# Diversity dilemmas: uncovering gender and nationality biases in graduate
admissions across top North American computer science programs

GKGhazal Kalhor  TZTanin Zeraati  BBBehnam Bahrak  School of Electrical and
Computer Engineering, College of Engineering, University of Tehran, Tehran,
Iran  Tehran Institute for Advanced Studies, Tehran, Iran

###### Abstract

Although different organizations have defined policies towards diversity in
academia, many argue that minorities are still disadvantaged in university
admissions due to biases. Extensive research has been conducted on detecting
partiality patterns in the academic community. However, in the last few
decades, limited research has focused on assessing gender and nationality
biases in graduate admission results of universities. In this study, we
collected a novel and comprehensive dataset containing information on
approximately 14,000 graduate students majoring in computer science (CS) at
the top 25 North American universities. We used statistical hypothesis tests
to determine whether there is a preference for students’ gender and
nationality in the admission processes. In addition to partiality patterns, we
discuss the relationship between gender/nationality diversity and the
scientific achievements of research teams. Consistent with previous studies,
our findings show that there is no gender bias in the admission of graduate
students to research groups, but we observed bias based on students’
nationality.

graduate admission,

statistical analysis,

partiality patterns,

gender equality,

nationality diversity,

###### keywords:

Research

## Introduction

Every year, many students from all over the world apply to pursue their
graduate studies at top universities in North America [1]. Despite the
committee-based nature of admission to many of these universities, professors
still play a prominent role in accepting students and providing them with
financial support [2]. As a result, students often directly contact faculty
members to enhance their chances of admission. Furthermore, students who are
admitted by a committee must find an academic advisor and research group, and
faculty members have the authority to approve or reject these requests.
Consequently, their research group may demonstrate a preference for accepting
students of similar gender, country of origin, or previous universities. In
this study, we aim to examine the existing biases in interactions with
computer science faculty members at top North American universities and their
preferences regarding nationality and gender when selecting graduate students
for their research group.

In addition to establishing fair admission systems, it is crucial to enhance
diversity in academia. Promoting diversity within universities enables them to
have a greater impact on societies [3]. This is because institutions aim to
address social issues, which cannot be effectively achieved without embracing
diversity [4]. Furthermore, it is argued that being in a diverse environment
can broaden students’ horizons [5].

Most prestigious universities typically strive to ensure fairness in the
admission process for their graduate programs. Various factors, such as merit,
gender equality, and diversity, contribute to establishing a fair graduate
admission system [6, 2]. However, it is argued that admitting a greater number
of marginalized students for graduate education at U.S. universities remains a
contentious issue [7].

To the best of our knowledge, only three studies have focused on assessing
gender or nationality bias in graduate admissions, and all of them were
conducted prior to 2000. Bickel and Hammel [8] analyzed admission results from
various schools at the University of California, Berkeley to examine the
presence of a gender gap. They found statistically significant favoritism
towards female applicants. Maxwell and Jones [9] employed adjustment
techniques to compare admission rates between women and men in four graduate
programs at the University of North Carolina, Chapel Hill. Their findings
suggested that gender was not a significant factor in admission decisions.
Subsequently, the authors of [10] discussed the influence of demographic
attributes, such as gender and country of citizenship, on graduate admission
decisions at top-ranked American universities. Their results indicated that
these universities placed greater emphasis on admitting U.S. students, and
female applicants received some degree of preference. Our work builds upon
these studies by addressing questions regarding gender/nationality bias in
more recent and comprehensive graduate admissions data. The dataset we
collected for this study encompasses a larger number of students and includes
a greater number of universities.

Some studies have examined the impact of gender/nationality diversity on the
performance of research teams. In [11], the authors investigated the level of
cultural diversity at which a research group achieves the highest performance.
AlShebli et al. [12] analyzed author lists of research papers to explore the
influence of diversity in characteristics such as gender and ethnicity on the
success of research teams. Llorens et al. [13] demonstrated the existence of
gender bias throughout scholars’ academic careers, affecting aspects such as
career opportunities, promotion, and grant allocation. They also proposed
solutions at various levels to enhance diversity, highlighting its importance
for scientific success. The authors of [14] examined different facets of
gender diversity and reported its positive impact on creativity and
performance in scientific domains. Kamerlin [15] addressed bias issues in
academia and presented strategies to promote gender diversity in academic
environments. Powell [16] utilized citation count to quantify the success of
research papers and investigated its relationship with various aspects of
diversity, such as gender, age, ethnicity, and affiliation, among the authors.
In addition to citation count, we consider faculty members’ h-index and
publication count as measures of success for their research groups.

Many initiatives have been undertaken to enhance diversity in computer
science. The author of [17] emphasized that these efforts should not be
limited to achieving gender equity alone. Wilson [18] highlighted how his team
in Hour of Code decided to translate their lectures into multiple languages
and establish branches in more countries to promote diversity in computer
science. One of the primary objectives of their program is to globalize
computer science [19]. Increasing students’ awareness of diversity and
inclusion is a crucial step towards fostering a more diverse community of
computer scientists [20]. These studies collectively underscore the
significance of addressing diversity issues in academia.

In this study, we aim to address the following questions:

  * •

Do professors exhibit a preference for admitting students of the same gender
to their research group?

  * •

Are they inclined to accept students who share their country of origin?

  * •

How do these bias patterns evolve over time?

  * •

Is there any correlation between the diversity of gender or nationality among
team members and the research team’s productivity?

Our contributions can be summarized as follows:

  1. 1.

We provide a comprehensive description of the dataset collected for this
study, highlighting its various features.

  2. 2.

We analyze the gender distributions of students and faculty members and
conduct hypothesis tests to examine the presence of gender bias in the
selection of students for graduate study.

  3. 3.

We investigate the distributions of advisors and students’ home countries and
explore the existence of bias in this variable.

  4. 4.

We construct an advisor-student relationship network using our dataset and
calculate centrality metrics to identify the most influential countries in
higher education.

  5. 5.

We examine the trends in gender/nationality biases and diversities among
advisor-student pairs over time using Mann-Kendall tests.

  6. 6.

We assess the correlations between academic success and diversity measures to
analyze the relationship between gender/nationality diversity and the
performance of research groups.

The rest of this paper is structured as follows. The ”Materials and methods”
section offers an overview of the data collection process and delineates the
diverse features within our dataset. Following this, our discoveries are
outlined and analyzed in the ”Results and discussion” section. In the ”Future
work” section, potential directions for future research are proposed. Finally,
the ”Conclusion” section succinctly summarizes the key takeaways of the paper.

## Materials and methods

In this section, we define the techniques and metrics that we use in answering
our research questions. Moreover, we describe the dataset that we collected
for this study.

### Methods

In this part, we introduce the algorithms and statistical tests utilized in
our study.

#### Disparity filter

The disparity filter is a graph sparsification algorithm utilized to
effectively reduce the number of edges in a network while preserving its
multi-scale nature [21]. We apply this algorithm to remove insignificant edges
from the advisor-student relationship network. Figure 1 provides an example of
the application of the disparity filter algorithm.

Figure 1: A sampled subgraph of the advisor-student relationship network.

#### Louvain community detection

The Louvain community detection algorithm is utilized to identify communities
within a large-scale network by optimizing the modularity. This algorithm aims
to maximize the difference between the expected edge counts within a community
and the actual edge counts. It employs a greedy approach with heuristics to
solve the problem efficiently in polynomial time [22]. We apply this algorithm
to detect communities within the advisor-student relationship network.

#### Leiden community detection

The Leiden community detection algorithm is an advancement of the Louvain
algorithm. It employs a fast local move approach and iteratively refines
partitions to ensure the connectedness of all detected communities. Compared
to the Louvain algorithm, it offers improved speed and provides more accurate
partitions [23]. We use this algorithm to identify communities within the
advisor-student relationship network.

Figure 2 shows the examples of the Louvain and Leiden community detection
algorithms.

Figure 2: The sampled subgraph of the advisor-student relationship network
with specified communities.

### Statistical analysis

In this part, we provide a description of the statistical methods employed in
this study.

#### Proportion hypothesis test

The proportion hypothesis test is a statistical method that compares the ratio
of an attribute in a population with a reference proportion. It also
establishes a range of values that are likely to include the population
proportion [24]. We utilize this technique to assess our research questions
regarding biases in graduate admission.

#### Mann-Kendall test

The Mann-Kendall test is a nonparametric method that assesses the presence and
direction of trends. It is particularly suitable for detecting monotonic
trends that exhibit consistent increases or decreases over time [25]. We
employ this technique to evaluate the trends of variables such as
gender/nationality diversity over time.

### Metrics

In this part, we provide the definitions of the measures that we calculate in
this study.

#### Weighted degree centrality

Weighted degree centrality is defined for each node in a network by summing
the weights of the edges connected to that node. The formula for weighted
degree centrality is as follows:

$\displaystyle WD(u)=\sum_{v}{w(v,u)},$ (1)

where $v$ is a neighbor of $u$, and $w(v,u)$ is the weight of the edge between
$v$ and $u$ [26]. We employ this measure to examine the faculty members from
which countries accept a greater number of students from other countries.

#### Closeness centrality

For each node, closeness centrality is defined as the average distance between
that node and all other nodes in the network. The formula for closeness
centrality is as follows:

$\displaystyle C(u)=\frac{n-1}{\sum_{v=1}^{n-1}d(v,u)},$ (2)

where $n$ represents the number of vertices that node $u$ is reachable from,
and $d(v,u)$ denotes the geodesic distance between nodes $v$ and $u$ [27]. We
utilize this metric to determine which countries are closer to the rest of the
world in terms of admission results.

#### Entropy

The entropy of a variable is defined as the average uncertainty of that
variable based on its probability distribution. The formula for entropy is as
follows:

$\displaystyle E(X)=-\sum_{v=1}^{n}p_{i}\log p_{i},$ (3)

where the base of the logarithm is $e$, and $p_{i}$ represents the probability
of the $i$-th outcome in variable $X$ [28]. We employ this measure to
calculate the diversity of an advisor’s research team.

### Dataset

Data collection was the most challenging aspect of this study. We collected
data from multiple websites, each with its own unique structure, using a
combination of manual and automated approaches.

The data collection procedure consists of four steps: manual data gathering,
data collection using crawlers, removal of unnecessary data, and
preprocessing. We collected data from the top 25 universities in North
America, as ranked by Quacquarelli Symonds (QS) in 2021 for computer science
[29].

#### Manual data collection

Among all the faculty members in the computer science departments of each
university, we randomly selected approximately 30 professors. We collected
information such as the professor’s academic rank, home country, gender,
research areas, and academic performance metrics (h-index and citation count).
We also completed the prior universities (alma maters) column by referring to
the professors’ resumes and information available on their websites, LinkedIn,
and Google Scholar. To determine the gender, we relied on images or pronouns
specified on their websites. If the birthplace was not explicitly stated, we
used the location of their undergraduate university to determine their home
country. We also gathered academic records, such as citation counts and
h-indexes, from Google Scholar.

The academic rank of faculty members, including Assistant Professor, Associate
Professor, and Professor, was typically available on the university’s website.
Table 1 presents the key information about faculty members that we collected
from the university homepage and the professors’ personal pages.

Table 1: Essential professor information. University | ID |  | Academic
---
Rank
Gender |  | Home
---
Country
| Previous
---
Universities
| Citation
---
Count
h-index |  | Publication
---
Count
| First Paper
---
Year
CMU | advisor37 | Professor | Female | United States | MIT/MIT/MIT | 34062 | 57 | 328 | 1979
Stanford | advisor311 |  | Associate
---
Professor
Male | Slovenia |  | University of
---
Ljubljana/CMU
103427 | 125 | 477 | 1999

For the professor’s field column, we initially obtained the professor’s
research interests from their website, resume, or in some cases, from Google
Scholar. Next, we manually determined whether the professor’s research
interests were associated with one or more of the 13 primary fields of the
Association for Computing Machinery (ACM) computer science field category
[30]. Table 2 presents a sample mapping between professor interests and ACM
subareas within our dataset.

Table 2: Mapping between professor interests and ACM subareas. ID | Field | Standard Field
---|---|---
advisor96 | AI,ML,Optimization | Computing methodologies
advisor445 |  | cloud computing,databases,distributed
---
systems
| Software and its engineering/
---
Information systems
advisor518 |  | natural language processing,computational
---
social sciences,machine learning
| Computing methodologies/
---
Information systems

After obtaining all the necessary information for each professor, we proceeded
to gather the names of their students and any additional available information
from their profiles. If any student-related information was available, we used
it to populate the corresponding column; otherwise, we left it blank and
planned to update it later with data collected from our crawlers in the next
stage. Furthermore, after running the crawlers, we manually cross-checked the
data to fill in any gaps using information available from other sources. The
process of finding the information and collecting the data proved to be
challenging and time-consuming, leading to the development of crawlers for
different sections. Table 3 presents the student information available in our
dataset.

Table 3: Student information in our dataset. ID | Degree | Start Year | Gender | Home Country | Previous Universities
---|---|---|---|---|---
student2760 | MS | 2019 | Female | South Korea | Yonsei University
student3575 | PhD | 2020 | Male | China | Tsinghua University
student10241 | PhD | 2021 | Male | United States | Columbia University

#### Data collection using crawlers

We used the list of all students as input for the Google search engine to
locate their websites and resumes, including their LinkedIn accounts. The next
challenge was to automatically extract the required data from these websites
and resumes to populate the information columns, such as degree, admission
year, and alma maters. We also performed data cleaning on the output from the
crawler and merged it with the primary dataset to ensure consistency and
completeness.

We used the Name2GAN website [31] to label a person’s gender, if it was not
manually identified. We checked the results of this tool for 3000 previously
labeled data. The results show that the gender detection tool has an accuracy
higher than 90%. We used manual labeling for cases that gender detection
uncertainty was high to enrich the quality of our dataset.

#### Irrelevant data removal

Since we recorded information about all students associated with each
randomly-selected professor, including visiting students, undergraduates,
postdocs, masters, and PhD students, it was important to filter out irrelevant
data and include only graduate students for our analysis. The final version of
the dataset was completed on August 2, 2022, and it consisted of a total of
13,936 graduate students.

#### Preprocessing

The preprocessing stage consists of two phases:

  1. 1.

Preparing the input for the crawlers.

  2. 2.

Preparing the data for analysis.

The most crucial component of the preprocessing stage was creating a
consistent list of institutions that could be used for analysis and for the
Google Maps crawler. We also double-checked the address results for each
university to ensure that the mapping between university and address was
unique. As mentioned earlier, we used these addresses to identify the
students’ countries of origin. In some cases, the home countries of students
were improperly reported as a state rather than the country, and we corrected
this during the preprocessing stage. Once the home country column was filled
out, we standardized the names of the countries and prepared them for
analysis. Additionally, the admission year column required cleaning, as there
were specific irrational values that were quickly corrected.

Students’ home country is determined based on explicit specifications, if
available. If not explicitly specified, we first consider the country from
which they earned an associate degree. If that information is not available,
we use the location of their undergraduate university to determine their home
country. Additionally, we utilized a crawler for the Google Maps API to search
for the location of universities and schools, which provided us with the
necessary addresses for further analysis.

### Data exploration

In this part, we present an overview of the key features of our dataset in
order to gain insights into their distributions.

#### Advisors’ gender

In this part, we examine the distributions of advisors’ gender across other
attributes. Figure 3 displays the mosaic plot depicting the relationship
between advisors’ gender and their academic rank. The majority of advisors in
our dataset hold the professor rank, and the highest proportion of male
advisors is also observed at the professor level. This finding aligns with the
results of [32], which suggest that men have a greater likelihood of being
promoted to the professor rank compared to women.

Figure 3: Mosaic plot of advisors’ academic rank and gender

Figure 4 illustrates the distribution of female and male faculty members
across different subfields of computer science in our dataset. The graph shows
that the computing methodologies subfield has the highest number of advisors.
This observation can be attributed to the growing significance of Artificial
Intelligence, which falls under the computing methodologies category and is an
interdisciplinary field [33, 34]. The theory of computation and computer
systems organization subfields represent the second and third largest groups,
respectively.

Figure 4:  Back-to-back bar plot of advisors’ gender and their research
fields.

Figure 5 shows the distribution of gender among computer science faculty
members across different universities.

Figure 5:  Gender-disaggregated bar plot showing the count of advisors for
each university.

#### Advisors’ academic performance metrics

In this part, we present the dispersion of academic performance metrics of the
faculty members, including publication count, h-index, and citation count,
which are crucial indicators of the success of their research teams. Figure 7
displays the boxplots of advisors’ publication counts for each university. To
enhance the resolution, advisors with more than 1000 publications were
excluded from this plot.

Figure 7 illustrates the distribution of citation counts for faculty members
at each university. To improve the clarity of the diagram, faculty members
with a citation count exceeding 100,000 were excluded.

Figure 6:  Boxplots of advisors’ publication counts for each university.

Figure 7:  Boxplots of advisors’ citation counts for each university.

Figure 8 presents the boxplots of h-indexes for faculty members at each
university. It is worth noting that the h-index metric has fewer outliers
compared to the previous metrics, indicating that it may be a better indicator
for assessing the success of research groups [35].

Figure 8:  Boxplots of advisors’ h-indexes for each university.

#### Students’ gender

In this part, we illustrate the distribution of students’ gender against other
features. Figure 9 presents a mosaic plot depicting the distribution of
students’ gender based on the degree they are pursuing (or have pursued) under
the supervision of their advisor. The plot reveals that there are fewer women
in graduate computer science programs, which aligns with the findings of Cuny
and Aspray’s study [36]. Additionally, the female-to-male ratio decreases as
the degree level progresses from masters to doctorate, potentially indicating
a lower tendency among women to pursue higher education [37].

Figure 9: Mosaic plot of students’ degree and gender.

Figure 10 displays the gender distribution of CS students across different
universities.

Figure 10:  Gender-disaggregated bar plot showing the count of students for
each university.

#### Nationality distributions

In this part, we explore the distribution of nationalities among students and
faculty members. Figure 11 presents the distribution of students’ citizenship
for each degree. It shows that the majority of students apply for doctoral
programs, and the percentage of international students is higher than that of
American and Canadian students. This finding is in line with the result of a
study by Okahana and Zhou [38], which states that in Fall 2015, approximately
55% of students majoring in computer science or related programs were
international students.

Figure 11: Mosaic plot of students’ degree and citizenship.

Figure 13 displays the distribution of students’ nationalities on the world
map. The United States and Canada have been excluded to focus solely on
international students. The map reveals that the majority of international
students are from China, India, and Iran, respectively. This finding aligns
with the results of [39], which indicate that graduate programs are
predominantly composed of Chinese and Indian students.

Figure 13 displays the distribution of faculty members’ home countries on the
world map. The map reveals that the majority of advisors originated from the
United States, followed by India, China, and Canada, respectively.

Figure 12: Distribution of students’ nationalities.

Figure 13: Distribution of advisors’ countries of origin.

Figure 14 depicts the sorted bar plots of the 15 most common countries among
faculty members and students.

Figure 14:  Distributions of countries of origin among faculty members and
students.

## Results and discussion

In this section, we provide a comprehensive explanation of our analyses and
interpret the results we obtained.

### Assessing gender partiality

In this part, we evaluate the presence of gender bias in admission decisions.
We conduct a two-sided hypothesis test with a significance level of 0.05 to
examine whether there is gender bias in the acceptance of graduate students
into advisors’ research groups. To accomplish this, we employ a simulation-
based approach with 500 iterations [24]. In each iteration, we generate 13,759
advisor-student pairs, where the gender of each component is selected based on
the observed ratio in our dataset. Specifically, the probability of an advisor
being male is 0.788, and the probability of a student being male is 0.771.
This simulation yields an approximately normal distribution with a mean of
0.6562 and a standard deviation of 0.0212, as depicted in Figure 15. It is
important to note that this distribution represents the values for the ratio
of advisor-student pairs with the same gender, assuming no gender bias in
admitting graduate students. In our dataset, the observed ratio of advisor-
student pairs with the same gender is 0.6896. We will now test whether this
observed value is likely to occur in the simulated distribution. Thus, our
hypothesis test is formulated as follows:

$\displaystyle\begin{gathered}H_{0}:p_{common\>gender\>ratio}=0.6562\\\
H_{a}:p_{common\>gender\>ratio}\neq 0.6562\end{gathered}$ (6)

Figure 15: Histogram of advisor-student common gender proportion.

Using a z-test, we obtained a p-value of 0.1152, which is higher than the
significance level of 0.05. Therefore, we cannot reject the null hypothesis.
In other words, the data does not provide strong evidence of gender bias in
the admissions of graduate students. This finding is consistent with the
results of Maxwell’s study [9], which also concluded that gender is not a
significant factor in graduate student acceptance.

### Evaluating nationality bias

In this part, we aim to investigate the presence of nationality bias in
advisor-student relationships. We conduct a two-sided hypothesis test with a
significance level of 0.05 to assess the existence of such bias. Similar to
the previous analysis, we employ a simulation-based approach with 500
iterations. For this analysis, we only consider international students who are
not from the United States or Canada. At each iteration, we generate 4,839
advisor-student pairs, where the nationality of each individual is selected
with a probability equal to the observed ratio in the dataset. In each
iteration, we calculate the ratio of advisor-student pairs with the same
nationality. The resulting distribution, shown in Figure 16, approximates a
normal distribution with a mean of 0.0682 and a standard deviation of 0.0113.
In our dataset, the proportion of advisor-student pairs with the same
nationality is 0.1593. To assess the likelihood of observing such a ratio in
the simulated distribution, we formulate the following hypothesis test:

$\displaystyle\begin{gathered}H_{0}:p_{common\>nationality\>ratio}=0.0682\\\
H_{a}:p_{common\>nationality\>ratio}\neq 0.0682\end{gathered}$ (9)

Figure 16: Histogram of advisor-student common nationality ratio.

Using a z-test, we obtain a p-value of $p<10^{-15}$, which is significantly
lower than the chosen significance level of 0.05. Therefore, we reject the
null hypothesis and conclude that there is strong evidence of nationality bias
in admitting international graduate students. This bias may be attributed to
advisors’ familiarity with universities in their home country and their
potential to make more accurate assessments of students who have graduated
from those universities.

### Advisor-student relationship network

In this part, we present a cross-country advisor-student relationship network
based on our dataset. The network is constructed by connecting the
nationalities of students and their advisors with weighted edges. We apply the
disparity filter algorithm [21] to eliminate insignificant edges and remove
isolated nodes from the network. Figure 18 provides an overview of the
advisor-student relationship network. In this visualization, the size of the
nodes and labels corresponds to the weighted degree and closeness
centralities, respectively. The thickness of the edges represents their
weight, which indicates the number of advisor-student pairs between the
respective countries. Additionally, the nodes are color-coded based on their
community assignment, determined using the Louvain community detection
algorithm [22].

The countries with the highest values for both centrality metrics are the
United States, India, China, Canada, and Iran, respectively. This observation
aligns with the previous findings that faculty members from these countries
are prevalent in top universities. It serves as further evidence of the
potential existence of nationality bias in advisor-student relationships.

In Figure 18, the advisor-student relationship network is depicted with
similar settings, but the Leiden algorithm [23] is utilized for community
detection. According to the results, Sweden and Romania are assigned to
different communities compared to Figure 18.

Figure 17: Cross-country advisor-student relationship network, with
communities detected via Louvain algorithm.

Figure 18: Cross-country advisor-student relationship network, with
communities detected via Leiden algorithm.

### Exploring time effect

In this part, we analyze the changes in bias patterns over time. Specifically,
we examine admissions from 2000 to 2021. For each year, we calculate the
ratios of advisor-student pairs with the same gender and nationality. Figure
20 illustrates the time series of the identical gender ratio. The results of a
Mann-Kendall test indicate that this time series exhibits a statistically
significant decreasing trend ($p<0.01$).

Figure 20 illustrates the proportions of advisor-student pairs with the same
nationality across different acceptance years. We observe an increasing trend
in these proportions, which is consistent with the results of a Mann-Kendall
test ($p<0.01$).

Figure 19: Time series of advisor-student identical gender ratio.

Figure 20: Time series of advisor-student similar nationality ratio.

### Investigating relationship between academic success and diversity

In this part, we aim to investigate whether there is a correlation between
diversity in advisors’ research groups and their academic success. To assess
this relationship, we employ scientometrics, which are described in Table 4,
as measures of research group success.

Table 4: Scientometrics and their explanations. Metric | Explanation
---|---
h-index | Advisor’s h-index
Mean Citation Count |  | Advisor’s number of citations divided by
---
the years of her/his presence in academia
Mean Publication Count |  | Advisor’s number of publications divided by
---
the years of her/his research experience

Moreover, we consider the entropy of genders and nationalities among an
advisor’s students as measures of diversity within their research group. We
calculate the academic success and diversity measures for 737 advisors in our
dataset. Subsequently, we compute the correlations between these variables, as
shown in Figure 21. To assess the statistical significance of each
correlation, we conduct a hypothesis test with a significance level of 0.01.
Based on the results, the correlations between gender entropy and other
variables are close to zero and not statistically significant. This suggests
that there is no significant linear correlation between gender diversity and
the performance of research groups. On the other hand, nationality entropy
exhibits a moderate positive correlation with advisors’ h-index. This implies
that research teams with greater diversity in terms of nationality tend to
have higher research productivity. Additionally, there are weak positive
linear relationships between nationality diversity and the remaining academic
success metrics. It is important to note that the h-index is considered a more
reliable measure of academic success [35].

Figure 21: Correlogram of academic success measures and gender/nationality
diversity.

### Analyzing trends of diversity

In this section, we discuss how gender and nationality diversities have
changed over the past two decades. Once again, we employ the Mann-Kendall test
to assess the strength of the observed trend. Figure 23 illustrates the
increasing trend in gender entropy over time. According to the results of the
Mann-Kendall test, the observed trend is highly statistically significant
($p<10^{-5}$).

Figure 23 shows the time series of nationality entropy. As depicted, there has
been a decrease in nationality diversity over time. The decline from 2016 to
2020 is particularly noticeable. The results of the Mann-Kendall test confirm
that the observed trend is statistically significant ($p<0.01$).

Figure 22: Students’ gender entropy across admission years.

Figure 23: Students’ nationality entropy across admission years.

## Future work

While our work presents a novel study analyzing gender and nationality biases
in graduate admissions over recent decades, future research should aim to
explore other crucial factors influencing admission decisions. These factors
include academic background, religion, and politics, in order to provide a
more comprehensive understanding of bias in graduate admissions. To achieve
this, researchers could contemplate integrating our dataset with additional
sources, such as institutional reports and the social media profiles of
students and faculty members on platforms like Twitter, to glean fresh
insights on this matter.

Moreover, another promising avenue for future research involves evaluating
whether specific stages of the admissions process accentuate gender and
nationality biases, and how these biases manifest diversely across various
universities. For example, researchers could concentrate on distinct phases of
the admissions process, such as committee decisions, to discern differing bias
patterns.

Additionally, future investigations might delve into the correlation between
gender and nationality diversity within computer science faculty and observed
biases in graduate admissions. This analysis could yield insights into
potential strategies for addressing these biases effectively.

Lastly, a valuable topic for future research could be assessing whether
significant variations in gender and nationality biases exist across different
subfields within computer science (e.g., artificial intelligence, systems,
theory). Furthermore, exploring how these biases correlate with broader trends
could provide valuable insights into the dynamics of bias within the field.

## Conclusion

In this study, we analyzed the distribution of genders and nationalities among
students and their advisors. We conducted two-sided hypothesis tests to
examine the presence of bias in gender and home country within advisor-student
relationships. Our findings indicate that there is no gender bias in admission
results. However, our results confirm the existence of bias against
international applicants based on nationality. Additionally, we explored
centrality metrics in the advisor-student relationship network, revealing that
the United States, India, and China are the dominant countries in CS academia,
influencing the composition of students and faculty members in top North
American universities. We investigated the trends in gender and nationality
bias over time and observed a reduction in gender bias, while nationality bias
has shown an increasing pattern. Furthermore, we established a positive
relationship between diversity in the nationalities of research group members
and their academic performance. Lastly, we demonstrated an increase in gender
diversity over time, alongside a decline in nationality diversity.

We acknowledge a limitation regarding the data collected for this study. We
cannot guarantee that each faculty member consistently includes all
individuals on their webpage. While the majority of computer science
professors at high-ranking universities update their homepage at least once a
year, some faculty members may not update information about newly admitted
students as frequently.

Universities can utilize the findings of this study to formulate and implement
policies aimed at promoting diversity and equality among their graduate
students. Furthermore, they can raise awareness among faculty members
regarding the benefits, particularly in terms of scientific achievement, that
arise from having a diverse research team. Universities can also encourage
faculty members to actively consider admitting students from a variety of
nationalities.

## Abbreviations

CS, Computer Science; QS, Quacquarelli Symonds; ACM, Association for Computing
Machinery; MS, Master of Science; Berkeley, University of California,
Berkeley; Caltech, California Institute of Technology; CMU, Carnegie Mellon
University; Columbia, Columbia University; Cornell, Cornell University;
Georgia Tech, Georgia Institute of Technology; Harvard, Harvard University;
McGill, McGill University; MIT, Massachusetts Institute of Technology; NYU,
New York University; Princeton, Princeton University; Stanford, Stanford
University; U of T, University of Toronto; UBC, University of British
Columbia; UChicago, University of Chicago; UCLA, University of California, Los
Angeles; UCSD, University of California, San Diego; UIUC, University of
Illinois at Urbana-Champaign; UMich, University of Michigan-Ann Arbor; UPenn,
University of Pennsylvania; USC, University of Southern California; UT Austin,
University of Texas at Austin; UW, University of Washington; Waterloo,
University of Waterloo; Yale, Yale University.

## Availability of data and materials

The dataset generated and analyzed during the current study is available in
the Advisor Student Data repository, https://github.com/kalhorghazal/Advisor-
Student-Data.

## Competing interests

The authors declare that they have no competing interests.

## Funding

No funding was received for conducting this study.

## Authors’ contributions

GK: Data curation, Formal analysis, Methodology, Investigation, Validation,
Visualization, Writing- Original draft.

TZ: Data curation, Investigation, Validation, Software, Writing- Original
draft.

BB: Conceptualization, Project administration, Supervision, Writing- Reviewing
and Editing.

## Acknowledgements

Thanks to Baharan Khatami for providing some helpful ideas for this research.

## References

  * [1] Sharaievska, I., Kono, S., Mirehie, M.S.: Are we speaking the same language? the experiences of international students and scholars in north american higher education. SCHOLE: A Journal of Leisure Studies and Recreation Education 34(2), 120–131 (2019)
  * [2] Posselt, J.R.: Toward inclusive excellence in graduate education: Constructing merit and diversity in phd admissions. American Journal of Education 120(4), 481–514 (2014)
  * [3] Bollinger, L.: Why diversity matters. The Education Digest 73(2), 26 (2007)
  * [4] Smith, D.G.: Diversity’s promise for higher education: Making it work. (2020)
  * [5] Maruyama, G., Moreno, J.F.: University faculty views about the value of diversity on campus and in the classroom. Does diversity make a difference? Three research studies on diversity in college classrooms, 9–35 (2000)
  * [6] Pitman, T.: Understanding ‘fairness’ in student selection: are there differences and does it make a difference anyway? Studies in Higher Education 41(7), 1203–1216 (2016)
  * [7] Barrera, C.R.: Making u.s. graduate education more diverse. Science 313, 614–614 (2006)
  * [8] Bickel, P.J., Hammel, E.A., O’Connell, J.W.: Sex bias in graduate admissions: Data from berkeley: Measuring bias is harder than is usually assumed, and the evidence is sometimes contrary to expectation. Science 187(4175), 398–404 (1975)
  * [9] Maxwell, S.E., Jones, L.V.: Female and male admission to graduate school: An illustrative inquiry. Journal of Educational Statistics 1(1), 1–37 (1976)
  * [10] Attiyeh, G., Attiyeh, R.: Testing for bias in graduate school admissions. Journal of Human Resources, 524–548 (1997)
  * [11] Barjak, F., Robinson, S.: International collaboration, mobility and team diversity in the life sciences: impact on research performance. Social geography 3(1), 23–36 (2008)
  * [12] AlShebli, B.K., Rahwan, T., Woon, W.L.: The preeminence of ethnic diversity in scientific collaboration. Nature Communications 9(1) (2018)
  * [13] Llorens, A., Tzovara, A., Bellier, L., Bhaya-Grossman, I., Bidet-Caulet, A., Chang, W.K., Cross, Z.R., Dominguez-Faus, R., Flinker, A., Fonken, Y., et al.: Gender bias in academia: A lifetime problem that needs solutions. Neuron 109(13), 2047–2074 (2021)
  * [14] Nielsen, M.W., Bloch, C.W., Schiebinger, L.: Making gender diversity work for scientific discovery and innovation. Nature human behaviour 2(10), 726–734 (2018)
  * [15] Kamerlin, S.C.L.: When we increase diversity in academia, we all win. EMBO reports 21(12) (2020)
  * [16] Powell, K.: These labs are remarkably diverse–here’s why they’re winning at science. Nature 558(7708), 19–23 (2018)
  * [17] Larsen, E.A., Stubbs, M.L.: INCREASING DIVERSITY IN COMPUTER SCIENCE: ACKNOWLEDGING, YET MOVING BEYOND, GENDER. Journal of Women and Minorities in Science and Engineering 11(2), 139–170 (2005)
  * [18] Wilson, C.: Hour of code. ACM Inroads 5(4), 22–22 (2014)
  * [19] Partovi, H.: A comprehensive effort to expand access and diversity in computer science. ACM Inroads 6(3), 67–72 (2015)
  * [20] Garcia-Holgado, A., Vazquez-Ingelmo, A., Verdugo-Castro, S., Gonzalez, C., Gomez, M.C.S., Garcia-Penalvo, F.J.: Actions to promote diversity in engineering studies: a case study in a computer science degree. In: 2019 IEEE Global Engineering Education Conference (EDUCON). IEEE, ??? (2019)
  * [21] Serrano, M.Á., Boguná, M., Vespignani, A.: Extracting the multiscale backbone of complex weighted networks. Proceedings of the national academy of sciences 106(16), 6483–6488 (2009)
  * [22] Blondel, V.D., Guillaume, J.-L., Lambiotte, R., Lefebvre, E.: Fast unfolding of communities in large networks. Journal of Statistical Mechanics: Theory and Experiment 2008(10), 10008 (2008)
  * [23] Traag, V.A., Waltman, L., van Eck, N.J.: From louvain to leiden: guaranteeing well-connected communities. Scientific Reports 9(1) (2019)
  * [24] Diez, D.M., Barr, C.D., Cetinkaya-Rundel, M.: OpenIntro Statistics. OpenIntro Boston, MA, USA:, ??? (2012)
  * [25] Mann, H.B.: Nonparametric tests against trend. Econometrica 13(3), 245 (1945)
  * [26] Wei, D., Li, Y., Zhang, Y., Deng, Y.: Degree centrality based on the weighted network. In: 2012 24th Chinese Control and Decision Conference (CCDC). IEEE, ??? (2012)
  * [27] Freeman, L.C.: Centrality in social networks conceptual clarification. Social networks 1(3), 215–239 (1978)
  * [28] Rényi, A.: On the dimension and entropy of probability distributions. Acta Mathematica Academiae Scientiarum Hungarica 10(1), 193–215 (1959)
  * [29] QS World University Rankings for Computer Science and Information Systems 2021 — topuniversities.com. https://www.topuniversities.com/university-rankings/university-subject-rankings/2021/computer-science-information-systems. [Accessed 30-Jan-2023]
  * [30] Computing Classification System — dl.acm.org. https://dl.acm.org/ccs. [Accessed 30-Jan-2023]
  * [31] Acua: Audience, customer, and user analytics — acua.qcri.org. https://acua.qcri.org/tool/Name2GAN. [Accessed 30-Jan-2023]
  * [32] Li, B., Jacob-Brassard, J., Dossa, F., Salata, K., Kishibe, T., Greco, E., Baxter, N.N., Al-Omran, M.: Gender differences in faculty rank among academic physicians: a systematic review and meta-analysis. BMJ open 11(11), 050322 (2021)
  * [33] Liu, J., Kong, X., Xia, F., Bai, X., Wang, L., Qing, Q., Lee, I.: Artificial intelligence in the 21st century. IEEE Access 6, 34403–34421 (2018)
  * [34] Schönemann, P.H.: On artificial intelligence. Behavioral and Brain Sciences 8(2), 241–242 (1985)
  * [35] Sharma, B., Boet, S., Grantcharov, T., Shin, E., Barrowman, N.J., Bould, M.D.: The h-index outperforms other bibliometrics in the assessment of research performance in general surgery: a province-wide study. Surgery 153(4), 493–501 (2013)
  * [36] Cuny, J., Aspray, W.: Recruitment and retention of women graduate students in computer science and engineering: results of a workshop organized by the computing research association. ACM SIGCSE Bulletin 34(2), 168–174 (2002)
  * [37] Berg, H.M., Ferber, M.A.: Men and women graduate students: Who succeeds and why? The Journal of higher education 54(6), 629–648 (1983)
  * [38] Okahana, H., Feaster, K., Allum, J.: Graduate enrollment and degrees: 2005 to 2015. Washington, DC: Council of Graduate Schools (2016)
  * [39] Sun, Q., Nguyen, T.D., Ganesh, G.: Exploring the study abroad journey: Chinese and indian students in u.s. higher education. Journal of International Consumer Marketing 32(3), 210–227 (2019)

## Figure Legends

Figure 1. The subgraph is shown before (a) and after (b) the application of
the disparity filter algorithm.

Figure 2. Communities are detected using Louvain (c) and Leiden (d) community
detection algorithms.

Figure 3. The numbers displayed on the bars indicate the percentage of
advisors’ gender, while the numbers on the gray section represent the
percentage of faculty members in each rank. The width of each bar corresponds
to the number of faculty members with a specific rank.

Figure 5. The numbers on each column represent the percentage of different
genders in that specific university.

Figure 9. The numbers on the bars represent the percentage of students of each
gender, while the numbers on the gray sections indicate the percentage of
students in each degree category. The width of each bar is proportional to the
number of students in that particular degree category.

Figure 10. The numbers on each column represent the percentage of different
genders in that particular university.

Figure 11. The numbers on each bar represent the percentage of different
citizenships, and the numbers on the gray section indicate the percentage of
students in each degree. The width of each bar is proportional to the number
of students in that particular degree.


# On real analytic functions on closed subanalytic domains

Armin Rainer  Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-
Platz 1, A-1090 Wien, Austria<EMAIL_ADDRESS>
###### Abstract.

We show that a function $f:X\to\mathbb{R}$ defined on a closed uniformly
polynomially cuspidal set $X$ in $\mathbb{R}^{n}$ is real analytic if and only
if $f$ is smooth and all its composites with germs of polynomial curves in $X$
are real analytic. The degree of the polynomial curves needed for this is
effectively related to the regularity of the boundary of $X$. For instance, if
the boundary of $X$ is locally Lipschitz, then polynomial curves of degree $2$
suffice. In this Lipschitz case, we also prove that a function
$f:X\to\mathbb{R}$ is real analytic if and only if all its composites with
germs of quadratic polynomial maps in two variables with images in $X$ are
real analytic; here it is not necessary to assume that $f$ is smooth.

###### Key words and phrases:

Real analyticity on closed sets, cuspidality of sets, Bochnak–Siciak theorem,
subanalytic sets, uniformly polynomially cuspidal sets

###### 2020 Mathematics Subject Classification:

26E05, 26E10, 32B20, 58C20, 58C25

Supported by FWF-Project P 32905-N

##  1\. Introduction

In this note, we are interested in Hartogs-type characterizations of real
analytic functions. Let us recall two fundamental results due to Bochnak and
Siciak: let $U\subseteq\mathbb{R}^{n}$ be a nonempty open set and
$f:U\to\mathbb{R}$ any function.

######  Theorem A ([2, 16]).

The function $f$ is real analytic if and only if $f$ is smooth and the
restriction of $f$ to each affine line that meets $U$ is real analytic.

######  Theorem B ([3, 4]).

The function $f$ is real analytic if and only if the restriction of $f$ to
each affine $2$-plane that meets $U$ is real analytic.

In Theorem A, the assumption that $f$ is smooth cannot be omitted. Recently,
Bochnak, Kollár, and Kucharz [1] proved a global version of Theorem B: a
function $f:M\to\mathbb{R}$ on a real analytic manifold $M$ of dimension
$n\geq 3$ is real analytic if $f|_{N}$ is real analytic for every real
analytic submanifold $N\subseteq M$ that is homeomorphic to the $2$-sphere.

We will investigate versions of these results on closed fat subsets $X$ of
$\mathbb{R}^{n}$ with cusps (that $X$ is _fat_ means that it is contained in
the closure of its interior, i.e., $X=\overline{X^{\circ}}$). Even if $X$ has
Lipschitz boundary, we will have to compose $f$ with germs of quadratic
polynomial maps instead of just affine maps. It will turn out that the maximal
degree of the polynomial maps needed to detect real analyticity is strongly
related to the regularity of the boundary (i.e., to the sharpness of the
cusps).

A recent result of Kucharz and Kurdyka [9] (see C below) shows that, for
subanalytic functions on real analytic manifolds, real analyticity can be
recognized by restriction to real analytic subsets of dimension one. We will
discuss a variant on suitable closed fat subsets $X$ of $\mathbb{R}^{n}$.

This note is a natural continuation of our papers [13] and [14]. Before we can
state the results, some terminology must be introduced.

###  1.1. Plot-analytic functions

Let $d,m,n$ be positive integers. Let $X\subseteq\mathbb{R}^{n}$ be nonempty.
By an _$(m,d)$ -plot in $X$_ we mean the germ $[p]$ at $0$ of a polynomial map
$p=(p_{1},\ldots,p_{n}):\mathbb{R}^{m}\to\mathbb{R}^{n}$

  * •

of degree at most $d$, i.e., $p_{i}\in\mathbb{R}[x_{1},\ldots,x_{m}]$ and
$\max_{1\leq i\leq n}\deg p_{i}\leq d$,

  * •

and image contained in $X$, i.e., there exists a neighborhood $U$ of $0$ in
$\mathbb{R}^{m}$ such that $p(U)\subseteq X$.

We will write $p$ instead of $[p]$; this slight abuse of notation will lead to
no confusion.

Let $\mathcal{P}_{m,d}(X)$ denote the set of all $(m,d)$-plots in $X$. We will
also consider

$\mathcal{P}_{m}(X):=\bigcup_{d=1}^{\infty}\mathcal{P}_{m,d}(X)$

and call its elements _polynomial $m$-plots in $X$_.

For any function $f:X\to\mathbb{R}$ and any $p\in\mathcal{P}_{m}(X)$ it is
meaningful to consider the function germ $f\circ p$ at $0$ in
$\mathbb{R}^{m}$. We define $\mathscr{C}^{\omega}_{m,d}(X)$ to be the set of
all functions $f:X\to\mathbb{R}$ such that

$f\circ p\text{ is a real analytic germ at }0\in\mathbb{R}^{m},\text{ for all
}p\in\mathcal{P}_{m,d}(X).$

Furthermore, we set

$\mathscr{C}^{\omega}_{m}(X):=\bigcap_{d=1}^{\infty}\mathscr{C}^{\omega}_{m,d}(X).$

We call the elements of $\mathscr{C}^{\omega}_{m,d}(X)$ _$(m,d)$ -plot-
analytic functions_. Recall that $f:X\to\mathbb{R}$ is called _arc-analytic_
if $f\circ c$ is real analytic for each germ $c$ of a real analytic arc in
$X$. The arc-analytic functions on $X$ form a subset of
$\mathscr{C}^{\omega}_{1}(X)$.

Let $X\subseteq\mathbb{R}^{n}$ be closed. Let $\mathcal{C}^{\omega}|_{X}$
(resp. $\mathcal{C}^{\infty}|_{X}$) be the set of all functions
$f:X\to\mathbb{R}$ such that there exist an open neighborhood $U$ of $X$ in
$\mathbb{R}^{n}$ and a real analytic (resp. smooth) function
$F:U\to\mathbb{R}$ such that $F|_{X}=f$. Note that each
$f\in\mathcal{C}^{\omega}|_{X}$ is the restriction to $X$ of a holomorphic
function defined on an open neighborhood of $X$ in $\mathbb{C}^{n}$.

###  1.2. UPC sets

Let us recall (cf. [12]) that a closed set $X\subseteq\mathbb{R}^{n}$ is
called _uniformly polynomially cuspidal_ (UPC) if there exist positive
integers $m,D$ and a constant $M>0$ such that for each $x\in X$ there is a
polynomial curve $h_{x}:\mathbb{R}\to\mathbb{R}^{n}$ of degree at most $D$
such that

  1. (1)

$h_{x}(0)=x$,

  2. (2)

$\operatorname{dist}(h_{x}(t),\mathbb{R}^{n}\setminus X)\geq M\,t^{m}$ for all
$x\in X$ and $t\in[0,1]$.

In that case, we say that the UPC set $X$ has the _characteristic_ $(m,D)$.
(The constant $M$ will not be important for us.) Let $\operatorname{char}(X)$
be the set of all characteristics of $X$. Note that if
$(m,D)\in\operatorname{char}(X)$ then
$(m,D)+\mathbb{N}^{2}\subseteq\operatorname{char}(X)$. We define

$d(X):=2\min_{(m,D)\in\operatorname{char}(X)}\max\\{m,D\\}.$

Any UPC set $X$ is fat. We say that $X$ is _simple_ if each $x\in\partial X$
has a basis of neighborhoods $\mathscr{U}$ such that $U\cap X^{\circ}$ is
connected for all $U\in\mathscr{U}$. We shall see in Example 3.3 that,
regarding our results, to be simple is a natural and indispensable condition.

An important class of UPC sets is the collection of all compact fat
subanalytic sets $X\subseteq\mathbb{R}^{n}$, see Pawłucki and Plesniak [12].

###  1.3. Hölder and Lipschitz sets

Let $\alpha\in(0,1]$ and $r,h>0$. The set

$\Gamma^{\alpha}_{n}(r,h):=\Big{\\{}(x^{\prime},x_{n})\in\mathbb{R}^{n-1}\times\mathbb{R}:|x^{\prime}|<r,\,\big{(}\tfrac{|x^{\prime}|}{r}\big{)}^{\alpha}<\tfrac{x_{n}}{h}<1\Big{\\}}$

is a _truncated open $\alpha$-cusp_. By an _$\alpha$ -set_ we mean a closed
fat set $X\subseteq\mathbb{R}^{n}$ such that $X^{\circ}$ has the _uniform
$\alpha$-cusp property_: for each $x\in\partial X$ there exist $\epsilon>0$, a
truncated open $\alpha$-cusp $\Gamma=\Gamma^{\alpha}_{n}(r,h)$, and an
orthogonal linear map $A\in O(n)$ such that $y+A\Gamma\subseteq X^{\circ}$ for
all $y\in X\cap B(x,\epsilon)$. A bounded open set in $\mathbb{R}^{n}$ has the
uniform $\alpha$-cusp property if and only if it has $\alpha$-Hölder boundary;
cf. [14, Remark 2.1]. We say that $X$ is a _Hölder set_ if $X$ is an
$\alpha$-set for some $\alpha\in(0,1]$; $1$-sets are also called _Lipschitz
sets_. Note that Hölder sets are always simple; see [13, Proposition 3.9].

By definition, any compact $\alpha$-set $X$ is a UPC set of characteristic
$(\lceil\alpha^{-1}\rceil,1)$ so that $d(X)\leq 2\lceil\alpha^{-1}\rceil$,
where $\lceil x\rceil$ is the smallest integer $m\geq x$. If $X$ is not a
$\beta$-set with $\lceil\beta^{-1}\rceil<\lceil\alpha^{-1}\rceil$, then
$d(X)=2\lceil\alpha^{-1}\rceil$. For instance, we have

$d(\overline{\Gamma}^{1/m}_{n}(r,h))=2m$

for the closure $\overline{\Gamma}^{1/m}_{n}(r,h)$ of $\Gamma^{1/m}_{n}(r,h)$,
where $m\in\mathbb{N}_{\geq 1}$.

###  1.4. Results

####  1.4.1. Smooth plot-analytic functions

The first result extends A to simple closed UPC sets with a precise control of
the degree of the required polynomial $1$-plots in terms of $d(X)$.

######  Theorem 1.1.

Let $X\subseteq\mathbb{R}^{n}$ be a simple closed UPC set and let $d:=d(X)$.
Then

$\mathscr{C}^{\omega}_{1,d}(X)\cap\mathcal{C}^{\infty}|_{X}=\mathcal{C}^{\omega}|_{X}.$

We shall see in Example 3.1 that $d=d(X)$ is optimal: in general,
$\mathscr{C}^{\omega}_{1,d}(X)\cap\mathcal{C}^{\infty}|_{X}\not\subseteq\mathcal{C}^{\omega}|_{X}$
if $d<d(X)$.

If we do not restrict the degree of the polynomial plots, we may infer:

######  Corollary 1.2.

Let $X\subseteq\mathbb{R}^{n}$ be a simple closed set such that for each
$z\in\partial X$ there exists a closed UPC set $X_{z}$ with $z\in
X_{z}\subseteq X$. Then

$\mathscr{C}^{\omega}_{1}(X)\cap\mathcal{C}^{\infty}|_{X}=\mathcal{C}^{\omega}|_{X}.$

For instance, the corollary applies to the compact subset
$X=X^{\prime}\cup\bigcup_{m=1}^{\infty}X_{m}$ of $\mathbb{R}^{2}$, where
$X_{m}:=\overline{\Gamma}^{1/m}_{2}(\frac{1}{m^{2}},\frac{1}{m})+(2\sum_{\ell=1}^{m}\frac{1}{\ell^{2}}-\frac{1}{m^{2}},-\frac{1}{m})$
and $X^{\prime}=[0,\tfrac{\pi^{2}}{3}]\times[0,1]$. It is simple, since the
cusps $X_{m}$ converge to the point $(\tfrac{\pi^{2}}{3},0)$. The spikes
$X_{m}$ can be replaced by suitable horn-like sets.

####  1.4.2. Plot-analytic functions that are not presupposed to be smooth

Next we will discuss extensions of B to closed fat sets. In the following, we
will not assume that $f$ is smooth.

By a _simplex_ in $\mathbb{R}^{n}$ we mean the convex hull of any collection
of $n+1$ affinely independent points in $\mathbb{R}^{n}$.

######  Theorem 1.3.

Let $X\subseteq\mathbb{R}^{n}$, $n\geq 2$, be a simple closed set such that
for each $z\in\partial X$ there exists a simplex $X_{z}$ with $z\in
X_{z}\subseteq X$. Then

$\mathscr{C}^{\omega}_{2,2}(X)=\mathcal{C}^{\omega}|_{X}.$

This is best possible; see Example 3.2. The assumption is fulfilled for all
Lipschitz sets. At this stage, we do not know if there is an analogue of
Theorem 1.3 for Hölder sets or simple fat closed subanalytic sets. But in
dimension two, we can give a fairly complete answer. Here (for technical
reasons) we need to work with

$d^{\prime}(X):=2\min_{(m,D)\in\operatorname{char}(X)}m\cdot D.$

For all compact Hölder sets $X$ we have $d^{\prime}(X)=d(X)$; in general,
$d^{\prime}(X)\geq d(X)$.

######  Theorem 1.4.

Let $X\subseteq\mathbb{R}^{2}$ be a simple compact fat subanalytic or Hölder
set and let $d^{\prime}:=d^{\prime}(X)$. Then

$\mathscr{C}^{\omega}_{2,d^{\prime}}(X)=\mathcal{C}^{\omega}|_{X}.$

It is likely that in this statement $d^{\prime}=d^{\prime}(X)$ is not optimal.

######  Corollary 1.5.

Let $X\subseteq\mathbb{R}^{2}$ be a simple closed set such that for each
$z\in\partial X$ there exists a compact fat subanalytic set $X_{z}$ with $z\in
X_{z}\subseteq X$. Then

$\mathscr{C}^{\omega}_{2}(X)=\mathcal{C}^{\omega}|_{X}.$

Note that for each Hölder set $X^{\prime}_{z}$ with $z\in
X^{\prime}_{z}\subseteq X$ we may find a compact fat subanalytic set $X_{z}$
with $z\in X_{z}\subseteq X^{\prime}_{z}$; the converse is not always
possible.

####  1.4.3. Curve-analytic functions

We give an application to curve-analytic functions on closed fat sets which
extends a recent result of Kucharz and Kurdyka [9], see C below.

Let $X\subseteq\mathbb{R}^{n}$ be nonempty and $f:X\to\mathbb{R}$ a function.
We say that $f$ is _curve-analytic_ if for each real analytic arc
$c:(-\epsilon,\epsilon)\to\mathbb{R}^{n}$ with image contained in $X$ there
exist $\delta\in(0,\epsilon]$ and a real analytic function $F:U\to\mathbb{R}$
defined on an open neighborhood $U$ of $c(0)$ in $\mathbb{R}^{n}$ such that
$c(t)\in U$ and $F(c(t))=f(c(t))$ for all $t\in(-\delta,\delta)$.

######  Remark 1.6.

If $X$ is a real analytic manifold, then $f:X\to\mathbb{R}$ is curve-analytic
if and only if $f|_{C}$ is real analytic for every locally irreducible real
analytic set $C$ of dimension $1$ in $X$; see [9, Lemma 2.2]. (That $f|_{C}$
is real analytic means that for each $x\in C$ there is a neighborhood $U$ of
$x$ in $X$ and a real analytic function $F:U\to\mathbb{R}$ such that
$f|_{C\cap U}=F|_{C\cap U}$.)

######  Theorem C ([9]).

A subanalytic function $f:X\to\mathbb{R}$ on a real analytic manifold $X$ is
real analytic if and only if it is curve-analytic.

It is an open question, if the assumption that $f$ is subanalytic is
necessary.

By definition, if $f:X\to\mathbb{R}$ is curve-analytic, then $f$ is arc-
analytic, in particular, $f\in\mathscr{C}^{\omega}_{1}(X)$. Thus, any _smooth_
curve-analytic $f:X\to\mathbb{R}$ is the restriction of a real analytic
function, for all $X$ satisfying the assumptions of Corollary 1.2. More
interestingly, without presupposing that $f$ is smooth, we have:

######  Theorem 1.7.

Let $X\subseteq\mathbb{R}^{n}$ be a simple closed set such that one of the
following conditions is satisfied:

  1. (1)

$n$ is arbitrary and for each $z\in\partial X$ there is a simplex $X_{z}$ with
$z\in X_{z}\subseteq X$.

  2. (2)

$n=2$ and for each $z\in\partial X$ there is a compact fat subanalytic set
$X_{z}$ with $z\in X_{z}\subseteq X$.

Let $f:X\to\mathbb{R}$ be any function such that $f|_{X_{z}}$ is subanalytic
for all $X_{z}$ appearing in 1 or 2. Then $f$ is curve-analytic if and only if
$f\in\mathcal{C}^{\omega}|_{X}$.

All the results are proved in Section 2. The examples in Section 3 show that
the connection between the degree of polynomial plots that recognize real
analyticity and the regularity of the boundary is optimal and complement the
investigation.

##  2\. Proofs

###  2.1. Localization of the problems

Proposition 2.2 shows that it suffices to study the problems locally at
boundary points. First we recall a lemma from [13].

######  Lemma 2.1 ([13, Lemma 6.1]).

Let $X\subseteq\mathbb{R}^{n}$ be closed and $U\subseteq\mathbb{R}^{n}$ open
with $U\cap X\neq\emptyset$. Then there exists an open subset $U_{0}\subseteq
U$ with $U_{0}\cap X=U\cap X$ such that for all $x\in U_{0}$ and all $a\in X$
that realize the distance of $x$ to $X$, i.e.,
$\operatorname{dist}(x,a)=\operatorname{dist}(x,X)$, the line segment $[x,a]$
is contained in $U_{0}$.

######  Proposition 2.2.

Let $X\subseteq\mathbb{R}^{n}$ be a closed fat set. Let
$f:X^{\circ}\to\mathbb{R}$ be real analytic. Suppose that for all
$x\in\partial X$ there exists a neighborhood $U_{x}$ of $x$ in
$\mathbb{R}^{n}$ and a real analytic function $F_{x}:U_{x}\to\mathbb{R}$ such
that $F_{x}|_{U_{x}\cap X^{\circ}}=f|_{U_{x}\cap X^{\circ}}$. Then there
exists a real analytic extension of $f$ to an open neighborhood of $X$.

###### Proof.

Let us show that $f$ and the $F_{x}$ glue coherently to a global real analytic
extension of $f$. Invoking Lemma 2.1, we replace $U_{x}$ by the connected
component of $(U_{x})_{0}$ that contains $x$. Then the open cover
$\\{U_{x}:x\in\partial X\\}\cup\\{X^{\circ}\\}$ of $X$ has the property that
for each $z\in U_{x}$ and each $a\in X$ that realizes the distance of $z$ to
$X$ the segment $[z,a]$ lies in $U_{x}$. Let $V$ be a connected component of
$U_{x}\cap U_{y}$. For each $z\in V$ and each $a\in X$ that realizes the
distance of $z$ to $X$ the line segment $[z,a]$ lies in $V$. Since $X$ is fat,
$V$ has nonempty intersection with $X^{\circ}$, and on this intersection
$F_{x}$ and $F_{y}$ coincide with $f$, by assumption. By the identity theorem,
$F_{x}$ and $F_{y}$ coincide on $V$ and hence on $U_{x}\cap U_{y}$, since the
connected component $V$ was arbitrary. ∎

###  2.2. Strongly injective homomorphisms

We will use the following result.

######  Theorem 2.3 ([5, 6]).

Let
$\Phi:\mathbb{K}\\{X_{1},\ldots,X_{n}\\}\to\mathbb{K}\\{Y_{1},\ldots,Y_{k}\\}$
be a homomorphism of convergent power series rings over the field
$\mathbb{K}$. Then the following conditions are equivalent:

  1. (1)

$\Phi$ is _strongly injective_ , i.e., if $\hat{\Phi}$ denotes the natural
extension of $\Phi$ to formal power series, then
$\hat{\Phi}(f)\in\mathbb{K}\\{Y_{1},\ldots,Y_{k}\\}$ implies
$f\in\mathbb{K}\\{X_{1},\ldots,X_{n}\\}$.

  2. (2)

The generic rank of $\Phi$ is $n$.

###  2.3. Proof of Theorem 1.1 and Corollary 1.2

######  Lemma 2.4.

Let $X\subseteq\mathbb{R}^{n}$ be nonempty and closed,
$f\in\mathcal{C}^{\infty}|_{X}$, and $d$ a positive integer. Then the
following conditions are equivalent.

  1. (1)

$f\in\mathscr{C}^{\omega}_{1,d}(X)$.

  2. (2)

$f\in\mathscr{C}^{\omega}_{m,d}(X)$ for all $m\geq 1$.

  3. (3)

$f\in\mathscr{C}^{\omega}_{m,d}(X)$ for some $m\geq 1$.

###### Proof.

(1) $\Rightarrow$ (2) Let $f\in\mathscr{C}^{\omega}_{1,d}(X)$, $m\geq 1$, and
$p\in\mathcal{P}_{m,d}(X)$. Fix a representative of $p$, also denoted by $p$.
So there exists an open neighborhood $U$ of $0\in\mathbb{R}^{m}$ such that
$p(U)\subseteq X$. Let $x\in U$, $v\in\mathbb{S}^{n-1}$, and $q(t):=x+tv$ for
$t$ near $0\in\mathbb{R}$ such that $q(t)\in U$. Then the germ of $q$ belongs
to $\mathcal{P}_{1,1}(U)$ and that of $p\circ q$ to $\mathcal{P}_{1,d}(X)$.
Thus, $f\circ p\circ q$ is real analytic, by (1). Since $f\circ p$ is smooth,
$f\circ p$ is real analytic, by A. (In fact, it is clear from the above that
$f\circ p|_{\ell\cap U}$ is real analytic, where $\ell$ is the affine line
generated by $q$.)

(2) $\Rightarrow$ (3) Trivial.

(3) $\Rightarrow$ (1) This follows easily from the fact that
$\mathcal{P}_{1,d}(X)$ can be identified with a subset of
$\mathcal{P}_{m,d}(X)$ (viewing $x_{1}\mapsto p(x_{1})$ as
$(x_{1},\ldots,x_{m})\mapsto p(x_{1})$). ∎

###### Proof of Theorem 1.1.

The inclusion
$\mathcal{C}^{\omega}|_{X}\subseteq\mathscr{C}^{\omega}_{1,d}(X)\cap\mathcal{C}^{\infty}|_{X}$
is clear. Suppose that
$f\in\mathscr{C}^{\omega}_{1,d}(X)\cap\mathcal{C}^{\infty}|_{X}$. By A,
$f|_{X^{\circ}}$ is real analytic. By Proposition 2.2, it suffices to show
that for each $x\in\partial X$ there exist a fat set $\Pi_{x}$ such that
$x\in\partial\Pi_{x}$ and $\Pi_{x}^{\circ}\subseteq X^{\circ}$, a neighborhood
$U_{x}$ of $x$ in $\mathbb{R}^{n}$, and a real analytic function
$F_{x}:U_{x}\to\mathbb{R}$ such that
$f|_{U_{x}\cap\Pi_{x}^{\circ}}=F_{x}|_{U_{x}\cap\Pi_{x}^{\circ}}$. Indeed,
since $X$ is simple, we may assume (after possibly shrinking $U_{x}$) that
$U_{x}\cap X^{\circ}$ is connected. Thus, $f|_{U_{x}\cap
X^{\circ}}=F_{x}|_{U_{x}\cap X^{\circ}}$, by the identity theorem, and
Proposition 2.2 implies the statement.

Fix $x\in\partial X$. Since $X$ is a UPC set with $d=d(X)$, we have
$d=2\max\\{m,D\\}$ for some $(m,D)\in\operatorname{char}(X)$. So there is a
polynomial curve $h_{x}:\mathbb{R}\to\mathbb{R}^{n}$ of degree at most $D$
such that $h_{x}(0)=0$ and

$\operatorname{dist}(h_{x}(t),\mathbb{R}^{n}\setminus X)\geq M\,t^{m},\quad
x\in X,\,t\in[0,1].$

We have $h_{x}(t)=t^{p}\tilde{h}_{x}(t)$, for some integer $p\geq 1$, where
$\tilde{h}_{x}(0)\neq 0$. Let $v_{1}:=\tilde{h}_{x}(0)/|\tilde{h}_{x}(0)|$ and
choose unit vectors $v_{2},\ldots,v_{n}$ in $\mathbb{R}^{n}$ such that
$v_{1},\ldots,v_{n}$ forms a basis of $\mathbb{R}^{n}$. Then

$\theta_{x}(s_{1},\ldots,s_{n}):=h_{x}(s_{1})+s_{2}^{m}v_{2}+\cdots+s_{n}^{m}v_{n}$

is a polynomial map $\theta_{x}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ of degree at
most $\max\\{m,D\\}$ which takes the set $C:=\\{(s_{1},\ldots,s_{2}):0\leq
s_{1}\leq 1,\,|s_{i}|\leq\big{(}\tfrac{M}{2(n-1)}\big{)}^{1/m}s_{1}\text{ for
}i\geq 2\\}$ into $X$. Indeed,

$\displaystyle\operatorname{dist}(\theta_{x}(s_{1},\ldots,s_{n}),\mathbb{R}^{n}\setminus
X)$ $\displaystyle\geq\operatorname{dist}(h_{x}(s_{1}),\mathbb{R}^{n}\setminus
X)-|s_{2}|^{m}-\cdots-|s_{n}|^{m}$ $\displaystyle\geq
Ms_{1}^{m}-\sum_{j=2}^{n}\tfrac{M}{2(n-1)}s_{1}^{m}$
$\displaystyle=\tfrac{M}{2}s_{1}^{m},$

if $(s_{1},\ldots,s_{n})\in C$. Since $C$ is a $1$-set and $0\in\partial C$,
there is a linear isomorphism $\ell:\mathbb{R}^{n}\to\mathbb{R}^{n}$ such that
$\ell(S)\subseteq C$, where $S$ is the convex hull of $0,e_{1},\ldots,e_{n}$.
There is an open neighborhood $U$ of $0$ in $\mathbb{R}^{n}$ such that
$q:\mathbb{R}^{n}\to\mathbb{R}^{n}$ given by

$q(x_{1},\ldots,x_{n}):=(x_{1}^{2},\ldots,x_{n}^{2})$

takes $U$ into $S$. Thus the polynomial map
$\psi_{x}:=\theta_{x}\circ\ell\circ q:\mathbb{R}^{n}\to\mathbb{R}^{n}$ of
degree at most $d$ takes $U$ into $X$ and so we may consider the composite
$f\circ\psi_{x}|_{U}$. By Lemma 2.4, $f\circ\psi_{x}|_{U}$ is real analytic.
Since $f$ is smooth, by assumption, we have its formal Taylor series
$F_{x}=T_{x}f$ at $x$. Then the power series
$F_{x}\circ\psi_{x}=T_{0}(f\circ\psi_{x})$ has positive radius of convergence.
By Theorem 2.3, $F_{x}$ has positive radius of convergence and thus defines a
real analytic function, again denoted by $F_{x}$, on an open neighborhood
$U_{x}$ of $x$ in $\mathbb{R}^{n}$. Clearly,
$f|_{U_{x}\cap\Pi_{x}}=F_{x}|_{U_{x}\cap\Pi_{x}}$, where
$\Pi_{x}:=\psi_{x}(U)$. This shows the claim and hence completes the proof. ∎

###### Proof of Corollary 1.2.

This follows easily from Theorem 1.1 (applied to each $X_{z}$) and Proposition
2.2. Note that $X$ is fat. ∎

###  2.4. Proof of Theorem 1.3

We need a variant of Lemma 2.4, where $f:X\to\mathbb{R}$ is not necessarily
smooth.

######  Lemma 2.5.

Let $X\subseteq\mathbb{R}^{n}$ be nonempty, $f:X\to\mathbb{R}$ any function,
and $d$ a positive integer. Then the following conditions are equivalent.

  1. (1)

$f\in\mathscr{C}^{\omega}_{2,d}(X)$.

  2. (2)

$f\in\mathscr{C}^{\omega}_{m,d}(X)$ for all $m\geq 2$.

  3. (3)

$f\in\mathscr{C}^{\omega}_{m,d}(X)$ for some $m\geq 2$.

###### Proof.

(1) $\Rightarrow$ (2) The proof is analogous to the one of Lemma 2.4 except
that now $q$ is a $(2,1)$-plot that parameterizes a piece of an affine
$2$-plane contained in $U$ and we use B.

(2) $\Rightarrow$ (3) Trivial.

(3) $\Rightarrow$ (1) $\mathcal{P}_{2,d}(X)$ can be seen as a subset of
$\mathcal{P}_{m,d}(X)$ if $m\geq 2$. ∎

######  Lemma 2.6.

Let $q:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be the map defined by
$q(x_{1},\ldots,x_{n}):=(x_{1}^{2},\ldots,x_{n}^{2})$. Let
$I_{r}:=(-r,r)^{n}\subseteq\mathbb{R}^{n}$. Then $Q_{r}:=q(I_{r})$ is a
neighborhood of $0$ in the first orthant $Q:=\\{x\in\mathbb{R}^{n}:x_{i}\geq
0\text{ for all }1\leq i\leq n\\}$. For any function $f:Q_{r}\to\mathbb{R}$
such that $f\circ q:I_{r}\to\mathbb{R}$ is real analytic there exists a real
analytic function $F:U\to\mathbb{R}$ defined in a neighborhood $U$ of $0$ in
$\mathbb{R}^{n}$ such that $F|_{U\cap Q_{r}}=f|_{U\cap Q_{r}}$.

###### Proof.

This is a consequence of a theorem of Luna [10], since
$x_{1}^{2},\ldots,x_{d}^{2}$ are the basic invariants of the natural action of
the group $G=\mathbb{Z}_{2}\times\cdots\times\mathbb{Z}_{2}$ on
$\mathbb{R}\times\cdots\times\mathbb{R}$.

Let us sketch an alternative proof. Take a smooth function
$\chi:\mathbb{R}^{n}\to\mathbb{R}$ such that $\chi\equiv 1$ in a neighborhood
of $0$ and $\chi\equiv 0$ outside $I_{s}$, where $s<r^{2}$. Then
$\tilde{f}:=\chi f$ extends by zero to a function defined on all of
$Q=q(\mathbb{R}^{n})$ and $\tilde{f}\circ q:\mathbb{R}^{n}\to\mathbb{R}$ is a
$G$-invariant smooth function. Thus, by [7] or [15], there is a smooth
function $g:\mathbb{R}^{n}\to\mathbb{R}$ such that $\tilde{f}\circ q=g\circ
q$, i.e., $g$ extends $\tilde{f}$ to $\mathbb{R}^{n}$. In particular, $f$ has
a smooth extension to a neighborhood of $0$ in $\mathbb{R}^{n}$. If $F$
denotes the Taylor series at $0$ of $f$ (equivalently, of $g$), then $F\circ
q=T_{0}(f\circ q)$ has positive radius of convergence, since $f\circ q$ is
real analytic. By Theorem 2.3, $F$ converges and defines a real analytic
function with the desired properties. ∎

###### Proof of Theorem 1.3.

We have to show
$\mathscr{C}^{\omega}_{2,2}(X)\subseteq\mathcal{C}^{\omega}|_{X}$. Let
$f\in\mathscr{C}^{\omega}_{2,2}(X)$. By Lemma 2.5,
$f\in\mathscr{C}^{\omega}_{n,2}(X)$. Fix $z\in\partial X$ and let $X_{z}$ be a
simplex such that $z\in X_{z}\subseteq X$. There is an affine isomorphism
$\ell:\mathbb{R}^{n}\to\mathbb{R}^{n}$ with $\ell(0)=z$ such that $X_{z}$ is
the image of the convex hull of $0,e_{1},\ldots,e_{n}$ under $\ell$. So there
is an open neighborhood $U$ of $0\in\mathbb{R}^{n}$ such that $f\circ\ell\circ
q:U\to\mathbb{R}$ is well-defined and real analytic. It follows from Lemma 2.6
that $f\circ\ell|_{q(U)}$ has a real analytic extension to some neighborhood
of $0$ and thus $f|_{\ell(q(U))}$ has a real analytic extension to some
neighborhood of $z$. Since $z$ was arbitrary, the theorem now follows from
Proposition 2.2. ∎

###  2.5. Proof of Theorem 1.4 and Corollary 1.5

Let $d\geq 3$ and consider the map
$\sigma=(\sigma_{1},\sigma_{2}):\mathbb{R}^{2}\to\mathbb{R}^{2}$ defined by

$\sigma_{1}(x,y):=x^{2}+y^{2},\quad\sigma_{2}(x,y):=\mathrm{Re}((x+iy)^{d}).$

Then $\sigma(\mathbb{R}^{2})=\\{(x,y)\in\mathbb{R}^{2}:x\geq 0,\,|y|\leq
x^{d/2}\\}=:\Sigma$. In fact, in polar coordinates
$\sigma_{1}(re^{i\theta})=r^{2}$ and
$\sigma_{2}(re^{i\theta})=r^{d}\cos(d\theta)$ so that $\sigma$ maps the circle
with radius $r$ around $0$ onto the vertical line segment between the points
$(r^{2},\pm r^{d})$. The polynomials $\sigma_{1}$ and $\sigma_{2}$ are the
basic invariants of the dihedral group $G=I^{d}_{2}$ consisting of the
orthogonal transformations of $\mathbb{R}^{2}$ that preserve the regular
$d$-gon.

######  Lemma 2.7.

Let $I_{r}=(-r,r)^{2}\subseteq\mathbb{R}^{2}$. Then
$\Sigma_{r}:=\sigma(I_{r})$ is a neighborhood of $0$ in $\Sigma$. For any
function $f:\Sigma_{r}\to\mathbb{R}$ such that
$f\circ\sigma:I_{r}\to\mathbb{R}$ is real analytic there exists a real
analytic function $F:U\to\mathbb{R}$ defined in a neighborhood $U$ of $0$ in
$\mathbb{R}^{2}$ such that $F|_{U\cap\Sigma_{r}}=f|_{U\cap\Sigma_{r}}$.

###### Proof.

Follow the proof of Lemma 2.6 and make the obvious modifications. ∎

###### Proof of Theorem 1.4.

We first assume that $X\subseteq\mathbb{R}^{2}$ is a compact Hölder set. Then
$d:=d(X)=d^{\prime}(X)$ and for each $x\in\partial X$ there is an affine
isomorphism $\ell:\mathbb{R}^{2}\to\mathbb{R}^{2}$ with $\ell(0)=x$ and $r>0$
such that $\ell(\Sigma_{r})\subseteq X$. If
$f\in\mathscr{C}^{\omega}_{2,d}(X)$, then $f\circ\sigma:I_{r}\to\mathbb{R}$ is
well-defined and real analytic. Now the result follows from Lemma 2.7 and
Proposition 2.2.

Let now $X\subseteq\mathbb{R}^{2}$ be a simple compact fat subanalytic set and
$d^{\prime}=d^{\prime}(X)$. Let $f\in\mathscr{C}^{\omega}_{2,d^{\prime}}(X)$.
Fix $x\in\partial X$. There exist $(m,D)\in\operatorname{char}(X)$ with
$d^{\prime}=2mD$, a polynomial curve $h_{x}$ of degree at most $D$, and a
basis $v_{1},v_{2}$ as in the proof of Theorem 1.1 (on page 2.3). Consider the
polynomial map $\Theta_{x}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ given by

$\Theta_{x}(s_{1},s_{2}):=h_{x}(s_{1})+s_{2}v_{2}.$

It has degree at most $D$ and takes the cusp $S=\\{(s_{1},s_{2}):0\leq
s_{1}\leq 1,\,|s_{2}|\leq\tfrac{M}{2}s_{1}^{m}\\}$ into
$X^{\circ}\cup\\{x\\}$. The function $f\circ\Theta_{x}|_{S}$ belongs to
$\mathscr{C}^{\omega}_{2,2m}(S)$ and hence it is the restriction of a real
analytic function defined on a neighborhood of $S$, by Theorem 1.4 for Hölder
sets (since $d^{\prime}(S)=d(S)=2m$). We may compute

$\partial_{s_{2}}^{k}(f\circ\Theta_{x})(s_{1},s_{2})=d^{k}_{v_{2}}f(\Theta_{x}(s_{1},s_{2})),\quad
k\in\mathbb{N},$

for $(s_{1},s_{2})\in S^{\circ}$ (since $f$ is real analytic in $X^{\circ}$,
by B) and letting $(s_{1},s_{2})\to(0,0)$ we see that the directional
derivatives on the right-hand side extend continuously to $x$. This remains
true for all $v_{2}$ in a small neighborhood of the chosen $v_{2}$. By the
proof of [13, Theorem 1.14] or [14, Theorem C], we find that $f$ is the
restriction to $X$ of a $\mathcal{C}^{\infty}$-function defined on
$\mathbb{R}^{2}$. The cited proof shows that the derivatives of all orders
extend continuously to the boundary of $X$. Since compact fat subanalytic sets
are Whitney $p$-regular, for some positive integer $p$, these derivatives
define a Whitney jet of class $\mathcal{C}^{\infty}$ on $X$ which extends to a
smooth function on $\mathbb{R}^{2}$.

Now it suffices to invoke Theorem 1.1 and Lemma 2.4 (since $d^{\prime}(X)\geq
d(X)$) in order to complete the proof of Theorem 1.4. ∎

Corollary 1.5 follows easily from Theorem 1.4 and Proposition 2.2.

###  2.6. Proof of Theorem 1.7

If $f$ is the restriction of a real analytic function, then clearly $f$ is
curve-analytic. So suppose that $f$ is curve-analytic.

######  Lemma 2.8.

Let $m\geq 1$. Let $p\in\mathcal{P}_{m}(X)$ be such that $f\circ p$ is
subanalytic (for instance, if the image of $p$ is contained in $X_{z}$ for
some $z$). Then $f\circ p$ is real analytic.

###### Proof.

Let $c:(-\epsilon,\epsilon)\to\mathbb{R}^{m}$ be a real analytic arc in the
domain of definition of (a representative of) $p$. Since $f$ is curve-
analytic, there exist $\delta\in(0,\epsilon]$ and a real analytic function
$F:U\to\mathbb{R}$ on a neighborhood $U$ of $p(c(0))$ in $\mathbb{R}^{n}$ such
that $p(c(t))\in U$ and $F(p(c(t)))=f(p(c(t)))$ for all
$t\in(-\delta,\delta)$. Thus $f\circ p$ is curve-analytic and subanalytic so
that $f\circ p$ is real analytic, by C. ∎

By Lemma 2.8, $f|_{X_{z}}\in\mathscr{C}^{\omega}_{2}(X_{z})$ and hence
$f|_{X_{z}}\in\mathcal{C}^{\omega}|_{X_{z}}$, thanks to Theorem 1.3 and
Theorem 1.4. Then Theorem 1.7 follows from Proposition 2.2.

######  Remark 2.9.

An important step in the proof of C in [9] is the criterion that an arc-
analytic subanalytic function $f$ defined on an open subset $U$ of
$\mathbb{R}^{n}$ is real analytic at a point $x\in U$ if and only if $f$
satisfies the _$(p,q)$ -test at $x$_ for all pairs $(p,q)$ of positive
integers. That means that for every pair of linearly independent vectors
$\xi,\nu\in\mathbb{R}^{n}$ and all $a,b\in\mathbb{R}$,
$\varphi(t):=f(x+at^{p}\xi+bt^{q}\nu)$ is a real analytic germ at
$0\in\mathbb{R}$ and the support of $\varphi$ is contained in
$\mathbb{N}p+\mathbb{N}q$. Theorem 1.7 suggests that often not all
$(p,q)$-tests are necessary to detect real analyticity.

##  3\. Examples

The first example shows that in general the degree of polynomial test plots in
Theorem 1.1 is optimal.

######  Example 3.1.

Let $D$ be a positive integer. The horn $H=\\{(x,y)\in\mathbb{R}^{2}:0\leq
x\leq 1,\,x^{D}\leq y\leq 2x^{D}\\}$ is a UPC set with characteristic $(D,D)$
and $d(H)=2D$. Indeed, every polynomial curve $h:\mathbb{R}\to\mathbb{R}^{2}$
with $h(0)=0$ and $h([0,1])\subseteq H$ must be of degree at least $D$ and
$\operatorname{dist}(h(t),\mathbb{R}^{2}\setminus H)\lesssim t^{D}$ for
$t\in[0,1]$. Every polynomial $1$-plot in $H$ through $0$ must be of degree at
least $2D$.

The function $f:H\to\mathbb{R}$ given by $f(x,y):=e^{-1/(x^{2}+y^{2})}$ for
$(x,y)\neq(0,0)$ and $f(0,0):=0$ belongs to $\mathcal{C}^{\infty}|_{H}$ and to
$\mathscr{C}^{\omega}_{1,2D-1}(H)$, simply because $f$ is real analytic away
from $0$ and for all $p\in\mathcal{P}_{1,2D-1}(H)$ we have $p(0)\neq 0$.

On the other hand, $f\not\in\mathscr{C}^{\omega}_{1,2D}(H)$ either by Theorem
1.1 or by observing that for $p(t):=(t^{2},t^{2D})$, which lies in $H$ for all
$t$ near $0$, the composite

$(f\circ p)(t)=\exp\Big{(}-\frac{1}{t^{4}(1+t^{4(D-1)})}\Big{)}$

has no analytic extension to $t=0$.

The following easy example shows that Theorem 1.3 is best possible.

######  Example 3.2.

Let $X=\\{x\in\mathbb{R}^{n}:x_{i}\geq 0\text{ for all }1\leq i\leq n\\}$ be
the first orthant. A plot $p\in\mathcal{P}_{m,1}(X)$ ($m$ arbitrary) can
satisfy $p(0)=0$ only if $p\equiv 0$. In other words, an arbitrary function
$f\in\mathscr{C}^{\omega}_{m,1}(X)$ is unrestricted at $0$.

The next example shows that the assumption of being simple is essential.

######  Example 3.3.

Consider the set $X_{1}:=\\{(x,y)\in\mathbb{R}^{2}:x\geq 0,\,x^{\sqrt{2}}\leq
y\leq x^{\sqrt{2}}+x^{2}\\}$. Every smooth curve $c$ in $X_{1}$ must vanish to
infinite order on $c^{-1}(0)$; see [14, Example 6.7]. Thus
$\mathscr{C}^{\omega}_{m}(X_{1})$ ($m$ arbitrary) contains elements that do
not have a real analytic extension.

Let $X_{2}:=\\{(x,y)\in\mathbb{R}^{2}:x\leq 0\\}$ be the left half-plane and
consider $X:=X_{1}\cup X_{2}$ which is not simple. But for each point $z$ of
$\partial X$ there exists a simple fat compact subanalytic set $X_{z}$ such
that $z\in X_{z}\subseteq X$. Again it is easy to find (even smooth) elements
of $\mathscr{C}^{\omega}_{m}(X)$ that do not admit a real analytic extension.

Note that it was wrongly stated in [13, Corollary 1.17] that, in a related
result, the assumption that the set is simple is not needed.

There are algebraic sets $X$ in $\mathbb{R}^{2}$ for which
$\mathscr{C}^{\omega}_{m}(X)$ ($m$ arbitrary) is essentially bigger than
$\mathcal{C}^{\omega}|_{X}$:

######  Example 3.4.

Consider the algebraic set $X=\\{(x,y)\in\mathbb{R}^{2}:x^{3}=y^{2}\\}$ and
the function $\varphi:X\to\mathbb{R}$ defined by $\varphi(x,y)=y^{1/3}$. Then
$\varphi\in\mathscr{C}^{\omega}_{m}(X)$ for all $m\geq 1$. Indeed, let
$p=(p_{1},p_{2})\in\mathcal{P}_{m}(X)$. The composite $u=\varphi\circ p$
satisfies $u^{2}=p_{1}$ and $u^{3}=p_{2}$ so that $u$ is smooth, by a result
of Joris [8], and in turn real analytic, by [11, Proposition VI.3.11].

In fact, we can completely describe $\mathscr{C}^{\omega}_{m}(X)$: for all
$m\geq 1$,

$\mathscr{C}^{\omega}_{m}(X)=\varphi^{*}\mathcal{C}^{\omega}(\mathbb{R}):=\\{g\circ\varphi:g\in\mathcal{C}^{\omega}(\mathbb{R})\\}.$

That $\varphi^{*}\mathcal{C}^{\omega}(\mathbb{R})$ is contained in
$\mathscr{C}^{\omega}_{m}(X)$ is an easy consequence of the fact that
$\varphi\in\mathscr{C}^{\omega}_{m}(X)$. Conversely, suppose that
$f\in\mathscr{C}^{\omega}_{m}(X)$. Then $g(t):=f(t^{2},t^{3})$ defines an
element $g\in\mathcal{C}^{\omega}(\mathbb{R})$ satisfying
$g(\varphi(x,y))=f(y^{2/3},y)=f(x,y)$.

## References

  * [1] J. Bochnak, J. Kollár, and W. Kucharz, _Checking real analyticity on surfaces_ , Journal de Mathématiques Pures et Appliquées 133 (2020), 167–171.
  * [2] J. Bochnak and J. Siciak, _Analytic functions in topological vector spaces_ , Studia Math. 39 (1971), 77–112.
  * [3] by same author, _Analytic functions in topological vector spaces_ , mimeographed preprint, IHES, 1971.
  * [4] by same author, _A characterization of analytic functions of several variables_ , Ann. Polon. Math. (2018), DOI:10.4064/ap180119-26-3.
  * [5] P. M. Eakin and G. A. Harris, _When $F(f)$ convergent implies $f$ is convergent_, Math. Ann. 229 (1977), no. 3, 201–210.
  * [6] A. M. Gabrièlov, _Formal relations among analytic functions_ , Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 1056–1090.
  * [7] G. Glaeser, _Fonctions composées différentiables_ , Ann. of Math. (2) 77 (1963), 193–209.
  * [8] H. Joris, _Une ${\mathcal{C}}^{\infty}$-application non-immersive qui possède la propriété universelle des immersions_, Arch. Math. (Basel) 39 (1982), no. 3, 269–277.
  * [9] W. Kucharz and K. Kurdyka, _Analytic functions and Nash functions along curves_ , Selecta Math. (N.S.) 29 (2023), no. 1, Paper No. 16, 12.
  * [10] D. Luna, _Fonctions différentiables invariantes sous l’opération d’un groupe réductif_ , Ann. Inst. Fourier (Grenoble) 26 (1976), no. 1, ix, 33–49.
  * [11] B. Malgrange, _Ideals of differentiable functions_ , Tata Institute of Fundamental Research Studies in Mathematics, No. 3, Tata Institute of Fundamental Research, Bombay, 1967.
  * [12] W. Pawłucki and W. Pleśniak, _Markov’s inequality and $C^{\infty}$ functions on sets with polynomial cusps_, Math. Ann. 275 (1986), no. 3, 467–480.
  * [13] A. Rainer, _Arc-smooth functions on closed sets_ , Compos. Math. 155 (2019), 645–680.
  * [14] by same author, _Arc-smooth functions and cuspidality of sets_ , Journal d'Analyse Mathématique (2023), to appear, arXiv:2112.14163.
  * [15] G. W. Schwarz, _Smooth functions invariant under the action of a compact Lie group_ , Topology 14 (1975), 63–68.
  * [16] J. Siciak, _A characterization of analytic functions of $n$ real variables_, Studia Math. 35 (1970), 293–297.