,f_{n})\in K[X]^{n}$ where each $f_{i}$ is a homogeneous
polynomial of degree $k.$ The first space $K[X]_{0}^{n}$ is just $K^{n}.$ The
second space $K[X]_{1}^{n}$ is isomorphic to the matrix space $M_{n\times
n}(K)$ over $K$ by the jacobian map

$\displaystyle\left(\sum_{j=1}^{n}a_{1j}x_{j},\cdots,\sum_{j=1}^{n}a_{nj}x_{j}\right)\mapsto(a_{ij}).$
(1)

With above notations, every series $\varphi\in K[[X]]^{n}$ has a unique
_homogeneous decomposition_

$\varphi=\sum_{k=0}^{\infty}\varphi_{k}$

with $\varphi_{k}\in K[X]_{k}^{n}.$ For convenience, we write $\varphi_{\leq
m}$ for $\sum_{i=0}^{m}\varphi_{i}.$ Also, we use the notation $K[[X]]_{\leq
a}^{n}$ for $\bigoplus_{i=0}^{a}K[X]_{i}^{n}.$ We always use Greek letters to
represent series in $K[[X]]^{n}$ and use $f$ or $g$ to represent the
components of series. Unless otherwise specified, these notations will be used
throughout this paper.

Note that for two series $\varphi=(f_{1},\cdots,f_{n}),\phi\in K[[X]]^{n},$
the series composition

$\varphi\circ\phi=(f_{1}(\phi),\cdots,f_{n}(\phi))$

is a product of $K[[X]]^{n}.$ It is easy to see that the set $G_{\infty}(n,K)$
defined in the introduction is multiplicatively closed. The classical result
(see, for example [22, Theorem 1.1.2]) about formal series shows that every
element in $G_{\infty}(n,K)$ has a unique inverse, that is, $G_{\infty}(n,K)$
is an infinite dimensional group. From now on, we always write $\varphi\phi$
for the composition $\varphi\circ\phi.$ Since $\varphi_{1}$ is just the
jacobian matrix of $\varphi$ by (1), the map
$(\cdot)_{1}:G_{\infty}(n,K)\rightarrow K[X]_{1}^{n}$ is indeed a group
homomorphism, i.e. the equality $(\varphi\phi)_{1}=\varphi_{1}\phi_{1}$ holds
for all $\varphi,\phi\in G_{\infty}(n,K).$

Next we explain that $G_{\infty}(n,K)$ is a limit of a sequence of finite-
dimensional Lie groups. For each positive integer $k,$ let

$S_{k}:=\\{\varphi\in K[[X]]^{n}|\ \varphi_{i}=0\ \mbox{for every }0\leq i\leq
k\\}.$

For convenience, we also write $S_{\infty}=0$ to match
$G_{\infty}(n,K)=G_{\infty}(n,K)\ \mathrm{mod}\ S_{\infty}.$ We say
$\varphi\equiv\phi\ \mathrm{mod}\ S_{k}$ if $\varphi_{j}=\phi_{j}$ for all
$j\leq k.$

######  Definition 6.

For each $k\geq 1,$ we define the $k$-th transformation group over $K$ of
order $n$ to be

$G_{k}(n,K):=G_{\infty}(n,K)\ \mathrm{mod}\ S_{k}.$

The product of $\varphi=(f_{1},\cdots,f_{n}),\phi=(g_{1},\cdots,g_{n})\in
G_{k}(n,K)$ is given by

$\varphi\phi=(f_{1}(g_{1},\cdots,g_{n}),\cdots,f_{n}(g_{1},\cdots,g_{n}))\
\mathrm{mod}\ S_{k}$

that is well-defined. The identity element $id$ is $(x_{1},\cdots,x_{n}).$

When $k<\infty,$ a direct calculation shows that the dimension of $G_{k}(n,K)$
over $K$ is

$\mathrm{dim}\,G_{k}(n,K)=n\sum_{i=1}^{k}\binom{n+i-1}{i}.$

If $k=1,$ the initial transformation group $G_{1}(n,K)$ is just the general
linear group $GL_{n}(K).$ So it is a generalization of the general linear
group. There is a natural homomorphism between any two transformation groups

$\pi_{k,m}:G_{k}(n,K)\rightarrow G_{m}(n,K),\ \varphi\mapsto\varphi\
\mathrm{mod}\ S_{m}$

for $k\geq m.$ In particular, $\pi_{k,1}(\varphi)=\varphi_{1}.$ If $\varphi$
is a polynomial vector with $\varphi(0)=0,$ then $\varphi_{1}$ can be viewed
as the tangent map of $\varphi:K^{n}\rightarrow K^{n}$ at $0.$

######  Remark 7.

There are other groups of special formal power series, for example, the group
of formal symmetric series [6, 23] and the group of formal Laurent series [9].

The infinite dimensional group $G_{\infty}(n,K)$ is actually the inverse limit
of $G_{k}(n,K),$ that is to say, $G_{\infty}(n,K)$ is isomorphic to

$\lim_{\longleftarrow}G_{k}(n,K)=\\{(A_{1},A_{2},\cdots)|A_{j}\in
G_{j}(n,K),\pi_{i,j}(A_{i})=A_{j},\forall i>j\\}.$

######  Remark 8.

We attach a positive integer $T(\varphi)$ to each $\varphi\in G_{\infty}(n,K)$
with $\varphi\neq\varphi_{1}$ by

$T(\varphi)=\sup\left\\{m\in\mathbb{N}|\
\varphi=\sum_{k=0}^{\infty}\varphi_{mk+1}\right\\}.$

If $\varphi=\varphi_{1},$ define $T(\varphi)=0.$ For instance, in $n=1$
dimension, $T(x+x^{2})=1$ and $T(x+x^{3}+x^{5})=2.$ By the recurrence
relations (2), one can see that $T(\varphi)=T(\varphi^{-1}).$

######  Example 9.

The inverse of a quadratic function $\varphi=x+ax^{2}\in G_{\infty}(1,K)$ with
$a\neq 0$ is given by

$\varphi^{-1}=\frac{\sqrt{4ax+1}-1}{2a}=x-ax^{2}+2a^{2}x^{3}-5a^{3}x^{4}+14a^{4}x^{5}-42a^{5}x^{6}+\cdots$

At the end of this section we describe the product of $G_{\infty}(n,K)$ in
another way. Now assume $\mathrm{char}\,K=0.$ For any $k\geq 2,$ consider the
linear space $\bigoplus_{i=2}^{k}K[X]_{i}^{n}$ over $K$ with a set of
monomials $e_{1},\cdots,e_{s}$ as its basis. Then the product of $G_{k}(n,K)$
gives an integer polynomial vector
$P_{k}=(P_{k,1},\cdots,P_{k,s})\in\mathbb{Z}[x_{1},\cdots,x_{s},y_{1},\cdots,y_{s}]^{s}$
by

$id+\sum_{i=1}^{s}P_{k,i}(x_{1},\cdots,x_{s},y_{1},\cdots,y_{s})e_{i}\equiv\left(id+\sum_{i=1}^{s}x_{i}e_{i}\right)\left(id+\sum_{i=1}^{s}y_{i}e_{i}\right)\
\mathrm{mod}\ S_{k}$

with respect to the basis $e_{1},\cdots,e_{s},$ which is independent of the
field $K.$ The existence of inverses in $G_{k}(n,K)$ implies that there exists
an integer polynomial vector
$I_{k}=(I_{k,1},\cdots,I_{k,s})\in\mathbb{Z}[x_{1},\cdots,x_{s}]^{s}$ defined
by the inverse equation

$id+\sum_{i=1}^{s}I_{k,i}(x_{1},\cdots,x_{s})e_{i}\equiv\left(id+\sum_{i=1}^{s}x_{i}e_{i}\right)^{-1}\
\mathrm{mod}\ S_{k}$

satisfying

$P_{k}(x_{1},\cdots,x_{s},I_{k}(x_{1},\cdots,x_{s}))=P_{k}(I_{k}(x_{1},\cdots,x_{s}),x_{1},\cdots,x_{s})=0.$

In fact, the collection of all polynomial vectors $\\{P_{k}\\}_{k=2}^{\infty}$
determines $\\{I_{k}\\}_{k=2}^{\infty},$ see Theorem 10 in section 3. Hence
$I_{k}$ is also independent of $K.$ These polynomial vectors $P_{k}$ and
$I_{k}$ are related to the incidence algebra [8, 12].

##  3 A recursive formula for inversion in $G_{\infty}(n,K)$

In this section we give a recursive description of the inverse of an
invertible series, which is simpler than Lagrange inversion formula [11, 14].

######  Theorem 10.

For every $\varphi\in G_{\infty}(n,K),$ the inverse
$\phi=\sum_{i=1}^{\infty}\phi_{i}$ of $\varphi$ is given by the following
recurrence relations,

$\displaystyle\phi_{1}=\varphi_{1}^{-1},\ \
\phi_{k+1}=-\varphi_{1}^{-1}\left(\varphi\left(\sum_{i=1}^{k}\phi_{i}\right)\right)_{k+1}.$
(2)

We next give some lemmas which will be used in the proofs of Theorem 10 and
other results of the present paper. A similar method has been used in [3].

######  Lemma 11.

Let $\varphi,\phi,\psi\in K[[X]]^{n}.$ We have

  * (i)

$\varphi_{1}(\phi+\psi)=\varphi_{1}\phi+\varphi_{1}\psi.$

  * (ii)

$(\phi+\psi)\varphi=\phi\varphi+\psi\varphi.$

###### Proof.

This lemma immediately follows by a direct calculation. ∎

Lemma 11 tells us that the right-multiplication of $K[[X]]^{n}$ and the left-
multiplication by $\varphi_{1}\in K[X]_{1}^{n}$ are linear. We then turn our
attention to the left-multiplication of $K[[X]]^{n}.$

Fix a nonzero degree $k$ homogenous series $\varphi.$ For each
$j\in\mathbb{N}_{\geq 1},$ the map

$K[[X]]_{\geq 1}^{n}\rightarrow K[X]_{j}^{n},\ \ \phi\mapsto(\varphi\phi)_{j}$

can be viewed as a map of infinite variables

$\Phi_{j}:\prod_{i=1}^{\infty}K[X]_{i}^{n}\rightarrow K[X]_{j}^{n},\ \
\Phi_{j}(\phi_{1},\phi_{2},\cdots)=(\varphi(\phi_{1}+\phi_{2}+\cdots))_{j}.$

Obviously, each $\Phi_{j}$ has only finite nontrivial variables, that is,
there exists an integer $r\in\mathbb{N}$ so that

$\Phi_{j}(\phi_{1},\phi_{2},\cdots)=\Phi_{j}(\phi_{1},\cdots,\phi_{r},0,0,\cdots)$

for any $(\phi_{1},\phi_{2},\cdots)\in\prod_{i=1}^{\infty}K[X]_{i}^{n}.$ In
other words, $(\varphi\phi)_{j}=(\varphi\phi_{\leq r})_{j}$ holds for any
$\phi\in K[[X]]_{\geq 1}^{n}.$ Such the smallest integer $r$ depends only on
$j$ and $k.$ We shall denote the smallest integer $r$ by $r(k,j).$ It is not
hard to see that

$\displaystyle r(k,j)=\mathrm{max}(j-k+1,0).$ (3)

######  Lemma 12.

Let $\varphi,\phi\in G_{\infty}(n,K)$ be two series with homogeneous
decompositions $\varphi=\sum_{i=1}^{\infty}\varphi_{i}$ and
$\phi=\sum_{i=1}^{\infty}\phi_{i},$ respectively. Then

$\displaystyle(\varphi\phi)_{\leq m}=\sum_{i=1}^{m}(\varphi_{i}\phi_{\leq
m+1-i})_{\leq m}.$

###### Proof.

This lemma follows immediately from the property (ii) in Lemma 11 and (3). ∎

######  Lemma 13.

For any $\varphi,\phi\in K[[X]]^{n},$ we have

$(\varphi\phi)_{\leq k+1}=(\varphi_{\leq k}\phi_{\leq k})_{\leq
k+1}+\varphi_{1}\phi_{k+1}+\varphi_{k+1}\phi_{1}.$

###### Proof.

By Lemma 11 and Lemma 12, the assersion follows from

$\begin{split}(\varphi\phi)_{\leq k+1}=&\ (\varphi_{\leq k}\phi)_{\leq
k+1}+(\varphi_{k+1}\phi)_{\leq k+1}\\\ =&\ (\varphi_{\leq k}\phi_{\leq
k})_{\leq k+1}+\varphi_{1}\phi_{k+1}+\varphi_{k+1}\phi_{1},\end{split}$

where the last equality holds due to the fact $(\varphi_{1}\phi_{\leq
k})_{k+1}=0.$ ∎

Now we are ready for the proof of Theorem 10.

_Proof of Theorem 10._ Now given a series $\varphi\in G_{\infty}(n,K).$ Let us
first calculate the necessary condition for $\varphi\phi=id$ in
$G_{\infty}(n,K).$ By $(\varphi\phi)_{1}=\varphi_{1}\phi_{1}=id,$ we see that
$\phi_{1}=\varphi_{1}^{-1},$ since $G_{1}(n,K)\cong GL_{n}(K)$ is a group.
Suppose we are given $\phi_{1},\phi_{2},\cdots,\phi_{k}$ with $\phi_{i}\in
K[X]_{i}^{n}$ such that $\phi=\sum_{i=1}^{k}\phi_{i}$ satisfies

$(\varphi\phi)_{\leq k}=(\phi\varphi)_{\leq k}=id.$

Now we solve the equation

$(\varphi(\phi+\phi_{k+1}))_{\leq k+1}=((\phi+\phi_{k+1})\varphi)_{\leq
k+1}=id$

about $\phi_{k+1}\in K[X]_{k+1}^{n}.$ By the property (i) in Lemma 11, left
multiplication by $\varphi_{1}$ is linear, and hence it follows from Lemma 12
that

$\begin{split}(\varphi(\phi+\phi_{k+1}))_{\leq k+1}=&\
(\varphi_{1}(\phi+\phi_{k+1}))_{\leq
k+1}+\sum_{i=2}^{k+1}(\varphi_{i}\phi)_{\leq k+1}\\\ =&\
\varphi_{1}\phi_{k+1}+(\varphi\phi)_{\leq k+1}.\end{split}$

As $(\varphi(\phi+\phi_{k+1}))_{k+1}=0,$ we obtain
$\phi_{k+1}=-\varphi_{1}^{-1}\cdot(\varphi\phi)_{k+1}.$ This finishes the
proof.

Next, we give an interesting corollary of Theorem 10:

######  Corollary 14.

Let $\varphi,\phi\in G_{\infty}(n,K)$ be two series with
$\varphi_{1}=\phi_{1}=id,$ satisfying

$(\varphi\phi)_{\leq k}=(\phi\varphi)_{\leq k}=id$

for some $k\in\mathbb{N}_{\geq 1}.$ Then
$(\varphi\phi)_{k+1}=(\phi\varphi)_{k+1}.$

###### Proof.

By Lemma 13, we find

$(\varphi\phi)_{k+1}=(\varphi_{\leq k}\phi_{\leq
k})_{k+1}+\phi_{k+1}+\varphi_{k+1},$ (4)

Assume $\psi\in G_{\infty}(n,K)$ is the inverse of $\varphi.$ It follows from
Theorem 10 that $\phi_{\leq k}=\psi_{\leq k}.$ Apply (4) to the products
$\varphi\psi$ and $\psi\varphi,$ respectively, we see that

$\displaystyle
0=(\varphi\psi)_{k+1}=(\varphi\phi)_{k+1}+\psi_{k+1}-\phi_{k+1},$
$\displaystyle
0=(\psi\varphi)_{k+1}=(\phi\varphi)_{k+1}+\psi_{k+1}-\phi_{k+1},$

and hence $(\varphi\phi)_{k+1}=(\phi\varphi)_{k+1}.$ ∎

##  4 Subgroups of $G_{k}(n,K)$

In this section we study a special kind of subgroups of $G_{k}(n,K).$ Now that
the projection $\pi_{k,m}$ is a homomorphism for $k\geq m,$ each $G_{k}(n,K)$
has $k-1$ natural subgroups. Sometimes we write $G_{k}$ for $G_{k}(n,K).$

Fix a dimension $n$ and a field $K.$ For each $k\in\mathbb{N}\cup\\{\infty\\}$
and $1\leq m\leq k,$ define the $m$-th subgroup $G_{k,m}$ of $G_{k}$ by

$G_{k,m}:=\mathrm{Ker}\,\pi_{k,m}=\\{\varphi\in K[[X]]^{n}/S_{k}|\
\varphi_{\leq m}\equiv id\ \mathrm{mod}\ S_{m}\\}.$

If we define $G_{k,0}:=G_{k}$ alone, then every transformation group $G_{k}$
has a filtration

$id=G_{k,k}\subset G_{k,k-1}\subset G_{k,k-2}\subset\cdots\subset
G_{k,1}\subset G_{k,0}=G_{k}.$

We remark that the dimension of $G_{k,m}(n,K)$ over $K$ is
$n\sum_{i=m+1}^{k}\binom{n+i-1}{i}.$ In particular,
$\mathrm{dim}\,G_{k,m}(1,K)=k-m.$

######  Theorem 15.

Let $k,m$ be integers with $k>m\geq\frac{k}{2}>0.$ Then the group $G_{k,m}$ is
commutative.

###### Proof.

We prove this assertion by induction on $k-m.$ First, when $k-m=1,$ for any
$\varphi,\phi\in G_{k,k-1}$ with homogeneous decomposition
$\varphi=id+\varphi_{k},\phi=id+\phi_{k},$ it follows that

$\begin{split}\varphi\phi=&\
id+(\varphi\phi)_{k}=id+((id+\varphi_{k})\phi)_{k}\\\ =&\
id+(\phi+\varphi_{k}\phi_{1})_{k}=id+\phi_{k}+\varphi_{k}=\phi\varphi.\end{split}$

Hence $G_{k,k-1}$ is commutative. Assume now that the group $G_{k,k-r}$ is
commutative for any $k,r$ with $1\leq r\leq[\frac{k}{2}].$ We then turn to the
case $k-m=r+1$ with $r+1\leq[\frac{k}{2}].$ For any $\varphi,\phi\in
G_{k,k-r-1}$ with
$\varphi=id+\sum_{i=0}^{r}\varphi_{k-i},\phi=id+\sum_{i=0}^{r}\phi_{k-i},$ it
follows by Lemma 12 that

$(\varphi\phi)_{\leq k}=\varphi_{k}+\phi_{k}+(\varphi_{\leq k-1}\phi_{\leq
k-1})_{\leq k}.$

Notice that $(\varphi_{\leq k-1}\phi_{\leq k-1})_{k}=0$ since $2r+1<k.$ Then

$(\varphi_{\leq k-1}\phi_{\leq k-1})_{\leq k}=(\varphi_{\leq k-1}\phi_{\leq
k-1})_{\leq k-1}.$

Using the assumption that $G_{k-1,k-r-1}$ is commutative, we have

$(\varphi_{\leq k-1}\phi_{\leq k-1})_{\leq k-1}=(\phi_{\leq k-1}\varphi_{\leq
k-1})_{\leq k-1},$

which implies that $(\varphi\phi)_{\leq k}=(\phi\varphi)_{\leq k}.$ Hence
$G_{k,k-r-1}$ is commutative. This finishes the proof by induction on $k-m.$ ∎

In particular, Theorem 15 implies that $G_{k,k-1},k>1,$ as a group, is
isomorphic to the vector space $K[X]_{k}^{n}.$ Moreover, for any $\varphi\in
G_{k,k-1},$ it follows that

$\displaystyle(\varphi^{m})_{\leq k}=id+m\cdot\varphi_{k}.$ (5)

Next we list all abelian subgroups of $G_{k}(1,K).$ The linear structure of
$K^{m}$ makes it an abelian group with the operator ’+’. However $K^{2}$ has
another group structure. Now we define a product on $K^{2}$ associated with a
constant $\lambda\in K$ by

$(x_{1},y_{1})\circ(x_{2},y_{2})=(x_{1}+x_{2},y_{1}+y_{2}+\lambda
x_{1}x_{2}).$

It is not difficult to verify that

$(x_{1},y_{1})\circ(x_{2},y_{2})\circ\cdots\circ(x_{s},y_{s})=\left(\sum
x_{i},\sum y_{i}+\lambda\sum_{i<j}x_{i}x_{j}\right).$

Hence $(K^{2},\circ)$ is an abelian group, and is denoted by
$\Omega_{\lambda}.$ In fact, $\Omega_{\lambda}$ is isomorphic to $K^{2}$ via a
group isomorphism

$(K^{2},+)\rightarrow(K^{2},\circ),\ \ (x,y)\mapsto(x,y+\frac{1}{2}\lambda
x^{2}).$

If $m\geq 1$ and $n=1,$ every polynomial vector in $G_{k,m}$ has the form
$x+\sum_{i=m+1}^{k}a_{i}x^{i}.$ So we can identify $G_{k,m}$ with $K^{k-m}$ in
a natural way.

######  Proposition 16.

When $n=1,$ all abelian groups in $\\{G_{i,j}\\}_{i\geq j}$ are given as
follows:

  * (i)

$G_{1,0}=K^{*},$

  * (ii)

$G_{2k,m}=K^{2k-m}$ with $2k\geq m\geq k\geq 1,$

  * (iii)

$G_{2k+1,k}=K^{k-1}\oplus\Omega_{k+1}$ with $k\geq 1,$

  * (iv)

$G_{2k+1,m}=K^{2k+1-m}$ with $2k\geq m-1\geq k\geq 1.$

###### Proof.

The proof is omitted here. It is easy to check them one by one. ∎

For example, let
$\varphi=x+\sum_{i=k+1}^{2k}a_{i}x^{i},\phi=x+\sum_{i=k+1}^{2k}b_{i}x^{i}\in
G_{2k,k}(1,K),$ then

$\varphi\phi\equiv\phi\varphi\equiv\varphi+\phi\ \mathrm{mod}\ x^{2k+1}.$

##  5 Periodic series

Recall that an element $g$ in an infinite group $G$ is periodic if there
exists a positive integer $s$ such that $g^{s}=e,$ where $e$ denotes the
identity element of $G,$ and let $\mathrm{ord}(g)$ denote the order of $g.$ We
say $g\in G$ is _$p$ -periodic_ if $\mathrm{ord}(g)=p.$

As mentioned in the introduction, the result about periodic series in
$G_{\infty}(1,K)$ is well known, see [2, 21, 5, 18]. We will discuss the
general case $n\geq 2.$ First we observe the following conclusion:

######  Theorem 17.

If $\mathrm{char}\,K=0$ and $\varphi\in G_{\infty}(n,K)$ is periodic, then
$\mathrm{ord}(\varphi)=\mathrm{ord}(\varphi_{1}).$

###### Proof.

We first prove the special case $\mathrm{ord}(\varphi_{1})=1.$ Now suppose
$\varphi^{s}=id$ for some positive integer $s,$ and let
$\varphi=\sum_{i=1}^{\infty}\varphi_{i}$ be the homogeneous decomposition of
$\varphi$ with $\varphi_{1}=id.$ We intend to prove $\varphi_{k}=0$ for any
$k\geq 2$ by induction on $k.$

  * (i)

When $k=2,$ it follows from (5) that $0=(\varphi^{s})_{2}=s\cdot\varphi_{2},$
and hence $\varphi_{2}=0.$

  * (ii)

If $\varphi_{2}=\varphi_{3}=\cdots=\varphi_{k}=0$ for $k\geq 2,$ then
$id+\varphi_{k+1}\in G_{k+1,k}.$ Using (5) again, we see that
$0=(\varphi^{s})_{k+1}=s\cdot\varphi_{k+1},$ which leads to $\varphi_{k+1}=0.$

We then prove the general case. Put $s=\mathrm{ord}(\varphi)$ and
$t=\mathrm{ord}(\varphi_{1}).$ Since the projection
$\pi_{\infty,1}:G_{\infty}\rightarrow G_{1}$ is a homomorphism, $t$ must
divide $s.$ Let $\phi=\varphi^{t}.$ Then $\phi_{1}=\varphi_{1}^{t}=id$ and
$\phi$ is periodic since $\phi^{\frac{s}{t}}=\varphi^{s}=id.$ Now apply the
above conclusion of the special case to $\phi,$ we find that $\phi=id,$ that
is, $\varphi^{t}=id.$ ∎

######  Remark 18.

From the above proof, it is not difficult to see that Theorem 17 still holds
for a commutative ring without torsion.

###  5.1 Orbits of periodic series

In this section we always assume that the period $p>1$ and
$p\notin\mathrm{char}\,K\cdot\mathbb{Z}.$ Let $\omega\in K$ be a primitive
$p$-th root of unit in $K$, i.e. $\mathrm{ord}(\omega)=p$ in the
multiplicative group $K^{*}.$ Every diagonal matrix in $GL_{n}(K)$ of order
$p$ has the form
$\mathrm{diag}(\omega^{\lambda_{1}},\cdots,\omega^{\lambda_{n}}),$ where
$\lambda=(\lambda_{1},\cdots,\lambda_{n})\in\mathbb{Z}^{n}$ is coprime to $p.$
We try to find all $p$-periodic series $\varphi$ in $G_{\infty}(n,K)$ with the
initial condition that $\varphi_{1}$ is a diagonal matrix.

We first give a simple lemma.

######  Lemma 19.

For any $\varphi\in K[[X]]^{n}$ and integers $m,k\geq 1,$ we have

$(\varphi^{m+1})_{k+1}=(\varphi^{m})_{k+1}\varphi_{1}+[(\varphi^{m})_{\leq
k}\varphi]_{k+1}.$

###### Proof.

By the fact $(\varphi^{m})_{\leq k+1}=(\varphi^{m})_{\leq
k}+(\varphi^{m})_{k+1}$ and Lemma 11, one has

$\displaystyle(\varphi^{m})_{\leq k+1}\varphi=(\varphi^{m})_{\leq
k}\varphi+(\varphi^{m})_{k+1}\varphi.$ (6)

Notice that $[(\varphi^{m})_{\leq k+1}\varphi]_{k+1}=(\varphi^{m+1})_{k+1},$
hence the lemma holds by taking terms of degree $k+1$ of the equality (6). ∎

Suppose $\varphi=\varphi_{1}+\sum_{i=2}^{k}\varphi_{i}$ satisfies
$(\varphi^{p})_{\leq k}=id$ for some $k\geq 1.$ Now we solve the equation
$((\varphi+\varphi_{k+1})^{p})_{k+1}=0$ about $\varphi_{k+1}\in
K[X]_{k+1}^{n}.$ Put $\phi=\varphi+\varphi_{k+1}.$ For the fixed integers $p$
and $k,$ we explore the relationship between $(\phi^{m+1})_{k+1}$ and
$(\phi^{m})_{k+1}.$ It follows from Lemma 13 and Lemma 19 that

$\begin{split}(\phi^{m+1})_{k+1}&=(\phi^{m}\phi)_{k+1}=((\varphi^{m})_{\leq
k}\varphi)_{k+1}+\varphi_{1}^{m}\varphi_{k+1}+(\phi^{m})_{k+1}\varphi_{1}\\\
&=(\varphi^{m+1})_{k+1}-(\varphi^{m})_{k+1}\varphi_{1}+\varphi_{1}^{m}\varphi_{k+1}+(\phi^{m})_{k+1}\varphi_{1}\end{split}$
(7)

By induction on $m,$ the equality (7) gives

$\displaystyle(\phi^{m})_{k+1}=(\varphi^{m})_{k+1}+\sum_{i=0}^{m-1}\varphi_{1}^{i}\varphi_{k+1}\varphi_{1}^{m-1-i}$
(8)

for any $m\geq 1.$ Here $\varphi_{1}^{0}=id.$ Putting $m=p$ in equality (8),
we see that $\varphi\in G_{\infty}(n,K)$ is $p$-periodic if and only if

$\displaystyle((\varphi_{\leq
k})^{p})_{k+1}+\sum_{i=0}^{p-1}\varphi_{1}^{i}\varphi_{k+1}\varphi_{1}^{p-1-i}=0$
(9)

for any $k\geq 1$ and $\mathrm{ord}(\varphi_{1})=p.$ This leads to the
following result:

######  Theorem 20.

Suppose $K,p$ are as above. Let $\varphi\in G_{\infty}(n,K)$ be a $p$-periodic
series. If $\varphi_{1}$ is a diagonalizable matrix in $GL_{n}(K),$ then
$\varphi$ is conjugate to $\varphi_{1},$ i.e. there exists $\phi\in
G_{\infty}(n,K)$ such that $\varphi=\phi^{-1}\varphi_{1}\phi.$

###### Proof.

Notice that $(A^{-1}\varphi A)_{1}=A^{-1}\varphi_{1}A$ for any $A\in
GL_{n}(K)$ and conjugacy is an equivalence relation, it is sufficient to prove
the case where $\varphi_{1}$ is a diagonal matrix.

We prove this theorem by constructing a series $\phi\in G_{\infty}(n,K)$ so
that $\varphi=\phi^{-1}\varphi_{1}\phi,$ or equivalently,
$(\phi\varphi-\varphi_{1}\phi)_{k}=0$ for any $k\in\mathbb{N}_{\geq 1}.$ We
divide our proof in six steps.

  * (i).

First, take $\phi_{1}=id.$ Now assume that
$\phi=\sum_{i=1}^{k}\phi_{i},\phi_{i}\in K[X]_{i}^{n}$ satisfies
$\varphi_{\leq k}=(\phi^{-1}\varphi_{1}\phi)_{\leq k}.$ It follows from Lemma
13 that

$[(\phi+\phi_{k+1})\varphi-\varphi_{1}(\phi+\phi_{k+1})]_{k+1}=(\phi\varphi)_{k+1}+\phi_{k+1}\varphi_{1}-\varphi_{1}\phi_{k+1}.$

Then our goal is to find $\phi_{k+1}\in K[X]_{k+1}^{n}$ such that

$\displaystyle(\phi\varphi)_{k+1}+\phi_{k+1}\varphi_{1}-\varphi_{1}\phi_{k+1}=0.$
(10)

  * (ii).

Next, consider the operator

$T:K[X]_{k+1}^{n}\rightarrow K[X]_{k+1}^{n},\ \
T(u)=[(\phi+u)^{-1}\varphi_{1}(\phi+u)]_{k+1}.$

We claim that

$T(u)=\varphi_{1}u-u\varphi_{1}+\varphi_{k+1}-(\phi\varphi)_{k+1}.$

By Theorem 10 and the assumption $\varphi_{\leq
k}=(\phi^{-1}\varphi_{1}\phi)_{\leq k},$ we have

$[(\phi+u)^{-1}\varphi_{1}(\phi+u)]_{\leq k+1}=\varphi_{\leq k}+T(u).$

To prove the above claim, it is sufficient to show that

$\displaystyle[\varphi_{1}(\phi+u)]_{k+1}=[(\phi+u)(\varphi_{\leq
k}+\varphi_{1}u-u\varphi_{1}+\varphi_{k+1}-(\phi\varphi)_{k+1})]_{k+1}.$ (11)

The left hand side of (11), denoted by $LHS,$ is given by

$LHS=\varphi_{1}u.$

Using the fact $\phi_{1}=id$ and Lemma 13, the right hand side of (11),
denoted by $RHS,$ is given by

$RHS=(\phi_{\leq k}\varphi_{\leq
k})_{k+1}+\varphi_{1}u+\varphi_{k+1}-(\phi\varphi)_{k+1}=\varphi_{1}u,$

and hence the equality (11) holds. The claim is proved.

  * (iii).

Let $t(u)=\varphi_{1}u-u\varphi_{1}$ be the linear term of $T(u).$ Since
$(\phi+u)^{-1}\varphi_{1}(\phi+u)$ is always $p$-periodic, we apply (9) to
$(\phi+u)^{-1}\varphi_{1}(\phi+u)$ and $\varphi$ to obtain that

$\begin{split}&((\varphi_{\leq
k})^{p})_{k+1}+\sum_{i=0}^{p-1}\varphi_{1}^{i}\varphi_{k+1}\varphi_{1}^{p-1-i}=0,\\\
&((\varphi_{\leq
k})^{p})_{k+1}+\sum_{i=0}^{p-1}\varphi_{1}^{i}T(u)\varphi_{1}^{p-1-i}=0.\end{split}$
(12)

Now define a linear map $R:K[X]_{k+1}^{n}\rightarrow K[X]_{k+1}^{n}$ by

$R(u)=\sum_{i=0}^{p-1}\varphi_{1}^{i}u\varphi_{1}^{p-1-i}.$

According to (12), we see that $R(T(u)-\varphi_{k+1})=0$ for any $u\in
K[X]_{k+1}^{n}.$

  * (iv).

Suppose
$\varphi_{1}=\mathrm{diag}(\omega^{\lambda_{1}},\cdots,\omega^{\lambda_{n}}),$
where $\omega$ is a primitive $p$-th root of unity and
$\lambda=(\lambda_{1},\cdots,\lambda_{n})\in\mathbb{Z}^{n}$ is coprime to $p.$
Note that the set of monomials $\\{X_{i,\alpha}\\}$ forms a basis for
$K[X]_{k+1}^{n},$ where the index $i$ runs over $\\{1,\cdots,n\\}$ and the
weight $\alpha$ runs over $\mathbb{N}^{n}$ with $|\alpha|=\sum\alpha_{j}=k+1.$
Then

$R(X_{i,\alpha})=\sum_{j=1}^{p}\omega^{-(\lambda,\alpha)}\cdot\omega^{(\lambda_{i}-(\lambda,\alpha))j}X_{i,\alpha}=\left\\{\begin{aligned}
0\ \ \ \ \ &,&\mbox{if}\ p\nmid(\lambda_{i}-(\lambda,\alpha)),\\\
p\omega^{-(\lambda,\alpha)}X_{i,\alpha}&,&\mbox{otherwise},\end{aligned}\right.$

where $(\lambda,\alpha)=\sum_{j=1}^{n}\lambda_{j}\alpha_{j}.$ Similarly, it
follows that
$t(X_{i,\alpha})=(\omega^{\lambda_{i}}-\omega^{(\lambda,\alpha)})X_{i,\alpha}.$
It is easy to see that $R\circ t=0.$ Since $R$ is linear, then
$R((\phi\varphi)_{k+1})=0.$

  * (v).

Now let $(\phi\varphi)_{k+1}=\sum a_{i,\alpha}X_{i,\alpha}$ and let $u=\sum
u_{i,\alpha}X_{i,\alpha}.$ When $p\nmid(\lambda_{i}-(\lambda,\alpha)),$ we
take
$u_{i,\alpha}=\frac{a_{i,\alpha}}{\omega^{\lambda_{i}}-\omega^{(\lambda,\alpha)}}.$
When $p|(\lambda_{i}-(\lambda,\alpha)),$ the condition
$R((\phi\varphi)_{k+1})=0$ forces that $a_{i,\alpha}=0,$ and we take
$u_{i,\alpha}=0.$ Hence $u_{0}=\sum u_{i,\alpha}X_{i,\alpha}$ satisfies the
equation $t(u_{0})-(\phi\varphi)_{k+1}=0,$ that is, $u_{0}$ is a solution to
(10).

  * (vi).

Finally, by induction on $k,$ we have constructed
$\phi=\sum_{k=1}^{\infty}\phi_{i}$ so that $\phi\varphi=\varphi_{1}\phi.$

∎

######  Remark 21.

Theorem 20 shows that every $p$-periodic series $\varphi\in G_{\infty}(n,K)$
with $\varphi_{1}$ diagonalizable is determined by an implicit function
$\phi(y)=\varphi_{1}\phi(x)$ of $x$ and $y$ for some $\phi\in
G_{\infty}(n,K).$ In particular, when $p=2$ and $\varphi_{1}=-id,$ the
equation $\phi(x)+\phi(y)=0$ is symmetric about $x$ and $y.$

######  Corollary 22.

If $\varphi\in G_{\infty}(n,\mathbb{C})$ is periodic, then $\varphi$ is
conjugate to $\varphi_{1}.$

###### Proof.

This is an immediate consequence of Theorem 20 and a fact that every periodic
matrix in $GL_{n}(\mathbb{C})$ is diagonalizable. ∎

We next consider a special case $\varphi_{1}=\omega\cdot id$ with
$\mathrm{ord}(\omega)=p.$ By (8), the equation $((\varphi_{\leq
k}+\varphi_{k+1})^{p})_{k+1}=0$ is equivalent to

$\displaystyle\left(\sum_{i=0}^{p-1}\omega^{-ki-1-k}\right)\varphi_{k+1}=-((\varphi_{\leq
k})^{p})_{k+1}.$ (13)

Simplify the sum in the left side of (13) as

$\sum_{i=0}^{p-1}\omega^{-ki}=\left\\{\begin{aligned} p\ &,&\ p|k,\\\ 0\ &,&\
\ \mbox{else}.\end{aligned}\right.$

Thus, $\varphi=\omega\cdot id+\sum_{i=2}^{\infty}\varphi_{i}\in
G_{\infty}(n,K)$ is $p$-periodic if and only if

$\displaystyle\varphi_{k+1}=-\frac{\omega}{p}(\varphi_{\leq k}^{p})_{k+1},\
\mbox{if }\ p|k;\ \ (\varphi_{\leq k}^{p})_{k+1}=0,\ \mbox{else}.$ (14)

We assert that the condition $(\varphi_{\leq k}^{p})_{k+1}=0,p\nmid k$ in (14)
can be removed, that is,

######  Lemma 23.

Suppose $K,p$ and $\omega$ are as above, and $\varphi=\omega\cdot
id+\sum_{i=2}^{pk}$ is $p$-periodic in $G_{pk}(n,K)$ for some $k\geq 1.$ Let
$\varphi_{pk+1}=-\frac{\omega}{p}(\varphi^{p})_{pk+1}.$ Then
$((\varphi+\varphi_{pk+1})^{p})_{r}=0$ for any $r=pk+2,pk+3,\cdots,pk+p.$

###### Proof.

We proof this lemma by induction on $r.$ Put $\phi=\varphi+\varphi_{pk+1}.$
When $r=pk+2,$ by (14) we see that $(\phi^{p})_{pk+1}=0,$ and hence $\phi$ is
$p$-periodic in $G_{pk+1}(n,K).$ Let $\psi$ be the inverse of $\phi^{p-1}$ in
$G_{\infty}(n,K).$ It is observed that $\psi_{\leq pk+2}=\phi+\psi_{pk+2}$
since $\phi^{p-1}$ has only one inverse in $G_{pk+1}(n,K).$ It follows from
$\phi^{p-1}\psi=\psi\phi^{p-1}=id$ that

$\displaystyle 0\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!(\phi^{p-1}\psi)_{pk+2}=(\phi^{p-1}(\phi+\psi_{pk+2}))_{pk+2}$
(15) $\displaystyle=$
$\displaystyle\\!\\!\\!\varphi_{1}^{p-1}\psi_{pk+2}+(\phi^{p})_{pk+2}$
$\displaystyle=$
$\displaystyle\\!\\!\\!\omega^{p-1}\psi_{pk+2}+(\phi^{p})_{pk+2},$
$\displaystyle 0\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!(\psi\phi^{p-1})_{pk+2}=((\phi+\psi_{pk+2})\phi^{p-1})_{pk+2}$
(16) $\displaystyle=$
$\displaystyle\\!\\!\\!(\phi^{p})_{pk+2}+\psi_{pk+2}\varphi_{1}^{p-1}$
$\displaystyle=$
$\displaystyle\\!\\!\\!\omega^{-2}\psi_{pk+2}+(\phi^{p})_{pk+2}.$

Relying on (15) and (16), we find that $\psi_{pk+2}=(\phi^{p})_{pk+2}=0.$

Now assume that $(\phi^{p})_{r}=0$ for $r=pk+2,\cdots,pk+s-1,$ where $2<s\leq
p.$ Then $\phi$ is $p$-periodic in $G_{pk+s-1}(n,K),$ and hence $\psi_{\leq
pk+s}=\phi+\psi_{pk+s}.$ With the same argument as above, we obtain

$\displaystyle 0\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!(\phi^{p-1}\psi)_{pk+s}=\omega^{p-1}\psi_{pk+s}+(\phi^{p})_{pk+s},$
(17) $\displaystyle 0\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!(\psi\phi^{p-1})_{pk+s}=\omega^{-s}\psi_{pk+s}+(\phi^{p})_{pk+s}..$
(18)

Then $\psi_{pk+s}=(\phi^{p})_{pk+s}=0$ as $\omega^{s-1}\neq 1.$ ∎

Thus, we have

######  Theorem 24.

Suppose $K,p,\omega$ are as above. Let $\varphi=\omega\cdot
id+\sum_{i=2}^{\infty}\varphi_{i}\in G_{\infty}(n,K).$ Then $\varphi$ is
$p$-periodic if and only if $\varphi_{pk+1}=-\frac{\omega}{p}((\varphi_{\leq
pk})^{p})_{pk+1}$ for any $k\geq 1.$

###### Proof.

It is an immediate consequence of (14) and Lemma 23. ∎

###  5.2 Examples

######  Example 25.

Consider the simplest case $n=1$ and $p=2.$ The recurrence relation for a
$2$-periodic series $\varphi=-x+\sum_{k=2}^{\infty}a_{k}x^{k}\in
G_{\infty}(1,\mathbb{Q})$ becomes $\varphi_{2k+1}=\frac{1}{2}((\varphi_{\leq
2k})^{2})_{2k+1}.$ The first few relations are

$\begin{split}a_{3}=&-a_{2}^{2},\\\ a_{5}=&2a_{2}^{4}-3a_{2}a_{4},\\\
a_{7}=&-13a_{2}^{6}+18a_{2}^{3}a_{4}-2a_{4}^{2}-4a_{2}a_{6},\\\
a_{9}=&145a_{2}^{8}-221a_{2}^{5}a_{4}+50a_{2}^{2}a_{4}^{2}+35a_{2}^{3}a_{6}-5a_{2}a_{8}-5a_{4}a_{6},\\\
a_{11}=&-2328a_{2}^{10}+3879a_{2}^{7}a_{4}-561a_{2}^{5}a_{6}-1263a_{2}^{4}a_{4}^{2}+61a_{2}^{3}a_{8}\\\
&+171a_{2}^{2}a_{4}a_{6}+55a_{2}a_{4}^{3}-6a_{2}a_{10}-3a_{6}^{2}-6a_{4}a_{8},\\\
\cdots&\end{split}$

######  Example 26.

Examine the case $n=1$ and $p=3.$ Let $\varphi=\omega
x+\sum_{k=2}^{\infty}a_{k}x^{k}$ be a $3$-periodic series in
$G_{\infty}(1,\mathbb{Q}[\omega]),$ where $\omega=e^{\frac{2\pi i}{3}}.$ The
first few relations are given by

$\begin{split}a_{4}=&\frac{1-4\omega}{3}a_{2}^{3}-\frac{7+8\omega}{3}a_{2}a_{3},\\\
a_{7}=&(3\omega-1)a_{3}^{3}+(4-8\omega)a_{2}^{2}a_{5}+\frac{116-68\omega}{9}a_{2}^{6}-\frac{10+14\omega}{3}a_{2}a_{6}\\\
&-\frac{13+11\omega}{3}a_{3}a_{5}-\frac{154+332\omega}{9}a_{2}^{4}a_{3}-\frac{79+74\omega}{9}a_{2}^{3}a_{3}^{2},\\\
a_{10}=&\frac{22586+5656\omega}{27}a_{2}^{9}+\frac{1461-14576\omega}{9}a_{2}^{7}a_{3}-\frac{14446+14501\omega}{27}a_{2}^{5}a_{3}^{2}\\\
&-\frac{8891+12490\omega}{27}a_{2}^{6}a_{3}^{2}+\frac{8-1604\omega}{27}a_{2}^{3}a_{3}^{3}-\frac{8101+8\omega}{27}a_{2}^{4}a_{3}^{3}\\\
&+\frac{374+505\omega}{3}a_{2}a_{3}^{4}+\frac{1247-182\omega}{3}a_{2}^{5}a_{5}-\frac{2131+4319\omega}{9}a_{2}^{3}a_{3}a_{5}\\\
&+\frac{-338+80\omega}{9}a_{2}a_{3}^{2}a_{5}+(7-15\omega)a_{2}a_{5}^{2}-\frac{103+1154\omega}{9}a_{2}^{4}a_{6}\\\
&-\frac{545+361\omega}{9}a_{2}^{2}a_{3}a_{6}-(7+22\omega)a_{3}^{2}a_{6}-\frac{16+17\omega}{3}a_{5}a_{6}\\\
&+(10-12\omega)a_{2}^{2}a_{8}-\frac{19+14\omega}{3}a_{3}a_{8}-\frac{13+20\omega}{3}a_{2}a_{9},\\\
\cdots&\end{split}$

######  Example 27.

Look at the case $n=1$ and $p=4.$ Let
$\varphi=ix+\sum_{k=2}^{\infty}a_{k}x^{k}$ be a $4$-periodic series in
$G_{\infty}(1,\mathbb{Q}[i]),$ where $i=\sqrt{-1}.$ The first two relations
are given by

$\begin{split}a_{5}=&\frac{-3+5i}{2}a_{2}^{4}+\frac{12+5i}{2}a_{2}^{2}a_{3}-\frac{3i}{2}a_{3}^{2}+(1-3i)a_{2}a_{4},\\\
a_{9}=&\frac{743+118i}{4}a_{2}^{8}-(35+578i)a_{2}^{6}a_{3}+\frac{-3504+1307i}{8}a_{2}^{4}a_{3}^{2}+\frac{172+215i}{4}a_{2}^{2}a_{3}^{3}\\\
&-\frac{65i}{8}a_{3}^{4}+\frac{-345+23i}{2}a_{2}^{5}a_{4}+\frac{119+332i}{2}a_{2}^{3}a_{3}a_{4}+\frac{2+99i}{2}a_{2}a_{3}^{2}a_{4}\\\
&+\frac{33-25i}{2}a_{2}^{2}a_{4}^{2}+\frac{32-7i}{2}a_{3}a_{4}^{2}-(35-25i)a_{2}^{3}a_{6}+(32+11i)a_{2}a_{3}a_{6}\\\
&-(1+5i)a_{4}a_{6}+\frac{24+27i}{2}a_{2}^{2}a_{7}-5ia_{3}a_{7}+(3-5i)a_{2}a_{8},\\\
\cdots&\end{split}$

##  6 An open problem

At the end of this paper, we propose a conjecture about periodic series over
$\mathbb{C}.$ Consider a special case in Example 25 . Let

$f_{a}(x)=-x+ax^{2}+\sum_{k=1}^{\infty}c_{k}a^{2k}x^{2k+1}$

be a $2$-periodic series in $G_{\infty}(1,\mathbb{C}),$ where
$\\{c_{k}\\}_{k=1}^{\infty}$ is a sequence of integers with first few items

$c_{1}=-1,\ c_{2}=2,\ c_{3}=-13,\ c_{4}=145,\ c_{5}=-2328,\ c_{6}=49784,\
\cdots$

See the sequence [13] for more integers. Take $a=1.$ The series
$\sum_{k=1}^{\infty}c_{k}x^{k}$ appears to be divergent for any
$x\in\mathbb{C}^{*}.$ Moreover, we conjecture that

######  Conjecture 28.

Let $p\geq 2$ and $\omega=\exp({\frac{2\pi i}{p}})\in\mathbb{C}.$ Suppose

$\varphi=\omega\cdot
id+\sum_{j=2}^{p}\varphi_{j}+\sum_{k=1}^{\infty}\varphi_{pk+1}\in
G_{\infty}(n,\mathbb{C})$

is $p$-periodic. If $\varphi\neq\varphi_{1},$ then the radius of convergence
of $\varphi$ is zero.

_Acknowledgements._ The author would like to thank Yue Chen for many useful
conversations and comments during this work.

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