lmbda = (itx[0]-n3[0]) / (n2[0]-n3[0])

if (lmbda<0 or lmbda>1):

return [-1,-1]

return itx

The next function, intersect_all(x), solves for the edge that the $x$-th nodal
ray intersects with. This is achieved by finding the intersections of the
$x$-th nodal ray with all other edges, throwing away invalid intersections,
and keeping the one with the shortest distance to the $x$-th vertex.

⬇

def intersect_all (x):

# solve the intersecting edge for the x-th nodal ray

global n

global nodes

# the variables for the intersecting edge

min_edge = x

min_itx = []

min_dis = math.inf

for i in range(n):

# i is adjacent to x

if (i==x or i==(x-1)%n):

continue

# the intersection of x-th nodal ray

# and i-th edge is invalid

itx = intersect_one(x,i)

if (itx == [-1,-1]):

continue

# maintain the closest intersection

dis = dist(nodes[x].vertex, itx)

if (dis < min_dis):

min_edge = i

min_itx = itx

min_dis = dis

return (min_edge, min_itx)

With the above foundations, the function mutate(x) calculates the polygon
after mutating the $x$-th nodal ray. It has two secondary helper functions
mutate_counterclockwise and mutate_clockwise, depending on whether the
intersecting edge is to the left or right of the the nodal ray. Here we
demonstrate the code for the former as the two are extremely similar.

⬇

def mutate_counterclockwise (head, tail, itx):

# mutate with nodal_ray < intersecting edge

global n

global nodes

mat = solve_matrix( nodes[head].nodal_ray,

nodes[head].nodal_ray,

nodes[head].edge,

nodes[(head-1)%n].edge )

# construct the new node

new_length = nodes[tail].affine_length

* dist(itx,

nodes[(tail+1)%n].vertex)

/ dist(nodes[tail].vertex,

nodes[(tail+1)%n].vertex)

new = node(itx, [-nodes[head].nodal_ray[0],

-nodes[head].nodal_ray[1]],

nodes[tail].edge, new_length)

nodes = np.insert(nodes, tail+1, new)

# adjust the head and tail node

nodes[tail].affine_length -= new_length

nodes[head-1].affine_length += nodes[head].affine_length

nodes = np.delete(nodes, head)

# update remaining nodes

for i in range(head, tail):

pre = nodes[(i-1)%n]

nodes[i].vertex[0] = pre.vertex[0]

+ pre.affine_length*pre.edge[0]

nodes[i].vertex[1] = pre.vertex[1]

+ pre.affine_length*pre.edge[1]

nodes[i].nodal_ray = dot(mat, nodes[i].nodal_ray)

nodes[i].edge = dot(mat, nodes[i].edge)

def mutate (x):

# mutate once by x-th nodal_ray

global n

global nodes

# y is the intersecting edge

# itx is the intersection point

(y, itx) = intersect_all(x)

if (x<y):

mutate_counterclockwise(x,y,itx)

return y

else:

mutate_clockwise(y,x,itx)

return y+1

Finally, we have two interface functions plot_nodes and print_embd that output
direct information for use. The first, plot_nodes, plots the polygon with
respect to edge length ratio; print_embd prints the staircase coordinate
$(z,\lambda)$ such that $E(1,z)\stackrel{{\scriptstyle
s}}{{\hookrightarrow}}P(\lambda,\lambda b)$. The embedding is constructed by
fitting the right triangle $\Delta OXY$ into the polygon. Below is an example
that gets the $v^{2}y^{2}$ example above.

⬇

b = (6 + 5 * D(30).sqrt()) / 12

init_polydisk(b)

mutate(2)

mutate(2)

mutate(1)

mutate(1)

plot_nodes()

print_embd()

It should be easy to generalize the initialization functions for other types
of polygons beyond rectangles, such as triangles and trapezoids. For a
complete file of the code, see the Github Repository.

## References

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  * [U2] M. Usher, private communication, 2022.


#  Climate-Resilient UAVs: Enhancing Energy-Efficient B5G Communication in
Harsh Environments

Abdu Saif1 Saeed Hamood Alsamhi2 Edward Curry2
1Faculty of Engineering  Taiz University  Taiz  Yemen
Email<EMAIL_ADDRESS>
2Insight Centre for Data Analytics  University of Galway  Galway  Ireland
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The deployment of Beyond Fifth Generation (B5G) networks is increasingly vital
yet challenging due to harsh environmental conditions exacerbated by climate
change. Unmanned Aerial Vehicles (UAVs) have emerged as critical enablers for
B5G communication in adverse weather conditions, including rain, fog, and
snow. This paper investigates the synergy between climate-resilient UAVs and
energy-efficient B5G communication. We assess UAV coverage and energy
efficiency across varying elevation angles in adverse weather. Our findings
highlight the pronounced impact of rainfall on UAV coverage and the
substantial influence of fog and snow on communication dynamics. Our paper
unveils significant improvements in energy efficiency attributed to reduced
interference, enhanced data transmission rates, and optimal channel gain under
diverse weather conditions. This paper addresses the challenges of harsh
environments, emphasizing the potential of climate-resilient UAVs to enhance
energy-efficient B5G communication. It underscores the importance of
technology in mitigating climate change’s impact on communication systems,
charting a path toward a more sustainable and resilient future.

###### Index Terms:

UAV-assisted communication, Harsh environments, Meteorological impacts, Energy
efficiency, Outage probability, Spectrum efficiency, B5G.

##  I Introduction

The deployment of Beyond Fifth Generation (B5G) networks represents a pivotal
advancement in contemporary communication systems, poised to usher in an era
of unparalleled connectivity and capabilities. The networks are designed to
transcend the boundaries of their predecessors, offering revolutionary
enhancements in data rates, network capacity, and latency reduction [1]. The
promises of B5G networks are far-reaching, encompassing applications that
range from immersive augmented reality experiences to seamless Internet of
Things (IoT) connectivity and mission-critical communications for autonomous
vehicles and industrial automation[2]. One of the most pressing issues is the
increasing prevalence and severity of adverse environmental conditions,
exacerbated by the changing climate. The effects of the conditions, such as
heavy rainfall, dense fog, and snowfall, pose formidable obstacles to the
reliable operation of wireless communication systems [3]. Rain, for instance,
can attenuate radio signals, leading to signal degradation and reduced
coverage area. Similarly, fog can scatter signals, causing signal loss and
impairing the quality of communication links. In snowy conditions, the
accumulation of ice and snow on communication equipment can disrupt signal
propagation and, in some cases, cause equipment failure. In response to these
challenges, Unmanned Aerial Vehicles (UAVs) have emerged as a disruptive
technology, offering a promising solution to enhance communication in harsh
environments. UAVs can swiftly navigate and adapt to challenging
meteorological conditions, thus making UAVs ideal candidates for providing
coverage and support to GNs in need [4],[5] offering an up-and-coming solution
to enhance communication in harsh environmental conditions. UAVs have
demonstrated exceptional adaptability and agility, which make them ideal
candidates for providing critical coverage and support to GNs operating in
challenging environments [4]. Unlike traditional fixed communication
infrastructure, UAVs can autonomously reposition themselves in real-time to
optimize signal transmission and reception, ensuring uninterrupted
communication even in adverse weather phenomena such as heavy rain, thick fog,
or snowstorms. This capability significantly enhances the reliability of
communication networks in harsh environments, where maintaining a consistent
and high-quality connection is paramount [4]. UAVs can significantly reduce
routing overhead. Traditional GNs in harsh environments often rely on complex
routing schemes to circumvent obstacles and maintain connectivity. With UAV’s
direct line-of-sight capability and adaptability, UAVs streamline routing and
reduce the complexity of data transmission, leading to more efficient network
operation [6].

Moreover, UAV-assisted cellular communication represents a vital technology
for fulfilling the dynamic communication requirements of modern society[7]. As
traditional communication infrastructure is susceptible to malfunction and
disruption during natural disasters and extreme weather events [8],[9] UAVs
provide swift recovery, ensuring that communication services remain available
despite adversity[10]. Furthermore, the adaptability and resilience of UAVs
extend to situations where communication infrastructure is temporarily
overloaded or incapacitated due to unexpected surges in network traffic, as
often occurs during major events or emergencies. In these cases, UAVs can be
rapidly deployed to offload traffic, ensuring that critical communication
services, including emergency calls and data transmission, continue to operate
smoothly [11]. This paper delves into the intersection of UAVs, adverse
meteorological conditions, and B5G communication, evaluating the role of UAVs
in enhancing energy-efficient communication amidst rain, fog, and snow. The
assessment encompasses a range of performance metrics, including energy
efficiency, outage probability, spectrum efficiency, and path loss. By
addressing these critical aspects, we aim to shed light on the potential of
UAV-assisted systems to thrive in harsh environments and to serve as viable
replacements for dysfunctional GNs in such scenarios.

###  I-A Motivation and contributions

The advent of B5G networks promises revolutionary advancements in
communication, enabling applications like ultra-low latency for autonomous
vehicles and seamless connectivity for IoT devices. The impact of harsh
environmental conditions brought on by climate change, such as heavy rain,
fog, and snow, which present significant obstacles to dependable wireless
communication, is limiting the effectiveness of B5G networks. This paper aims
to bridge this gap by exploring the role of climate-resilient UAVs in
enhancing energy-efficient B5G communication in adverse weather conditions,
unlocking the full potential of B5G networks for a wide range of
transformative applications. While B5G networks hold immense promise, a
critical gap exists in ensuring the resilience of communication infrastructure
to the adverse meteorological conditions intensified by climate change. Rain,
fog, and snow can severely degrade wireless communication quality, disrupting
services and hindering B5G network deployment. This paper fills this gap by
looking into how climate-resilient UAVs can be used to improve energy-
efficient B5G communication in harsh environments. It offers crucial insights
and solutions for keeping communication services running in bad weather,
ensuring critical applications do not stop working, and making B5G
communication systems more climate-resilient.

The paper aims to advance understanding of how UAVs can play a pivotal role in
enhancing the resilience and energy efficiency of B5G communication networks
in harsh environmental conditions. The contributions encompass demonstrating
UAVs as a climate-resilient solution and quantifying their impact on energy
efficiency and key performance metrics. The contributions of the paper are
summarised as follows:

  * •

We introduce climate-resilient UAVs as a significant contribution to the
field. It showcases how UAVs can be dependable for maintaining communication
services in adverse meteorological conditions (i.e., rain, fog, and snow). The
paper highlights UAVs as a critical element in addressing the challenges posed
by climate-induced weather conditions in communication infrastructure.

  * •

The paper quantifies and emphasizes the significant energy efficiency
improvements made possible by integrating UAVs into B5G communication networks
in harsh environments. The contributions highlight the reduced interference,
augmented data transmission rates, and optimal channel gain facilitated by
UAVs, underscoring their potential to optimize energy consumption in
communication systems.

  * •

We provide a holistic assessment of the proposed UAV assisted system by
evaluating key performance metrics. By addressing energy efficiency, outage
probability, spectrum efficiency, and path loss, it offers a comprehensive
understanding of the capabilities and limitations of UAVs in mitigating the
adverse effects of weather on B5G communication.

###  I-B Related work

The intersection of UAVs, adverse meteorological conditions, and wireless
communication has garnered considerable attention in recent research. Several
studies have laid the foundation for understanding the potential of UAVs in
mitigating the challenges posed by harsh environments. The authors of [6]
conducted pioneering research on wireless communications with UAVs,
elucidating the opportunities and challenges of UAV-assisted communication.
The authors highlighted the agility and adaptability of UAVs in providing on-
demand wireless links, particularly in scenarios where traditional
infrastructure is compromised[12]. While their study focused on general UAV
applications, it provided valuable insights into the feasibility of UAVs in
adverse environmental conditions. In [4], presented a comprehensive tutorial
on the applications, challenges, and open problems associated with UAVs in
wireless networks. The authors of [13] delved into the effects of fog and haze
on visible light communication. While not UAV-centric, their work highlighted
the need for resilient communication solutions in adverse weather. The study
emphasized the impact of weather conditions on communication quality and
reliability, aligning with our research objectives.

The authors of [14] explored the utilization of UAVs in 6G communication
networks, emphasizing their ability to provide rapid deployment and coverage
extension during extreme weather events. In [15] investigated the impact of
rainfall on millimetre-wave communication, a key component of B5G and 6G
networks. The findings highlighted the significance of weather-induced signal
attenuation and the potential for UAVs to act as dynamic relays, mitigating
the effects of rain on high-frequency communication. Furthermore, [16] delved
into the challenges of energy-efficient communication in adverse weather. The
work specifically examined energy-aware routing protocols for UAV-assisted
networks operating in foggy conditions, shedding light on the importance of
energy efficiency in challenging meteorological contexts.

While the above studies have made significant strides in understanding the
role of UAVs and weather effects in communication networks, our paper
contributes to this evolving landscape by conducting a comprehensive
assessment of UAV performance in B5G networks under the combined influence of
rain, fog, and snow. Our paper provides insights into energy efficiency,
outage probability, spectrum efficiency, and path loss, addressing specific
challenges related to climate-resilient and energy-efficient B5G communication
in harsh environments.

##  II Proposed System Model

The system model is tailored to address a scenario where GNs face formidable
challenges in maintaining wireless connectivity, primarily due to adverse
environmental conditions, encompassing rain, fog, snow, and other disruptive
factors. These conditions disrupt the conventional wireless coverage services
typically provided by Ground Base stations, necessitating an innovative
approach. We strategically deploy UAVs to reinstate and sustain essential
communication links with GNs in these demanding scenarios to overcome this
pervasive issue. In our proposed model, UAVs have advanced directional
antennas, a critical feature designed to optimize network coverage
performance. Furthermore, the UAVs employ dynamic altitude adjustments,
responding to factors such as antenna beamwidth and the density of nearby
structures, as quantified by the number of installations [17].

The altitude adaptation ensures efficient GN coverage, especially in harsh
environmental conditions. The UAVs’ capability to dynamically allocate GNs and
user devices within the coverage area significantly ensures reliable and
uninterrupted connectivity in challenging scenarios. The allocation strategy
enhances coverage and connectivity, even in adverse weather conditions. To
assess the efficacy of our proposed system, we employ a range of performance
parameters, including path loss, energy efficiency, and coverage area. These
metrics are vital for evaluating the system’s performance across diverse
weather conditions, ensuring reliable connectivity is maintained in the
harshest environments.

###  II-A Attenuation Models for Rain, Fog and Snow

Utilizing UAVs for telecommunications services in adverse weather conditions
presents a significant challenge. To establish a foundation for deploying UAV
communications in such dynamic environments, it is essential to carefully
investigate attenuation models for various typical weather conditions,
including rain, fog, and snow. The attenuation models for these weather
conditions are expressed as follows:

$\displaystyle{\gamma}=\begin{dcases*}k{R^{\alpha}}&Rain model\\\
{K_{1}}(f,T)M\quad({\text{dB}}/{\text{km}})&Fog model\\\
0.00349\frac{{R_{\text{s}}^{1.6}}}{{{\lambda^{4}}}}+0.00224\frac{{{R_{\text{s}}}}}{\lambda}\quad({\text{dB}}/{\text{km}})&Snow
model\end{dcases*}$ (1)

Rain Model: The rain attenuation model is characterized by the equation
${\gamma}=k{R^{\alpha}}$, where $R$ represents the rain rate in millimetres
per hour exceeded for 0.01

Snow Model: The snow attenuation model is described by the equation
$0.00349\frac{{R_{\text{s}}^{1.6}}}{{{\lambda^{4}}}}+0.00224\frac{{{R_{\text{s}}}}}{\lambda}$,
where $\lambda$ is the wavelength and $R_{\text{s}}$ is the snowfall speed.

Fog Model: The attenuation coefficient is given by
$\frac{{4.34f^{2}}}{{\lambda^{2}}}$ in units of $(dB/km)/(g/m^{3})$. Where,

$\eta$ is defined as
$\frac{2+\varepsilon^{\prime}}{\varepsilon^{\prime\prime}}$, and the complex
permittivity of water is expressed as ${\varepsilon^{\prime\prime}}(f)$ and
${\varepsilon^{\prime}}(f)$. For fog attenuation modelling, the equations
governing the complex permittivity of water (${\varepsilon^{\prime\prime}}(f)$
and ${\varepsilon^{\prime}}(f)$) are provided, with temperature ($T$) and
various constants determining their values. The specific attenuation
coefficient (${K_{1}}(f,T)$) is essential for quantifying fog-induced
attenuation and is directly related to the density of liquid water in the
cloud or fog ($M$).

${K_{1}}(f,T)=\frac{{0.819f}}{{{\varepsilon^{\prime\prime}}\left({1+{\eta^{2}}}\right)}}\quad({\text{dB}}/{\text{km}})/\left({{\text{g}}/{{\text{m}}^{3}}}\right),$
(2)

where $\eta=\frac{2+\varepsilon^{\prime}}{\varepsilon^{\prime\prime}}$ and the
$\varepsilon^{\prime\prime}(f)$ is given as:

$\displaystyle{\varepsilon^{\prime\prime}}(f)=\frac{{f\left({{\varepsilon_{0}}-{\varepsilon_{1}}}\right)}}{{{f_{p}}\left[{1+{{\left({f/{f_{p}}}\right)}^{2}}}\right]}}+\frac{{f\left({{\varepsilon_{1}}-{\varepsilon_{2}}}\right)}}{{{f_{s}}\left[{1+{{\left({f/{f_{s}}}\right)}^{2}}}\right]}},$
(3)
$\displaystyle{\varepsilon^{\prime}}(f)=\frac{{{\varepsilon_{0}}-{\varepsilon_{1}}}}{{\left[{1+{{\left({f/{f_{p}}}\right)}^{2}}}\right]}}+\frac{{{\varepsilon_{1}}-{\varepsilon_{2}}}}{{\left[{1+{{\left({f/{f_{s}}}\right)}^{2}}}\right]}}+{\varepsilon_{2}},$
(4)

Where $\varepsilon_{0}$ is calculated as $77.66+103.3(\theta-1)$,
$\varepsilon_{1}$ is derived as $0.0671$ times $\varepsilon_{0}$,
$\varepsilon_{2}$ is a constant with a value of $3.52$. Here, $\theta$ is
determined as $300/T_{\text{fog}}$, where $T_{\text{fog}}$ represents the
temperature during foggy weather conditions and is set to 293.15K.
Additionally, we define the primary relaxation frequency, $f_{p}$, as
$20.20-146(\theta-1)+316(\theta-1)^{2}$, and the secondary relaxation
frequency, $f_{s}$, as $39.8$ times $f_{p}$ (in GHz).

$T$ represents the temperature of liquid water, $K_{1}$ stands for the
specific attenuation coefficient (dB/km per $\text{g/m}^{3}$), and $M$ denotes
the density of liquid water in the cloud or fog ($\text{g/m}^{3}$), as
documented in [18]. Additionally, $Rs$ corresponds to the snowfall speed
measured in millimeters per hour, and $\lambda$ indicates the wavelength
measured in centimeters

The attenuation models are crucial for understanding and mitigating the
effects of rain, fog, and snow on wireless communication, particularly when
deploying UAVs in challenging meteorological environments, providing valuable
insights into how environmental conditions impact communication performance
and guide the development of resilient communication systems.

###  II-B Path Loss for Rain, Fog and Snow

Path loss propagation plays a pivotal role in shaping the wireless
communication channel between UAVs and GNs within the air-to-ground (A2G)
channel. Environmental parameters, including the distance between UAVs and
GNs, GN elevation angles, and UAV altitudes, profoundly influence the signal
path loss. To comprehensively model the A2G channel under multiple weather
conditions, we integrate the specific attenuation coefficients for rain, fog,
and snow into the path loss models. The resulting A2G channel models can be
expressed as:

$\begin{array}[]{c}P{L_{{\text{UAV}}}}=\left({P{L_{{\text{LoS}}}}\times{P_{{\text{LoS}}}}+P{L_{{\text{NLoS}}}}\times{P_{{\text{NLoS}}}}}\right)+\frac{{(\beta+\gamma)d}}{{1000}}\\\
=\left({\frac{A}{{1+a\exp\left({-b\left({\frac{{180}}{\pi}{{\tan}^{-1}}\left({\frac{h}{r}}\right)-a}\right)}\right)}}}\right.\\\
\left.{+20\log\frac{r}{{\cos\left({\frac{{180}}{\pi}{{\tan}^{-1}}\left({\frac{h}{r}}\right)}\right)}}+B}\right)+\frac{{(\beta+\gamma)d}}{{1000}},\end{array}$
(5)

Where, $PL_{\text{UAV}}$ represents the UAV path loss, $PL_{\text{LoS}}$ and
$PL_{\text{NLoS}}$ are path loss components for Line-of-Sight (LoS) and Non-
Line-of-Sight (NLoS) conditions, respectively. The variables $A$ and $\gamma$
account for specific attenuation coefficients related to various weather
scenarios in dB/km. Additionally, $f_{c}$ denotes the carrier frequency, $c$
is the speed of light, and $d_{[ki]}$ represents the distance from the UAV to
the GNs. The terms $\eta_{\text{LoS}}$ and $\eta_{\text{NLoS}}$ consider
excessive loss due to shadowing and scattering in LoS and NLoS links, while
$\beta$ represents atmospheric attenuation.

###  II-C Connectivity in UAV coverage area

To determine the optimal coverage area of the UAV, we derive the first
derivative of equation (16) concerning the variable $r$ as follows:

$\displaystyle 0$
$\displaystyle=-\frac{Aah}{r^{2}}\left(b\right)\left(\tan^{-1}\left(\frac{h}{r}\right)-a\right)$
(6)
$\displaystyle\quad\times\left(e^{-b\left(\tan^{-1}\left(\frac{h}{r}\right)-a\right)}\right)$
$\displaystyle\quad\times\left(1+ae^{-b\left(\tan^{-1}\left(\frac{h}{r}\right)-a\right)}\right)^{-2}$
$\displaystyle\quad\times\left(\frac{h^{2}}{r^{2}}+1\right)^{-1}$
$\displaystyle\quad+20\log\left(\cos\left(180\frac{1}{\pi}\arctan\left(\frac{h}{r}\right)\right)\right)^{-1}$
$\displaystyle\quad-3600\frac{\left(\log
h\right)h}{r\pi}\sin\left(180\frac{1}{\pi}\tan^{-1}\left(\frac{h}{r}\right)\right)$
$\displaystyle\quad\times\left(\cos\left(180\frac{1}{\pi}\arctan\left(\frac{h}{r}\right)\right)\right)^{-2}$
$\displaystyle\quad\times\left(\frac{h^{2}}{r^{2}}+1\right)^{-1}$

This derivative is instrumental in determining the optimal coverage area of
the UAV, taking into account critical parameters like altitude, distance, and
environmental factors. By findings from [19], we combine the propagation
attenuation effects of rain, fog, and snow under various weather conditions to
model wireless channels effectively. This integrated model helps us understand
how adverse weather impacts communication performance, particularly when
deploying UAVs.

The desired performance metrics for assessing UAV-assisted GN communications
encompass several crucial aspects, including achieving higher data rates,
enhancing energy efficiency, increasing network capacity, and ensuring service
availability during harsh environmental events [20]. Notably, network capacity
is evaluated based on the traffic that can be handled with a minimum bit error
rate, especially in challenging conditions. The derivative-based optimization
approach, coupled with a comprehensive understanding of weather-induced
attenuation, forms the basis for effective wireless communication system
design, focusing on achieving optimal performance metrics in UAV-assisted GN
communications, even under adverse environmental conditions.

###  II-D Energy Efficiency

In harsh environmental communication scenarios, the UAV takes on the crucial
role of a temporary communication relay, ensuring the timely and reliable
exchange of vital information. However, adverse weather conditions such as
rain, fog, and snow significantly impact the UAV’s energy resources,
particularly during refilling operations. The conditions consume valuable time
and deplete the UAV’s energy reserves. As a result, an urgent need exists to
enhance the energy efficiency of UAV-assisted communication systems. The
holistic evaluation of energy efficiency in the context of UAV-assisted
communication involves considering the entire instantaneous transmission
vector of the UAV, denoted as $EE^{k}_{UAV}$. This vector encompasses all the
constituent elements, representing every link between the UAV and the ground
nodes. It serves as a comprehensive metric to assess and optimize the system’s
energy efficiency, addressing the challenges posed by harsh environmental
conditions and ensuring the effective utilization of the UAV’s resources for
reliable communication.

$\mbox{EE}^{[k]}_{\text{UAV}}=\frac{B\cdot\mbox{log}_{2}(1+\frac{p_{i}{h}_{i}}{\sum_{m=1}^{M}p_{m}h_{m,i}+p_{j}{h}_{j}+\sigma^{2}})}{P_{tx}h},$
(7)

Where, $h$ represents the number of hops from the UAV to GN communications,
and $P_{tx}$ stands for the maximum transmission power used by the UAV for
downlink communications with GNs.

##  III Results and Discussion

In this section, we present an extensive array of simulation results that
vividly illustrate the performance of the proposed schemes. Our evaluation
encompasses critical parameters and metrics, focusing on LoS probability, path
loss, and energy efficiency in UAV-to-ground node communication. These
evaluations are conducted across harsh environmental scenarios, including
rain, fog, and snow.

Our simulations consider a scenario where ground nodes are randomly
distributed within the UAV’s coverage area, mirroring real-world conditions.
To achieve a comprehensive assessment, we vary ground node elevation angles
and adjust UAV altitudes accordingly. Considering the UAV’s coverage area, we
assume the ground nodes are scattered randomly in each scenario. The
transmission distance between the source and destination of the ground nodes
and the UAV is set from 100 meters to 1000 meters. Moreover, we explore a wide
range of elevation angles for the ground nodes, spanning from 0 degrees to 90
degrees. By conducting rigorous simulations, we aim to provide a comprehensive
understanding of how our proposed schemes perform under varying conditions and
scenarios. This empirical evidence will validate the efficacy of our
approaches and offer insights into their real-world applicability.

TABLE I: Simulation Parameters Parameters |  Values
---|---
B |  $5$ MHz
$\sigma^{2}$ & $-174$ dBm/Hz |
$f_{c}\&[28~{}GHz,60~{}GHz]$ |
Urban area factors |  $a=9.61$, $b=0.16$, $\eta_{\text{LoS}}=1$, $\eta_{\text{NLoS}}=20$
Medium rain |  $R=12.5$ mm/h
Medium fog |  $M=0.05$ g/m3
Snow |  $Rs=5$ mm/h
Weather conditions [dB/km] attenuation coefficient [21] |  Clear air = 0.43, Haze = 4.2,
|  Moderate rain (12.5 mm/h) = 5.8,
|  Heavy rain (25 mm/h) = 9.2,
|  Light fog = 20,
|  Moderate fog = 42.2,
|  Heavy fog = 125
UAV altitude |  $120$ m
UAV TX power |  5 W
GN |  $0^{\circ}$ to $90^{\circ}$

###  III-A Path Loss

Path loss propagation is a critical consideration when assessing communication
performance, and it is essential to understand how it varies under different
conditions. In the context of harsh environments, we observed a notable trend
in path loss as the elevation angle of the ground nodes (GNs) changed from
$0^{\circ}$ to $90^{\circ}$, as depicted in Fig. 1. The variation is primarily
attributed to weather conditions such as rain, fog, and snow. In particular,
when GNs are situated in harsh weather conditions, the path loss substantially
increases, ranging from 0 dB to 310 dB as the GN elevation angle varies. The
dramatic change in path loss indicates the challenges imposed by adverse
weather. Notably, in a rainy environment, the path loss experiences a more
modest increase, rising from 0 dB to 51 dB. In contrast, the fog environment
displays a path loss that escalates from 100 dB to 175 dB. Finally, for the
snow environment, path loss undergoes a considerable increase, ranging from 0
dB to 310 dB, as the GN elevation angle varies from $0^{\circ}$ to
$90^{\circ}$ due to the specific characteristics of a single city model.

Figure 1:  Variation in Path Loss with GN Elevation Angle in Different Harsh
Environments (Rain, Fog, Snow)

Data: Rain rate: $rain\\_rate\\_mm\\_hr$

Frequency: $frequency\\_GHz$

Temperature: $temperature\\_Kelvin$

Snowfall speed: $snowfall\\_speed\\_mm\\_hr$

Wavelength: $wavelength\\_cm$

$t_{max}$: Maximum number of iterations

$P_{max}$: Maximum transmission power of UAV

$d_{u,i}$: The distance between UAV and GNS

$\theta_{j,k}$: Elevation angles of GNs

$N$: length(weather_conditions) (’Rain’, ’Fog’, ’Snow’)

Result: Energy efficiency and UAV Coverage in conditions of (’Rain’, ’Fog’,
’Snow’)

for  _$t=1$ to $t_{max}$_ do

for  _$i=1$ to $N$_ do

for  _$j=1$ to $N$_ do

Calculate Path Loss Based on Eq(5)

Calculate UAV Coverage Area Based on Eq(6)

Calculate Energy Efficiency Based on Eq(7)

end for

end for

end for

Algorithm 1 Climate-Resilient UAV Communication Model Figure 2: Architecture
of the UAV to GN communication in harsh environments

###  III-B Energy Efficiency

In this context, we delve into the analysis of both the UAV and the GN energy
efficiency, shedding light on how these efficiency metrics behave under
varying conditions. Fig. 3 visually represents our findings. We observe that
energy efficiency experiences a decline as the distance between the GNs and
the UAV increases; a decrease in energy efficiency is a significant aspect to
consider, as it signifies that more energy is required to maintain
communication over longer distances. However, an exciting observation arises
as we examine energy efficiency across different scenarios. As the
transmission distance of the ground nodes increases, the energy efficiency
values for each scenario start to converge and become closely aligned. The
phenomenon suggests that as the ground node distance expands, the impact of
interference on energy efficiency diminishes. When GNs are farther apart,
interference plays a less significant role in reducing energy efficiency.

Increasing the transmission power is one effective way to counteract this
reduction in energy efficiency due to interference, particularly in scenarios
with longer transmission distances[22]. The signal can better overcome
interference by providing more power, leading to improved energy efficiency.
Furthermore, it is essential to highlight that co-channel interference
primarily affects energy efficiency, where multiple communication channels
overlap and interfere. In this context, directional antennas can be a valuable
strategy. Directional antennas can focus and concentrate the signal in a
specific direction, reducing interference and enhancing the overall efficiency
metrics. Hence, incorporating directional antennas is a practical approach to
mitigate the adverse effects of interference on energy efficiency in UAV-to-GN
communication scenarios.

Figure 3: Variation in Energy Efficiency with Ground Node Distance Across
Different Scenarios (Rain, Fog, Snow)

###  III-C UAV Communication Coverage

Fig 4 illustrates the impact of harsh environmental conditions, including
rain, fog, and snow, on UAV communication systems’ cell radius and operating
altitude. In scenarios with moderate rainfall, the coverage radius and the
optimal UAV altitude are relatively small, indicating the need for lower
operating altitudes for adequate coverage. Conversely, the coverage radius
significantly expands under light snow conditions, enabling UAVs to operate at
higher altitudes while providing extensive coverage. This demonstrates that
the severity of weather conditions plays a pivotal role in determining the
trade-off between coverage area and UAV altitude in UAV communication, with
milder conditions allowing for greater altitude and more extensive coverage
areas compared to scenarios with more moderate rain.

Figure 4: Variation in UAV Communication Coverage Radius with Altitude Across
Different Harsh Weather Conditions (Rain, Fog, Snow)

##  IV Conclusion

In this paper, we have explored the pivotal role of UAV-facilitated ground
node communication in challenging meteorological conditions, encompassing
rain, fog, and snow. Our assessment has focused on key performance metrics,
including energy efficiency, outage probability, spectrum efficiency, and path
loss. In our proposed system model, UAVs offer invaluable coverage support to
empower GNs operating in harsh environments, leading to enhanced network
scalability, reduced routing overhead, optimized throughput, and expanded
coverage. UAV-assisted cellular communication is poised to become a
cornerstone technology, catering to the evolving demands of dynamic and
diverse communication scenarios. Furthermore, UAVs are indispensable for swift
recovery when traditional communication infrastructure succumbs to dysfunction
during natural disasters and adverse environmental conditions. Our results
affirm the viability of the UAV-assisted system, showcasing its capability to
perform on par with GNs in harsh environments. UAVs emerge as a fitting
replacement for dysfunctional GNs in these challenging scenarios, offering
reliability and rapid restoration of communication services during critical
situations.

## Acknowledgment

This publication has emanated from research conducted with the financial
support of Science Foundation Ireland under Grant number SFI/12/RC/2289_P2.

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# On Mahler’s inequality and small integral generators of totally complex
number fields

Murray Child  and  Martin Widmer  Department of Mathematics
Royal Holloway, University of London
TW20 0EX Egham
UK<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We improve Mahler’s lower bound for the Mahler measure in terms of the
discriminant and degree for a specific class of polynomials: complex monic
polynomials of degree $d\geq 2$ such that all roots with modulus greater than
some fixed value $r\geq 1$ occur in equal modulus pairs. We improve Mahler’s
exponent $\frac{1}{2d-2}$ on the discriminant to $\frac{1}{2d-3}$. Moreover,
we show that this value is sharp, even when restricting to minimal polynomials
of integral generators of a fixed not totally real number field.

An immediate consequence of this new lower bound is an improved lower bound
for integral generators of number fields, generalising a simple observation of
Ruppert from imaginary quadratic to totally complex number fields of arbitrary
degree.

###### Key words and phrases:

Mahler measure, Mahler’s inequality, discrete logarithmic energy, small
generators, number fields, Weil height

###### 2010 Mathematics Subject Classification:

Primary 11R06; 30C10 Secondary 11G50; 11R04

##  1\. Introduction

In this short note we prove a new lower bound for the Mahler measure of a
monic polynomial in $\mathbb{C}[x]$ whose “large” roots come in pairs of equal
modulus. The bound is expressed in terms of the degree and the discriminant,
and the dependence on the discriminant is best-possible. The result implies a
new lower bound for the smallest integral generator of a totally complex
number field.

We refer the reader to Smyth’s survey article [14] (see also [9]) for a
detailed account on the Mahler measure and its significance.

Our proofs are completely elementary and straightforward but the results seem
to close a gap in the literature.

Let $f=a_{0}(x-\alpha_{1})\cdots(x-\alpha_{d})\in\mathbb{C}[x]$ be of degree
$d\geq 2$. The Mahler measure $M(f)$ and the discriminant $\Delta_{f}$ of $f$
are defined by

$\displaystyle M(f):=|a_{0}|\prod_{i=1}^{d}\max\\{{1,|\alpha_{i}|\\}},$
$\displaystyle\Delta_{f}:=a_{0}^{2d-2}\prod_{1\leq i<j\leq
d}(\alpha_{i}-\alpha_{j})^{2}.$

Mahler’s classical inequality bounds the former from below in terms of the
latter.

######  Theorem 1 (Mahler, 1964 [8]).

Let $f\in\mathbb{C}[x]$ be a polynomial of degree $d\geq 2$. Then

$M(f)^{2d-2}\geq d^{-d}|\Delta_{f}|.$

Moreover, Mahler showed that we have equality if and only if
$f=a_{0}x^{d}+a_{d}$ with $|a_{0}|=|a_{d}|>0$. Note that for irreducible
$f\in\mathbb{Z}[x]$ this only happens when $f=\pm(x^{d}+1)$ and $d$ is a power
of $2$. However, for primes $p$ the irreducible polynomials $f=(p+1)x^{d}-p$
in $\mathbb{Z}[x]$ satisfy111A simple calculation gives
$|\Delta_{f}|=d^{d}\left(p(p+1)\right)^{d-1}$ and $M(f)=p+1$.

$M(f)^{2d-2}=d^{-d}|\Delta_{f}|\left(\frac{p+1}{p}\right)^{d-1},$

and thus even when restricting to irreducible polynomials neither the exponent
${2d-2}$ nor the constant $d^{-d}$ can be improved. What if we restrict to
monic irreducible polynomials in $\mathbb{Z}[x]$? Using Swan’s discriminant
formula for trinomials [15, Theorem 2] we get, for the $p$-Eisenstein
polynomial $f=x^{d}+px^{d-1}+(-1)^{d+1}p$,

$|\Delta_{f}|=p^{d-1}\left(d^{d}+\left((d-1)p\right)^{d-1}\right)>(d-1)^{d-1}p^{2d-2}.$

On the other hand, $M(f)\leq(2+1/p)p<3p$ (using inequality (1.2)), and
therefore

$M(f)^{2d-2}<\left(\frac{9}{d-1}\right)^{d-1}|\Delta_{f}|.$

Hence, at least the exponent $2d-2$ is sharp, even when restricting to monic
irreducible polynomials in $\mathbb{Z}[x]$.

Next, let us consider a monic quadratic polynomial $f=x^{2}+bx+c$ with
coefficients $a,b\in\mathbb{R}$ and no real roots. Then, as noted by Ruppert
[12],

(1.1)
$M(f)=\max\left\\{1,\frac{b^{2}+|\Delta_{f}|}{4}\right\\}\geq\frac{|\Delta_{f}|}{4}.$

So instead of Mahler’s exponent ${2d-2}={2}$ one obtains the much better
exponent $1$. This raises the question of whether one can improve Mahler’s
exponent ${2d-2}$ when restricting to monic $f\in\mathbb{R}[x]$ with no real
roots - and in particular, what is the sharp exponent in this case? We have
been unable to find the answer in the literature but our next result shows
that for such polynomials the exponent can be improved to ${2d-3}$.

######  Theorem 2.

Let $m$ be non-negative and even, let $r\geq 1$, and let

$f(x)=\left(\prod_{i=1}^{\frac{m}{2}}(x-\alpha_{i})(x-\alpha_{i}^{\prime})\right)\left(\prod_{i=\frac{m}{2}+1}^{d-\frac{m}{2}}(x-\alpha_{i})\right)\in\mathbb{C}[x]$

be a monic polynomial of degree $d\geq 2$, with
$|\alpha_{i}|=|\alpha_{i}^{\prime}|$ for all $i=1,\ldots,\frac{m}{2}$ and
$|\alpha_{i}|\leq r$ for all $i=\frac{m}{2}+1,\ldots,d-\frac{m}{2}$. Then

$M(f)^{2d-3}\geq(2r)^{d(1-d)}|\Delta_{f}|.$

The exponent $2d-3$ in Theorem 2 is sharp, even when restricting to minimal
polynomials of integral generators of a given not totally real number field.

Let $K$ be a number field of degree $d$ and ${\mathcal{O}}_{K}$ its ring of
integers. For $\alpha\in K$ we write $f_{\alpha,\mathbb{Z}}$ for the minimal
polynomial222The minimal polynomial $f_{\alpha,\mathbb{Z}}$ of $\alpha$ over
$\mathbb{Z}$ is the unique polynomial in $\mathbb{Z}[x]$ of minimal degree
with positive leading coefficient and coprime coefficients that vanishes at
$\alpha$. of $\alpha$ over $\mathbb{Z}$. Recall that
$\alpha\in{\mathcal{O}}_{K}$ if and only if $f_{\alpha,\mathbb{Z}}$ is monic.

######  Proposition 1.

Let $K$ be a number field of degree $d$ with a non-real embedding. Then there
exists $c_{K}>0$ depending only on $K$ such that there are infinitely many
$\alpha\in{\mathcal{O}}_{K}$ with $K=\mathbb{Q}(\alpha)$,

$M(f_{\alpha,\mathbb{Z}})^{2d-3}\leq c_{K}|\Delta_{f_{\alpha,\mathbb{Z}}}|,$

and all roots of $f_{\alpha,\mathbb{Z}}$, except one pair of complex conjugate
roots, have modulus at most $c_{K}$.

Under the hypothesis that the number field $K$ possesses a real embedding one
can prove in the same way that there are infinitely many
$\alpha\in{\mathcal{O}}_{K}$ with $K=\mathbb{Q}(\alpha)$ and
$M(f_{\alpha,\mathbb{Z}})^{2d-2}\leq c_{K}|\Delta_{f_{\alpha,\mathbb{Z}}}|$,
and all roots of $f_{\alpha,\mathbb{Z}}$ except one have modulus at most
$c_{K}$.

Next we apply Theorem 2 to get a new lower bound for integral generators of
totally complex number fields.

Let $K$ be a number field. Then every $\alpha\in K$ with
$K=\mathbb{Q}(\alpha)$ is called a generator of $K$. It is natural to ask: how
“large” must a generator of $K$ be in terms of the degree $d$ and the modulus
of the discriminant $|\Delta_{K}|$? And what happens if we restrict to
integral generators? This problem has been studied by several authors,
including Cochrane et al. [2], Dubickas [3], Eldredge & Petersen [4], Kihel &
Lizotte [6], Pierce & Turnage-Butterbaugh & Wood [11], Ruppert [12], and
Vaaler & Widmer [17, 16].

A good measure for size here is the Mahler measure $M(f_{\alpha,\mathbb{Z}})$
of the minimal polynomial $f_{\alpha,\mathbb{Z}}$ of $\alpha$ over
$\mathbb{Z}$. Following Eldredge & Petersen [4] and Dubickas [3] we use the
following notation:

$M(K):=\text{min}\\{M(f_{\alpha,\mathbb{Z}}):\alpha\in
K,\mathbb{Q}(\alpha)=K\\},$

and

$M({\mathcal{O}}_{K}):=\text{min}\\{M(f_{\alpha,\mathbb{Z}}):\alpha\in{\mathcal{O}}_{K},\mathbb{Q}(\alpha)=K\\}.$

Writing $\|f\|_{1}$ for the $L^{1}$-norm of the coefficient vector of $f$ one
has by a result of Mahler [7]

(1.2) $2^{-d}\|f\|_{1}\leq M(f)\leq\|f\|_{1}$

for each $f\in\mathbb{C}[x]$ of degree $d\geq 1$. Hence there are only
finitely many polynomials $f\in\mathbb{Z}[x]$ of degree at most $D$ and
$M(f)\leq T$, irrespective of how large $D$ and $T$ are. This shows that the
minima in the definitions of $M(K)$ and $M({\mathcal{O}}_{K})$ exist.

Now recall that if $\alpha\in{\mathcal{O}}_{K}$ with $K=\mathbb{Q}(\alpha)$
then $f_{\alpha,\mathbb{Z}}$ is monic and
$\Delta_{f_{\alpha,\mathbb{Z}}}=\Delta_{K}[{\mathcal{O}}_{K}:\mathbb{Z}[\alpha]]^{2}$
where the positive integer $[{\mathcal{O}}_{K}:\mathbb{Z}[\alpha]]$ is the
index of the order $\mathbb{Z}[\alpha]$ in the maximal order
${\mathcal{O}}_{K}$. Therefore Mahler’s inequality (Theorem 1) implies

(1.3) $M({\mathcal{O}}_{K})\geq
d^{-\frac{d}{2d-2}}|\Delta_{K}|^{\frac{1}{2d-2}}.$

Silverman [13] proved this bound holds even for $M(K)$, and it is known333If
$p<q<2p$ are two primes and $K=\mathbb{Q}((p/q)^{1/d})$ then $M(K)\leq
q<(2pq)^{1/2}\leq 2^{1/2}|\Delta_{K}|^{1/(2d-2)}$. (cf. [12]) that the
exponent $\frac{1}{2d-2}$ on $|\Delta_{K}|$ is sharp for $M(K)$ and every
$d\geq 2$.

Ruppert [12] showed that $M(K)\ll_{d}|\Delta_{K}|^{\frac{1}{2d-2}}$ when $d=2$
and asked [12, Question 1] whether this remains true when $d>2$. This was
answered in the negative by Vaaler & Widmer [17] for composite $d$, and by
Dubickas [3] for prime $d\geq 3$. Ruppert [12, Question 2] asked also the
analogous question with exponent $1/2$ instead of $1/(2d-2)$. This question is
still open but has been affirmatively answered for not totally complex number
fields, and, conditionally under GRH, also for general number fields [16].

Less is known for $M({\mathcal{O}}_{K})$. The best general upper bound is

$M({\mathcal{O}}_{K})\leq|\Delta_{K}|$

which follows easily from Minkowski’s convex body theorem (cf. [10, Lemma
7.1]). The question of whether the exponent $1/(2d-2)$ in (1.3) is sharp seems
more delicate than its counterpart for $M(K)$. The cubic case was
affirmatively answered by Eldredge & Petersen [4, Theorem 1.1]. Motivated by a
different question Jones published a result [5, Theorem 1.5 (3)] that gives an
affirmative answer for all $d\in\\{3,4,5,7,9\\}$. He shows that if
$(d,w)\in\\{(3,1),(4,2),(5,1),(7,1),(9,2)\\}$ then there are infinitely many
primes $t$ such that

$P_{t}(x)=x^{d}-16d(dt+w)x^{d-1}+dt+w\in\mathbb{Z}[x]$

is irreducible and monogenic, i.e., ${\mathcal{O}}_{K}=\mathbb{Z}[\alpha]$
where $K=\mathbb{Q}(\alpha)$ and $\alpha$ is any root of $P_{t}(x)$. Hence,
$\Delta_{P_{t}}=\Delta_{K}$, and Swan’s discriminant formula shows that
$|\Delta_{P_{t}}|\gg t^{2d-2}$ while (1.2) implies that
$M({\mathcal{O}}_{K})\leq M(P_{t})\ll t$.

However, just as for $M(K)$, the exponent $2d-2$ is certainly not always sharp
when $d>2$. Eldredge & Petersen [4, Theorem 1.2] proved for cubic, and
Dubickas444Dubickas proves the sharper inequality
$(1-\epsilon)|\Delta_{K}|^{\frac{1}{d}}<M({\mathcal{O}}_{K})<|\Delta_{K}|^{\frac{1}{d}}$
for arbitray $\epsilon>0$. [3, Theorem 2] for arbitrary degrees $d\geq 2$,
that there are infinitely many number fields $K$ of degree $d$ such that

$\frac{1}{30}|\Delta_{K}|^{\frac{1}{d}}<M({\mathcal{O}}_{K})<\frac{4}{3}|\Delta_{K}|^{\frac{1}{d}}.$

These fields $K$ are very special and are all of the form
$\mathbb{Q}(p^{1/d})$ for certain primes $p$. It would be interesting to find
improved lower bounds that apply to more general families of number fields.
Ruppert [12] observed that for all imaginary quadratic fields

(1.4) $M({\mathcal{O}}_{K})\geq\frac{1}{4}|\Delta_{K}|.$

The bound (1.4) suggests that such improvements of Mahler’s exponent
$1/(2d-2)$ might hold for the family of totally complex fields of any fixed
degree $d\geq 2$.

Theorem 2 applied to the minimal polynomial of integral generators yields a
generalisation of Ruppert’s observation (1.4) to totally complex number fields
of arbitrary degree.

######  Corollary 1.

Let $K$ be a totally complex number field of degree $d\geq 2$. Then

$M({\mathcal{O}}_{K})\geq
2^{\frac{d(1-d)}{2d-3}}|\Delta_{K}|^{\frac{1}{2d-3}}.$

From number fields $K$ back to polynomials $f\in\mathbb{C}[x]$, here is
another application of Theorem 2. Mahler [8, Corollary] combined Theorem 1 and
(1.2) to get

(1.5) $|\Delta_{f}|\leq d^{d}\|f\|_{1}^{2d-2}.$

For polynomials $f$ as in Theorem 2 we get a better bound whenever $\|f\|_{1}$
is sufficiently large.

######  Corollary 2.

For polynomials $f\in\mathbb{C}[x]$ satisfying the conditions in Theorem 2 we
have

$|\Delta_{f}|\leq(2r)^{d(d-1)}\|f\|_{1}^{{2d-3}}.$

Note that if $K$ is a not totally real number field of degree $d$, and
$f=f_{\alpha,\mathbb{Z}}$ is as in Proposition 1, then

$\|f\|_{1}^{{2d-3}}\ll_{K}|\Delta_{f}|.$

##  2\. Discrete logarithmic energy for point configurations

Theorem 2 is proved by considering the complex roots of $f$ as points in
$\mathbb{R}^{2}$, and then bounding the logarithmic energy of this point
configuration in $\mathbb{R}^{2}$ from below in terms of the number of points
and the product of their Euclidean norms (ignoring those points inside the
unit disc). Our simple argument is agnostic to the dimension and thus works
for point configurations in arbitrary dimensions.

Let $k,d\in\mathbb{N}$, $d\geq 2$, let
${\boldsymbol{a_{1}}},\ldots,{\boldsymbol{a_{d}}}$ be points in
$\mathbb{R}^{k}$, and set
$\boldsymbol{\alpha}=({\boldsymbol{a_{1}}},\ldots,{\boldsymbol{a_{d}}})$. We
define

${\mathfrak{M}}(\boldsymbol{\alpha}):=\prod_{i=1}^{d}\max\\{1,|{\boldsymbol{a_{i}}}|\\},$

and

$|\Delta_{\boldsymbol{\alpha}}|:=\prod_{1\leq i<j\leq
d}|{\boldsymbol{a_{i}}}-{\boldsymbol{a_{j}}}|^{2}.$

Note that $-\log|\Delta_{\boldsymbol{\alpha}}|$ is the discrete logarithmic
energy $E_{\log}(\boldsymbol{\alpha})$ of the $d$-point configuration
$\boldsymbol{\alpha}$ in $\mathbb{R}^{k}$ (cf. [1]), and thus our goal is to
bound $E_{\log}(\boldsymbol{\alpha})$ from below in terms of
${\mathfrak{M}}(\boldsymbol{\alpha})$ and $d$. Mahler’s inequality shows that
$E_{\log}(\boldsymbol{\alpha})$ for a $d$-point configuration of points
$\boldsymbol{\alpha}$ on the closed unit disc in $\mathbb{R}^{2}$ is at least
$-d\log d$, and that this value is attained if and only if the points are
equidistributed on the unit circle. But finding the minimal logarithmic energy
$E_{0}(d;k)$ for a $d$-point configuration in the closed unit ball in
$\mathbb{R}^{k}$ for higher dimensions $k\geq 3$ is a difficult open problem
about which we have nothing to say.

Let $r\geq 1$ be a real number, and let $m\in\\{0,1,\ldots,d\\}$ be the number
of points ${\boldsymbol{a_{i}}}$ in $\boldsymbol{\alpha}$ with Euclidean
length strictly greater than $r$. We prove the following generalisation of
Theorem 2.

######  Theorem 3.

Suppose $m$ is even and that the points in $\boldsymbol{\alpha}$ with
Euclidean length greater than $r$ occur in pairs of equal length. Then

(2.6)
${\mathfrak{M}}(\boldsymbol{\alpha})^{2d-3}\geq(2r)^{d(1-d)}|\Delta_{\boldsymbol{\alpha}}|.$

Moreover, the exponent $2d-3$ is sharp.

The case $m=0$ corresponds to an arrangement of $d$ points in a closed ball of
radius $r$ in $\mathbb{R}^{k}$. In this case our theorem only yields the
trivial lower bound $d(1-d)\cdot\log(2r)$ for the logarithmic energy. On the
other hand it is clear from the proof that good lower bounds for
$E_{0}(d-m;k)$ can be used to refine the constant $(2r)^{d(1-d)}$ in our
theorem, at least if $d\geq m+2$. However, in this work we are only concerned
with the exponent of the discriminant and for simplicity we have decided to
record only the simplest possible explicit constant which is $(2r)^{d(1-d)}$.

###### Proof.

Without loss of generality, we can relabel the points
${\boldsymbol{a_{1}}},\ldots,{\boldsymbol{a_{d}}}$ in $\boldsymbol{\alpha}$ in
order of decreasing Euclidean length:

${\boldsymbol{\alpha_{1}}},{\boldsymbol{\alpha_{1}}}^{\prime},\ldots,{\boldsymbol{\alpha_{\frac{m}{2}}}},{\boldsymbol{\alpha_{\frac{m}{2}}}}^{\prime},{\boldsymbol{\alpha_{\frac{m}{2}+1}}},{\boldsymbol{\alpha_{\frac{m}{2}+2}}},\ldots,{\boldsymbol{\alpha_{d-\frac{m}{2}}}},$

where $|{\boldsymbol{\alpha_{i}}}|=|{\boldsymbol{\alpha_{i}}}^{\prime}|$
($i=1,\ldots,\frac{m}{2}$). Note that the points of Euclidean length strictly
greater than $r$ are exactly the $m$ points
${\boldsymbol{\alpha_{1}}},{\boldsymbol{\alpha_{1}}}^{\prime},\ldots,{\boldsymbol{\alpha_{\frac{m}{2}}}},{\boldsymbol{\alpha_{\frac{m}{2}}}}^{\prime}$,
so

$|{\boldsymbol{\alpha_{1}}}|=|{\boldsymbol{\alpha_{1}}}^{\prime}|\geq|{\boldsymbol{\alpha_{2}}}|=|{\boldsymbol{\alpha_{2}}}^{\prime}|\geq\ldots\geq|{\boldsymbol{\alpha_{\frac{m}{2}}}}|=|{\boldsymbol{\alpha_{\frac{m}{2}}}}^{\prime}|>r\geq|{\boldsymbol{\alpha_{\frac{m}{2}+1}}}|\geq\ldots\geq|{\boldsymbol{\alpha_{d-\frac{m}{2}}}}|.$

If any two points are equal then $|\Delta_{\boldsymbol{\alpha}}|=0$, and (2.6)
is trivially true. Hence, we can assume all of these points are distinct.

Recall that
$\boldsymbol{\alpha}=({\boldsymbol{a_{1}}},\ldots,{\boldsymbol{a_{d}}})=({\boldsymbol{\alpha_{1}}},{\boldsymbol{\alpha_{1}}}^{\prime},\ldots,{\boldsymbol{\alpha_{\frac{m}{2}}}},{\boldsymbol{\alpha_{\frac{m}{2}}}}^{\prime},{\boldsymbol{\alpha_{\frac{m}{2}+1}}},\ldots,{\boldsymbol{\alpha_{d-\frac{m}{2}}}})$,
so by definition of the discriminant we have

$\displaystyle|\Delta_{\boldsymbol{\alpha}}|$ $\displaystyle=\prod_{1\leq
i<j\leq d}|{\boldsymbol{a_{i}}}-{\boldsymbol{a_{j}}}|^{2}$ (2.7)
$\displaystyle=\prod_{i=1}^{\frac{m}{2}}\left(|{\boldsymbol{\alpha_{i}}}-{\boldsymbol{\alpha_{i}}}^{\prime}|^{2}\prod_{{\boldsymbol{\beta}}\in\beta_{i}}|{\boldsymbol{\alpha_{i}}}-{\boldsymbol{\beta}}|^{2}|{\boldsymbol{\alpha_{i}}}^{\prime}-{\boldsymbol{\beta}}|^{2}\right)\prod_{\frac{m}{2}+1\leq
i<j\leq
d-\frac{m}{2}}|{\boldsymbol{\alpha_{i}}}-{\boldsymbol{\alpha_{j}}}|^{2},$

where the set
$\beta_{i}:=\\{{\boldsymbol{\alpha_{i+1}}},{\boldsymbol{\alpha_{i+2}}},\ldots,{\boldsymbol{\alpha_{d-\frac{m}{2}}}},{\boldsymbol{\alpha_{i+1}}}^{\prime},{\boldsymbol{\alpha_{i+2}}}^{\prime},\ldots,{\boldsymbol{\alpha_{\frac{m}{2}}}}^{\prime}\\}$.

Let us start by evaluating the right-most product here, which we denote by
$C_{0}$ for convenience555Of course the sharp upper bound here is $C_{0}\leq
r^{(d-m)(d-m-1)}e^{-E_{0}(d-m;k)}$ where $E_{0}(d-m;k)$ denotes the minimal
logarithmic energy of a $(d-m)$-configuration in the unit ball in
$\mathbb{R}^{k}$.. We have
$|{\boldsymbol{\alpha_{i}}}|,|{\boldsymbol{\alpha_{j}}}|\leq r$ whenever
$\frac{m}{2}+1\leq i<j\leq d-\frac{m}{2}$, so by the triangle inequality for
all terms in this product,

$|{\boldsymbol{\alpha_{i}}}-{\boldsymbol{\alpha_{j}}}|^{2}\leq(|{\boldsymbol{\alpha_{i}}}|+|{\boldsymbol{\alpha_{j}}}|)^{2}\leq(2r)^{2}.$

Hence,

$\begin{split}C_{0}&\leq\prod_{m+1\leq i<j\leq n}(2r)^{2}=(2r)^{2{d-m\choose
2}}=(2r)^{(d-m)(d-m-1)}.\end{split}$

Now let us evaluate the left-most product in (2). Noting that
$|{\boldsymbol{\alpha_{i}}}|=|{\boldsymbol{\alpha_{i}}}^{\prime}|$
($i=1,\ldots,\frac{m}{2}$), we have for all $i=1,\ldots,\frac{m}{2}$,

$|{\boldsymbol{\alpha_{i}}}-{\boldsymbol{\alpha_{i}}}^{\prime}|^{2}\leq(|{\boldsymbol{\alpha_{i}}}|+|{\boldsymbol{\alpha_{i}}}^{\prime}|)^{2}=(2|{\boldsymbol{\alpha_{i}}}|)^{2}=2^{2}|{\boldsymbol{\alpha_{i}}}|^{2}.$

Likewise, noting that
$|{\boldsymbol{\beta}}|\leq|{\boldsymbol{\alpha_{i}}}|=|{\boldsymbol{\alpha_{i}}}^{\prime}|$
for all ${\boldsymbol{\beta}}\in\beta_{i}$ ($i=1,\ldots,\frac{m}{2}$), we have
for all ${\boldsymbol{\beta}}\in\beta_{i}$ ($i=1,\ldots,\frac{m}{2}$),

$|{\boldsymbol{\alpha_{i}}}-{\boldsymbol{\beta}}|^{2}\leq
2^{2}|{\boldsymbol{\alpha_{i}}}|^{2},$

and

$|{\boldsymbol{\alpha_{i}}}^{\prime}-{\boldsymbol{\beta}}|^{2}\leq
2^{2}|{\boldsymbol{\alpha_{i}}}|^{2}.$

Combining this all together, we get

$\begin{split}|\Delta_{\boldsymbol{\alpha}}|&\leq
C_{0}\cdot\prod_{i=1}^{\frac{m}{2}}\left(2^{2}|{\boldsymbol{\alpha_{i}}}|^{2}\prod_{{\boldsymbol{\beta}}\in\beta_{i}}2^{2}|{\boldsymbol{\alpha_{i}}}|^{2}\cdot
2^{2}|{\boldsymbol{\alpha_{i}}}|^{2}\right)=C_{0}\cdot\prod_{i=1}^{\frac{m}{2}}\left(2^{2}|{\boldsymbol{\alpha_{i}}}|^{2}\cdot(2^{4}|{\boldsymbol{\alpha_{i}}}|^{4})^{|\beta_{i}|}\right).\end{split}$

Now consider the number of elements in $\beta_{i}$. This is straightforward to
calculate:

$|\beta_{i}|=(d-\frac{m}{2}-i)+(\frac{m}{2}-i)=d-2i.$

So

$\begin{split}|\Delta_{\boldsymbol{\alpha}}|&\leq
C_{0}\cdot\prod_{i=1}^{\frac{m}{2}}\left(2^{2}|{\boldsymbol{\alpha_{i}}}|^{2}\cdot(2^{4}|{\boldsymbol{\alpha_{i}}}|^{4})^{d-2i}\right)\\\
&=C_{0}\cdot 2^{m}\cdot
2^{2dm}\cdot\prod_{i=1}^{\frac{m}{2}}\left(2^{-8i}|{\boldsymbol{\alpha_{i}}}|^{4d-8i+2}\right)\\\
&=C_{0}\cdot
2^{m(2d+1)}\cdot\prod_{i=1}^{\frac{m}{2}}\left(2^{-8i}\right)\prod_{i=1}^{\frac{m}{2}}\left(|{\boldsymbol{\alpha_{i}}}|^{4d-6-8i+8}\right)\\\
&=C_{0}\cdot
2^{m(2d-m-1)}\prod_{i=1}^{\frac{m}{2}}\left(|{\boldsymbol{\alpha_{i}}}|^{4d-6}\right)\prod_{i=1}^{\frac{m}{2}}\left(|{\boldsymbol{\alpha_{i}}}|^{-8i+8}\right).\end{split}$

Let

$C(d,m):=C_{0}\cdot 2^{m(2d-m-1)}.$

Then we see that

$\begin{split}C(d,m)&\leq(2r)^{d^{2}+m^{2}-2dm-d+m}\cdot 2^{m(2d-m-1)}\\\
&=(2r)^{d(d-1)+m(m+1-2d)}\cdot 2^{m(2d-m-1)}\\\ &\leq(2r)^{d(d-1)}\cdot
2^{m(m+1-2d+2d-m-1)}=(2r)^{d(d-1)},\end{split}$

where we used that $r\geq 1$ and $m+1-2d\leq d+1-2d\leq 0$. Hence,

$\begin{split}|\Delta_{\boldsymbol{\alpha}}|&\leq(2r)^{d(d-1)}\prod_{i=1}^{\frac{m}{2}}\left(|{\boldsymbol{\alpha_{i}}}|^{4d-6}\right)\prod_{i=1}^{\frac{m}{2}}\left(|{\boldsymbol{\alpha_{i}}}|^{-8i+8}\right)\\\
&=(2r)^{d(d-1)}\prod_{i=1}^{\frac{m}{2}}\left(|{\boldsymbol{\alpha_{i}}}|^{2}\right)^{2d-3}\prod_{i=1}^{\frac{m}{2}}\left(|{\boldsymbol{\alpha_{i}}}|^{-8i+8}\right)\\\
&=(2r)^{d(d-1)}{\mathfrak{M}}(\boldsymbol{\alpha})^{2d-3}\prod_{i=1}^{\frac{m}{2}}\left(|{\boldsymbol{\alpha_{i}}}|^{-8i+8}\right).\end{split}$

Consider the remaining product here. We have that
$|{\boldsymbol{\alpha_{i}}}|>r\geq 1$ for all $i=1,\ldots,\frac{m}{2}$; and
$-8i+8\leq 0$ also for all $i=1,\ldots,\frac{m}{2}$. Hence,
$0<|{\boldsymbol{\alpha_{i}}}|^{-8i+8}\leq 1$ for all
$i=1,\ldots,\frac{m}{2}$, and thus
$0<\prod_{i=1}^{\frac{m}{2}}|{\boldsymbol{\alpha_{i}}}|^{-8i+8}\leq 1$. We
conclude that

$|\Delta_{\boldsymbol{\alpha}}|\leq(2r)^{d(d-1)}{\mathfrak{M}}(\boldsymbol{\alpha})^{2d-3}$

which proves inequality (2.6).

Finally, to see that the exponent $2d-3$ is sharp it suffices to consider
$d-2$ distinct fixed points on the $(k-1)$-sphere of radius $r$ centred at the
origin and taking the remaining two points ${\boldsymbol{\alpha_{1}}}$ and
${\boldsymbol{\alpha_{1}}}^{\prime}=-{\boldsymbol{\alpha_{1}}}$ with
arbitrarily large Euclidean distance. This completes the proof of Theorem 3. ∎

##  3\. Upper bounds for the Mahler measure of minimal polynomials of integral
generators

In this section we prove Proposition 1. Let $K$ be a number field of degree
$d$. Let $\sigma_{i}:K\rightarrow\mathbb{R}$ for $1\leq i\leq r$ be the $r$
real embeddings, and let $\sigma_{i},\sigma_{i+s}:K\rightarrow\mathbb{C}$ for
$r+1\leq i\leq r+s$ be the $s$ pairs of complex conjugate embeddings, so that
$d=r+2s$. Let

$\begin{split}\boldsymbol{\sigma}:\
&K\rightarrow\mathbb{R}^{r}\times\mathbb{C}^{s}\\\
&\alpha\mapsto(\sigma_{1}(\alpha),\ldots,\sigma_{r}(\alpha),\sigma_{r+1}(\alpha),\ldots,\sigma_{r+s}(\alpha))\end{split}$

be the Minkowski embedding of $K$, and set
$\Lambda=\boldsymbol{\sigma}O_{K}\subseteq\mathbb{R}^{r}\times\mathbb{C}^{s}$.
Identifying $\mathbb{C}\cong R^{2}$ via the isomorphism
$\alpha\mapsto(\text{Re}(\alpha),\text{Im}(\alpha))$ turns $\Lambda$ into a
lattice in $\mathbb{R}^{d}$. For the convenience of the reader we recall
Proposition 1.

######  Proposition 2.

Let $K$ be a number field of degree $d$ with a non-real embedding (i.e. $s\geq
1)$. Then there exists $c_{K}>0$ depending only on $K$ such that there are
infinitely many $\alpha\in{\mathcal{O}}_{K}$ with $K=\mathbb{Q}(\alpha)$,

$M(f_{\alpha,\mathbb{Z}})^{2d-3}\leq c_{K}|\Delta_{f_{\alpha,\mathbb{Z}}}|,$

and all roots of $f_{\alpha,\mathbb{Z}}$, except one pair of complex conjugate
roots, have modulus at most $c_{K}$.

###### Proof.

For real $c,T\geq 1$ we define the subset $S_{c,T}$ of
$\mathbb{R}^{r}\times\mathbb{C}^{s}$ as the set of points ${\bf
x}=(x_{1},\ldots,x_{r+s})\in\mathbb{R}^{r}\times\mathbb{C}^{s}$ such that

$\begin{cases}(2i-2)c<x_{i}<(2i-1)c\quad&\text{for}\ i=1,\ldots,r\\\
(2i-2)c<\text{Re}(x_{i})<(2i-1)c\quad&\text{for}\ i=r+1,\ldots,r+s-1\\\
c<\text{Im}(x_{i})<2c\quad&\text{for}\ i=r+1,\ldots,r+s-1\\\
T<\text{Re}(x_{r+s})<T+c\\\ 3c+T<\text{Im}(x_{r+s})<T+4c.\\\ \end{cases}$

If we identify $\mathbb{R}^{r}\times\mathbb{C}^{s}$ with $\mathbb{R}^{d}$ like
before, then $S_{c,T}$ is a box with sides parallel to the axes, each of
length $c$. Therefore, there exists $c_{\Lambda}$ such that $S_{c,T}$ contains
a point of the lattice $\Lambda$ whenever $c\geq c_{\Lambda}$.

Furthermore, for all ${\bf x}=(x_{1},\ldots,x_{r+s})\in S_{c,T}$,

$|x_{i}-x_{j}|>c\quad(1\leq i<j\leq r+s).$

This implies that for $\alpha\in K$ and $\boldsymbol{\sigma}(\alpha)\in
S_{c,T}$,

(3.8) $|\sigma_{i}(\alpha)-\sigma_{j}(\alpha)|>c\quad(1\leq i<j\leq r+s).$

If $r+s+1\leq i<j\leq r+2s$, then
$|\sigma_{i}(\alpha)-\sigma_{j}(\alpha)|=|\overline{\sigma_{i-s}(\alpha)-\sigma_{j-s}(\alpha)}|=|\sigma_{i-s}(\alpha)-\sigma_{j-s}(\alpha)|>c$,
by (3.8). If $1\leq i\leq r+s$ and $j\geq r+s+1$ then

$|\sigma_{i}(\alpha)-\sigma_{j}(\alpha)|\geq|\text{Im}(\sigma_{i}(\alpha))-\text{Im}(\sigma_{j}(\alpha))|>|0-(-c)|=c.$

We have shown that if $\alpha\in K$ and $\boldsymbol{\sigma}(\alpha)\in
S_{c,T}$ then

(3.9) $|\sigma_{i}(\alpha)-\sigma_{j}({\alpha})|>c$

whenever $1\leq i<j\leq d$.

Take $c=c_{\Lambda}\geq 1$ as before. Then there exists
$\alpha\in{\mathcal{O}}_{K}$ such that
$\boldsymbol{\sigma}(\alpha)\in\Lambda\cap S_{c,T}$. Let
$f_{\alpha,\mathbb{Z}}\in\mathbb{Z}[x]$ be the minimal polynomial of $\alpha$.
Now (3.9) implies that $\sigma_{i}(\alpha)\neq\sigma_{j}(\alpha)$ for all
$1\leq i<j\leq d$. Hence, we conclude $\mathbb{Q}(\alpha)=K$. Finally, using
(3.9) we get

$\begin{split}|\Delta_{f_{\alpha,\mathbb{Z}}}|&=\prod_{i\neq
j}|\sigma_{i}(\alpha)-\sigma_{j}(\alpha)|\\\ &=\prod_{\begin{subarray}{c}i\neq
j\\\ i,j\neq
r+s,r+2s\end{subarray}}|\sigma_{i}(\alpha)-\sigma_{j}(\alpha)|\cdot\prod_{i\neq
r+s}|\sigma_{i}(\alpha)-\sigma_{r+s}(\alpha)|^{2}\cdot\prod_{\begin{subarray}{c}i\neq
r+2s\\\ i\neq
r+s\end{subarray}}|\sigma_{i}(\alpha)-\sigma_{r+2s}(\alpha)|^{2}\\\ &\geq
c^{(d-3)(d-2)}\cdot T^{2(d-1)}\cdot T^{2(d-2)}\\\ &=c^{(d-3)(d-2)}\cdot
T^{2(2d-3)}.\end{split}$

On the other hand, using $|z|\leq|\text{Re}(z)|+|\text{Im}(z)|$ for
$z\in\mathbb{C}$,

$\displaystyle M(f_{\alpha,\mathbb{Z}})$
$\displaystyle=\prod_{i=1}^{d}\text{max}\\{1,|\sigma_{i}(\alpha)|\\}$
$\displaystyle\leq(\left(2(r+s-1)+1)c\right)^{d-2}(2T+5c)^{2}$ (3.10)
$\displaystyle\leq(2dc)^{d-2}(3T)^{2}$ $\displaystyle\leq(2dc)^{d}T^{2},$

provided $T\geq 5c$. Hence,

$\begin{split}M(f_{\alpha,\mathbb{Z}})^{2d-3}&\leq(2dc)^{d(2d-3)}c^{-(d-3)(d-2)}|\Delta_{f_{\alpha,\mathbb{Z}}}|\\\
&\leq(2dc)^{d(2d-3)}|\Delta_{f_{\alpha,\mathbb{Z}}}|\\\
&=c_{K}|\Delta_{f_{\alpha,\mathbb{Z}}}|,\end{split}$

where $c_{K}:=(2dc)^{d(2d-3)}$ depends only on $K$, as $c_{\Lambda}$ depends
only on $\boldsymbol{\sigma}{\mathcal{O}}_{K}=\Lambda$.

Choosing a sequence of $T$’s, say $T_{1},T_{2},T_{3},\ldots$ with
$T_{i+1}>T_{i}+c$, yields a sequence of disjoint boxes $S_{c,T_{i}}$, each box
containing an admissible $\alpha_{i}\in{\mathcal{O}}_{K}$ with
$K=\mathbb{Q}(\alpha_{i})$ and $M(f_{\alpha_{i},\mathbb{Z}})^{2d-3}\leq
c_{K}|\Delta(f_{\alpha_{i},\mathbb{Z}})|$. Moreover, as observed in (3), all
roots of $f_{\alpha_{i},\mathbb{Z}}$ have modulus at most $2dc\leq c_{K}$
except one pair of complex conjugate roots. This proves the theorem. ∎

It is clear that the proof of Proposition 2 can be adapted to show that for
any given number field $K$ of degree $d\geq 2$ with at least one real
embedding there exist infinitely many $\alpha\in{\mathcal{O}}_{K}$ with
$K=\mathbb{Q}(\alpha)$,

$M(f_{\alpha,\mathbb{Z}})^{2d-2}\leq c_{K}|\Delta_{f_{\alpha,\mathbb{Z}}}|,$

and all roots of $f_{\alpha,\mathbb{Z}}$, except one, have modulus at most
$c_{K}$.

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