# Finite-size scaling at the edge of disorder in a time-delay Vicsek model

Viktor Holubec<EMAIL_ADDRESS>Charles University, Faculty of
Mathematics and Physics, Department of Macromolecular Physics, V Holešovičkách
2, CZ-180 00 Praha, Czech Republic  Daniel Geiss  Institut für Theoretische
Physik, Universität Leipzig, Postfach 100 920, D-04009 Leipzig, Germany  Max
Planck Institute for Mathematics in the Sciences, D-04103 Leipzig, Germany
Sarah A.M. Loos  Institut für Theoretische Physik, Universität Leipzig,
Postfach 100 920, D-04009 Leipzig, Germany  Klaus Kroy  Institut für
Theoretische Physik, Universität Leipzig, Postfach 100 920, D-04009 Leipzig,
Germany  Frank Cichos  Peter Debye Institute for Soft Matter Physics,
Universität Leipzig, 04103 Leipzig, Germany.

###### Abstract

Living many-body systems often exhibit scale-free collective behavior
reminiscent of thermal critical phenomena. But their mutual interactions are
inevitably retarded due to information processing and delayed actuation. We
numerically investigate the consequences for the finite-size scaling in the
Vicsek model of motile active matter. A growing delay time initially
facilitates but ultimately impedes collective ordering and turns the dynamical
scaling from diffusive to ballistic. It provides an alternative explanation of
swarm traits previously attributed to inertia.

Interacting assemblies of active elements ranging from neural networks in the
brain to forest fires and bird flocks can exhibit scale-free behavior [1, 2,
3, 4]. This might be indicative of an underlying powerful physical ordering
principle overwriting their inherent complexity. Finite-size scaling theory
[5] associates such behavior with a correlation length exceeding the system
size and conjectured to arise from a mechanism called self-organized
critically [6]. It is indeed an appealing idea that simple interaction rules,
such as an innate tendency of individual active elements to replicate the
action of their neighbors, can drive a non-equilibrium ensemble towards
criticality. Yet, it does not generally seem to apply to both natural systems
[7, 8] and their models [9, 10, 11].

Studying the emergent finite-size scaling in natural assemblies is thus vital
for their prospective modeling in the spirit of non-equilibrium many-body
systems, as successful models should be required to reproduce the observed
scaling [10] and correlations [12]. In this vein, the inertia spin model [12]
was proposed to overcome known deficiencies of the classical Vicsek model (VM)
[13] in comparison with empirical data for natural swarms and flocks. Inspired
by observations of birds and insects [4], which cannot turn instantaneously,
it adds inertia to the navigation rules for the individual motile spins. The
dynamical scaling and time correlation functions found in natural swarms [14]
of moderate size (dynamical exponent $z\approx 1.1$) could thereby be
reconciled with the classical VM prediction ($z=2$) [4].

However, it seems worthwhile to point out that, for motile ensembles, physical
inertia can have quite similar effects as delayed reactions due to finite
speeds of information transfer and processing, and actuation [15, 16, 17]. The
latter are indeed ubiquitously found both in nature and in engineering
practice, from insects to birds and various robotic systems [18, 19, 20], and
also thought to cause traffic jams [21]. Recent experiments [22, 23] with
feedback-driven artificial microswimmers [24] have moreover established the
role of delayed interactions also in the microscopic world of active Brownian
particles. Beyond oscillatory behavior, which is also a common trait of
inertial motion, time-delayed interactions can give rise to multistability,
instabilities, and even chaos [25, 26, 17, 27]. Conversely, intermediate time
delays may facilitate clustering compared to the classical VM [26] and
flocking in the Cucker–Smale model [28]. And recent indications that delay-
dependent optimizations play a role in artificial microswimmer assemblies [23]
seem reminiscent of the optimum run-and-tumble times of bacteria [29, 30] or
the improved localisation achieved with feedback cooling [31, 32] or feedback-
driving of robots [33].

Figure 1: Sketch of the delayed Vicsek model defined by Eqs. (1) and (2). The
conventional alignment rule with interaction range $R$ is executed with a time
delay $\tau$.

In this Letter, we demonstrate that by introducing retarded reactions in the
classical VM [13], we can obtain the same finite-size scaling and time
correlations near the ordering transition as previously obtained with the
inertia spin model [4]. This suggests that similar scaling and correlations as
observed for natural swarms can be expected for a wealth of systems with time-
delayed interactions, including over-damped Brownian particle assemblies. We
can also corroborate the observation that increasing delay times may have a
non-monotonic effect on the stability of coherent collective motion [26].

Model: The (classical) VM [13] arguably is the simplest model for motile
active-particle assemblies, ranging from bacteria to birds, and a central
paradigm in the field motile active matter [34]. In each discrete time step,
all particles advance with the same constant speed $v_{0}$. And they
instantaneously adapt their orientations to the previous average orientation
of their neighbors within an interaction sphere of radius $R$, up to some
random error contributed by a local noise term. The orientation is thus
clearly an overdamped variable, as it bears no inertia.

In the delay VM, depicted in Fig. 1, particle $i$ adapts at time $t$ to the
mean orientation of all particles that had distance less than $R$ from its
previous position, at time $t-1-\tau$, with an integer-valued time delay
$\tau\geq 0$. The discrete time step and the interaction radius $R=1$ serve as
units of time and length, respectively. The dynamics of the standard VM is
recovered for $\tau=0$. The velocity $\mathbf{v}_{i}$ and position
$\mathbf{r}_{i}$ of particle $i$ in three spatial dimensions (3D) thus obey
the set of equations [13]

$\displaystyle\mathbf{v}_{i}(t+1)$
$\displaystyle=v_{0}\mathcal{R}_{\alpha}\Theta\\!\left[\mathbf{v}_{i}(t)+\sum_{j}n_{ij}(t-\tau)\mathbf{v}_{j}(t-\tau)\right],$
(1) $\displaystyle\mathbf{r}_{i}(t+1)$
$\displaystyle=\mathbf{r}_{i}(t)+\mathbf{v}_{i}(t+1).$ (2)

The noise operator $\mathcal{R}_{\alpha}X$ randomly rotates its argument $X$
within a uniformly distributed solid angle $4\pi\alpha$ centered around $X$,
and $\Theta(\mathbf{v})\equiv\mathbf{v}/|\mathbf{v}|$ normalizes its argument.
We assume geometric interactions corresponding to the connectivity matrix
elements $n_{ij}(t)=1$ for $i\neq j$ if
$r_{ij}(t)=|\mathbf{r}_{i}(t)-\mathbf{r}_{j}(t)|<R$, and $n_{ij}(t)=0$
otherwise.

We simulated the delay VM with fixed speed $v_{0}=0.05$ and noise strength
$\alpha=0.45$ inside a cube with size $L^{3}$ and periodic boundary conditions
for six values of the particle number $N=2^{n}$, $n=6,\dots,11$. In this
setting, we repeated the analysis performed in Refs. [35, 14] for the static
and dynamic scaling and the correlation functions of the standard VM, for
delay times $\tau=0,\dots,20$. As control parameter, we used the average
nearest-neighbor distance $r_{\mathrm{1}}$ between the individual particles,
which we changed by varying $L$. Here, we present the main results from the
simulations. Further data, technical details, and some analytical discussion
can be found in the Supplemental Material [36].

The central object for our data analysis is the Fourier transformed spatio-
temporal correlation function (CF)

$C(k,t)=\left<\frac{1}{N}\sum_{i,j}^{N}\frac{\sin[k\,r_{ij}(t,t_{0})]}{kr_{ij}(t,t_{0})}\delta\hat{\mathbf{v}}_{i}(t_{0})\cdot\delta\hat{\mathbf{v}}_{j}(t_{0}+t)\right>$
(3)

of the normalized velocity fluctuations [36, 14, 4]

$\delta\hat{\mathbf{v}}_{i}=\frac{\delta\mathbf{v}_{i}}{\sqrt{N^{-1}\sum_{k}\delta\mathbf{v}_{k}\cdot\delta\mathbf{v}_{k}}},$
(4)

where $\delta\mathbf{v}_{i}=\mathbf{v}_{i}-\sum_{k}\mathbf{v}_{k}/N$ is the
deviation of the velocity of particle $i$ from the average velocity, and
$r_{ij}(t_{0},t)=|\mathbf{r}_{i}(t_{0})-\mathbf{r}_{j}(t)|$ the distance
between particles $i$ and $j$ at times $t$ and $t_{0}<t$. The average
$\left<\dots\right>$ is taken over $t_{0}$ [36].

Static scaling: At $t=0$, $C(k,0)$ exhibits a global maximum at
$k=k^{\star}\sim 1/\xi$, where $\xi$ corresponds to the correlation length.
Assuming proportionality between fluctuation and response, this value of the
CF is interpreted as a susceptibility $\chi\equiv C(k^{*},0)$ [36, 14, 4].

For given delay time $\tau$ and particle number $N$, the susceptibility $\chi$
exhibits a maximum $\chi^{*}=\chi^{*}(\tau,N)$ as function of the nearest
neighbour distance at $r_{1}^{*}=r_{1}^{*}(\tau,N)$. The system is found to be
ordered (large average velocity) for $r_{1}<r_{1}^{*}$ and disordered (small
average velocity) otherwise. For given $N$, the susceptibility $\chi^{*}$ at
the transition decreases monotonically with growing $\tau$ and eventually
saturates, see Figs. 2a and b and the supplemental Figs. S1 and S2 [36]. The
equal-time orientation correlations are thus generally reduced for retarded as
opposed to instantaneous interactions, which suggests that the sensitivity to
external perturbations decreases accordingly. For sufficiently large $\tau$
and $N$, the derivative of the susceptibility with respect to $r_{1}$ abruptly
increases at some $r_{1}<r_{1}^{*}$, see the vertical dotted line at
$r_{1}\approx 0.4$ in Fig. 2b. No such kink is observed for small $\tau$ [36].

Figure 2: Static finite-size scaling in the 3D delay VM. Each simulation
departed from a random initial state and was evolved for a transient period of
1500 time-steps before measurements started. Panels a) and b) show the
susceptibilities $\chi\equiv\max_{k}C(k,0)$ averaged over 11 trajectories of
$10^{4}$ time-steps for $N=64,128,256,512,1024,2048$ particles (from bottom to
top) and delay times $\tau=0$ [a), standard VM] and $\tau=15$ [b), $\tau
v_{0}/R=0.75$], respectively, over the mean nearest neighbour distance
$r_{1}$. Panels c) and d) illustrate the data collapse achieved for $N\geq
256$, e)-h) show box-plots [37] of the exponent ratio $\gamma/\nu$, the
individual exponents, and the extrapolated critical parameter $r_{C}$ for
$N\to\infty$, respectively, all for delay times $\tau$ from 0 to 20. Broken
dashed lines mark averages over the 11 realizations. The horizontal dotted
lines depict the values $\nu\approx 0.75$, $\gamma\approx 1.53$, and
$r_{C}\approx 0.41$ obtained for $\tau=0$, where the model reduces to the
standard VM, consistent with the data in Ref. [35]. In a) and b) black squares
mark susceptibility maxima $\chi(r_{1}^{*})$ and solid lines are computed
using the finite-size scaling relation
$\chi\sim(r_{\mathrm{1}}^{*}-r_{C})^{-\gamma}$ with parameters from e)-h). The
vertical dotted line in b) marks the abrupt changes in the slope of $\chi$.

For given $\tau$ and large enough $N$, the susceptibility in the vicinity of
the ordering transition [35] exhibits finite-size scaling according to [38]

$\displaystyle r_{\mathrm{1}}^{*}$ $\displaystyle\sim r_{C}+N^{-1/(3\nu)},$
(5) $\displaystyle\chi$ $\displaystyle\sim N^{\gamma/(3\nu)}.$ (6)

In other words, for any given $\tau$, the limiting location
$r_{C}=r_{\mathrm{1}}^{*}(\tau,\infty)$ of the transition for large (infinite)
particle numbers and the critical exponents $\gamma$ and $\nu$ of the
susceptibility $\chi\sim(r_{1}^{*}-r_{C})^{-\gamma}$ and the correlation
length $\xi\sim(r_{1}^{*}-r_{C})^{-\nu}$, respectively, can all be
extrapolated from a data collapse of the susceptibilities for different $N$.
The procedure is illustrated in Figs. 2c and d. The resulting exponents and
$r_{C}$ in Figs. 2e-h exhibit strong dependencies on $\tau$, which saturate as
$\tau v_{0}/R\approx 1/2$, as also suggested by an analytical argument [36].

Since the static critical exponents in the standard VM are known to depend
strongly on the density, speed and interaction radius [39], their absolute
values are of limited interest. Rather, their trends and dependencies are
revealing. The critical nearest-neighbor distance $r_{C}$ in Fig. 2h exhibits
a pronounced maximum at $\tau v_{0}/R\approx 0.2$, indicating that a system
with an intermediate delay time favors order already at lower densities as
compared to the system without delay. This somewhat counter-intuitive result
is in agreement with findings of Refs. [33, 26] that intermediate delays
stabilize collective motion. For larger delay times, the critical nearest-
neighbor distance $r_{C}$ drops sharply to a value below that for the standard
VM.

The exponents $\nu$ and $\gamma$ also display local maxima, but at somewhat
larger $\tau$. Their saturation values are much higher than the respective
critical exponents in all known universality classes, including the standard
VM. To attribute the observed finite-size scaling to a critical point in the
infinite-size limit according to the conventional scaling hypothesis [40]
would require an extraordinarily sharp divergence of the correlation length
and the susceptibility at criticality, approached extraordinarily slowly with
increasing particle number $N$, see the solid lines in Figs. 2a and b.

Figure 3: Dynamical scaling of the orientational correlations at the
susceptibility maximum displayed in Fig. 2. a) Normalized time-correlation
functions $\overline{C}(t)$ for delays $\tau=0,5,20$ and $N=2048$. b) The
upper envelopes $\overline{C}_{m}$ of the curves in a). c) The correlation
functions $\hat{C}(t)$ calculated from trajectories (under)sampled with
frequency $1/(\tau+1)$ for $N=2048$. d) - f) The normalized CFs
$\overline{C}$, $\overline{C}_{m}$ and $\hat{C}$ collapse upon measuring time
in units of the relaxation time $\tau_{R}$ for $\tau=0$ (d) and $\tau=20$ (e)
and (f) and system sizes $N=2^{n}$, $n=8,\dots,11$ ($50$ simulation runs for
each system size); g) – i) the corresponding slopes. Box-plots [37] in j), k)
show the dynamical exponent $z$ obtained from collapsing the CFs
$\overline{C}_{m}$, $\hat{C}$ by rescaling time as $t\xi^{z}$. The green
circles are from linear fits to $\log\tau_{R}(N)=-z\log k^{*}(N)$ for all 50
data sets with 95% confidence intervals of the fits (dashed).

Time correlation functions: The time dependence of the CFs (3) for the delay
VM at the transition, quantifying the temporal loss of orientational
correlations [14, 4], is strongly influenced by the delay. Figure 3a shows
that the normalized CFs $\overline{C}(t)\equiv C(k^{*},t)/C(k^{*},0)$ acquire
oscillations with period ($\tau+1$) and an amplitude increasing with $\tau$.
They reveal the transmission of orientational correlations over discrete time
steps $\tau+1$ and can be understood analytically by a spin-wave theory that
accounts for the delay [36].

In Fig. 3d, we show that logarithms of CFs for $N=2^{n}$, $n=8,\dots,11$ and
$\tau=0$ collapse onto the master curve $-t/\tau_{R}$ upon rescaling time by
the relaxation times $\tau_{R}$ obtained from Eq. (S14) [36]. Figure 3g shows
the corresponding increasingly negative time derivatives for $t\to 0$,
indicating the exponential loss of correlation in the standard VM [14].

Due to the delay-induced superimposed oscillations, the initial slope of the
CFs always steepens with increasing delay time $\tau$. In contrast, the
_overall_ decay flattens with $\tau$, as revealed by the upper envelopes
$\overline{C}_{m}(t)$ of $\overline{C}(t)$, in Fig. 3b. As intuitively
expected, the delayed interactions thus tend to increase the memory in the VM.
The data collapse of $\overline{C}_{m}(t/\tau_{R})$ in Fig. 3 confirms the
non-exponential relaxation, while its slope in Fig. 3h still increases for
$t\to 0$ (see Sec. S5 [36] for an approximate analytical result).

However, we note that the sampling rates used in practical measurements may
not always be sufficient to resolve delay induced oscillations [36], which
might moreover have a tendency to be washed out by a natural dispersion of the
delay times. To account for such undersampling, Fig. 3c shows normalized CFs
$\hat{C}(t)=C_{\rm C}(k^{*},t)/C_{\rm C}(k^{*},0)$ calculated from particle
positions that were (under-)sampled with frequency $1/(\tau+1)$, i.e. we
calculated the corresponding velocities as
$\mathbf{v}_{i}(t)=[\mathbf{r}_{i}(t)-\mathbf{r}_{i}(t-\tau-1)]/(\tau+1)$. In
Fig. S9 [36] we show that the undersampled CFs are independent of the sampling
rate as long as it is comparable to or smaller than $1/\tau$. The resulting
CFs are shown in Fig. 3c. The exponential initial decay for vanishing delay
time $\tau=0$ is seen to increasingly flatten with growing $\tau$, as also
corroborated by the data collapse in Fig. 3f. For $v_{0}\tau\gtrsim 0.5$, the
absolute slope $|\dot{\overline{C}}(t)|$ starts to decrease for $t\to 0$ as
shown in Fig. 3i and Figs. S6 and S8 [36].

The undersampled delay VM yields qualitatively the same relaxation of
orientational correlations as observed for natural swarms [14]. It thus
provides an alternative explanation for the data, which where so far
interpreted within the inertia spin model [41] — the version of the VM in
which the particle reorientation exhibits inertial dynamics, which amounts to
another form of time-delayed alignment interactions. The dynamics induced by a
discrete time delay appears to be more prone to developing oscillatory
patterns, as the ones illustrated in Fig. 3a, though.

Dynamical scaling: The dynamical scaling hypothesis [40] states that the
relaxation time $\tau_{R}$ diverges with the correlation length as
$\tau_{R}\sim\xi^{z}\sim(k^{*})^{-z}$, at a critical point. Directly fitting
the relation

$\log\tau_{R}(N)=-z\log k^{*}(N)$ (7)

for the relaxation times of $\overline{C}_{m}$ and $\hat{C}$ as functions of
time delay $\tau$ yield the dynamical exponent $z$ as depicted by circles in
Figs. 3j and k, respectively. The figures also show box-plots [37] resulting
from the best data collapse of CFs $\overline{C}_{m}$ and $\hat{C}$ using
$(k^{*})^{-z}$ with $z$ as a free parameter in place of the relaxation time
$\tau_{R}$, itself. The exponents $z$ obtained from these two approaches
nicely agree for each CF. While the values for $z$ obtained from the two
alternative CFs differ, they exhibit the same robust trend: a crossover from a
“diffusive” to a “ballistic” signature, i.e., from $z\approx 2$ for $\tau=0$
to about $z\approx 1.1$ for $\tau v_{0}/R\approx 1$. The saturation of
$z(\tau)$ for large $\tau$ follows from that of $\xi(\tau)$ (implied by the
saturation of the finite-size scaling parameters in Fig. 2) and
$\tau_{R}(\tau)$ (Fig. S11 [36]). These results can therefore reconcile the
standard VM predictions with the observations for natural swarms.

Conclusion: We analyzed the VM with retarded interactions, as they are
expected from natural delays between sensing and reaction. It provides an
alternative to the rotational inertia hypothesis for reconciling the
discrepancies in the dynamical scaling and relaxation between the standard VM
and natural swarms [41, 4]. While the navigation of insects and other flying
species is certainly influenced both by inertia and time delay, our focus on
delay could help to better understand their relation to feedback-driven
robotic [33] or micro-particle [22, 42, 23] swarms. Especially in the latter,
current experimental techniques [24] allow inertial effects to be suppressed
so that only the unavoidable time delay remains.

###### Acknowledgements.

Acknowledgements: We acknowledge funding through a DFG-GACR cooperation by the
Deutsche Forschungsgemeinschaft (DFG Project No 432421051) and by the Czech
Science Foundation (GACR Project No 20-02955J). V. H. was supported by the
Humboldt Foundation. D.G. acknowledges funding by International Max Planck
Research Schools (IMPRS). The time-correlation functions were computed at the
GPU-Cluster Clara at the Scientific Computing Center in Leipzig. We are
grateful to Lokrshi Dadhichi for his helpful remarks concerning the final
manuscript. Support by Kiril Panayotov Blagoev during a preliminary stage of
the project is also acknowledged.

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# On the Area of the Fundamental Region of a Binary Form Associated with
Algebraic Trigonometric Quantities

Anton Mosunov

###### Abstract

Let $F(x,y)$ be a binary form of degree at least three and non-zero
discriminant. We estimate the area $A_{F}$ bounded by the curve $|F(x,y)|=1$
for four families of binary forms. The first two families that we are
interested in are homogenizations of minimal polynomials of
$2\cos\left(\frac{2\pi}{n}\right)$ and $2\sin\left(\frac{2\pi}{n}\right)$,
which we denote by $\Psi_{n}(x,y)$ and $\Pi_{n}(x,y)$, respectively. The
remaining two families of binary forms that we consider are homogenizations of
Chebyshev polynomials of the first and second kinds, denoted $T_{n}(x,y)$ and
$U_{n}(x,y)$, respectively.

##  1 Introduction

Let $F(x,y)$ be a binary form with real coefficients of degree $n\geq 3$ and
non-zero discriminant $D_{F}$. The set

$\\{(x,y)\in\mathbb{R}^{2}\colon|F(x,y)|\leq 1\\},$

which represents the collection of all points bounded by the curve
$|F(x,y)|=1$, is called the _fundamental region_ of $F$. It was proved by
Mahler [7] that the area $A_{F}$ of the fundamental region of $F$ is finite.

In what follows, we restrict our attention to binary forms $F$ having integer
coefficients. The quantity $A_{F}$, which can be evaluated via the formula [1,
Section 3]

$A_{F}=\int\limits_{-\infty}^{+\infty}\frac{dx}{|F(x,1)|^{2/n}},$ (1)

plays a significant role in the analysis of certain Diophantine equations and
inequalities associated with $F$. Mahler [7] proved that, for a positive
integer $h$, the number of integer solutions $Z_{F}(h)$ to the _Thue
inequality_ $|F(x,y)|\leq h$ satisfies

$\left|Z_{F}(h)-A_{F}h^{2/n}\right|\ll_{F}h^{1/(n-1)}.$

In 2019, Stewart and Xiao [13] proved that the number of integers of absolute
value at most $h$ which are represented by the form $F$ is asymptotic to
$C_{F}h^{2/n}$, where $C_{F}$ is a positive number which depends on $F$ and is
a rational multiple of $A_{F}$.

The values $A_{F}$ for certain binary forms $F$ are intimately connected to
the values of the _beta function_

$B(x,y)=\int\limits_{0}^{1}t^{x-1}(1-t)^{y-1}dt,$ (2)

where $x$ and $y$ are complex numbers with positive real parts. An overview of
important properties of $B(x,y)$ is given in Section 2. For a matrix
$M=\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)$, with real
coefficients, define

$F_{M}(x,y)=F(ax+by,cx+dy).$

Two forms $F$ and $G$ are said to be _equivalent under_
$\operatorname{GL}_{2}(\mathbb{R})$ if $G=F_{M}$ for some
$M\in\operatorname{GL}_{2}(\mathbb{R})$. Analogously, we define the
equivalence of forms under $\operatorname{GL}_{2}(\mathbb{Z})$. In 1994, Bean
[1, Corollary 1] proved that
$A_{F}=3B\left(\frac{1}{3},\frac{1}{3}\right)=15.8997\ldots$ when $F$ is a
binary form that is equivalent under $\operatorname{GL}_{2}(\mathbb{Z})$ to
$xy(x-y)$. Furthermore, he proved that this value is the largest among all
binary forms with integer coefficients, degree at least three and non-zero
discriminant.

It was proved by Stewart and Xiao [13, Corollary 1.3] that if $a,b$ are fixed
non-zero integers and $F_{n}(x,y)=ax^{n}+by^{n}$ is a binomial form of degree
$n\geq 3$, then

$A_{F_{n}}=\begin{cases}\frac{1}{n|ab|^{1/n}}\left(2B\left(\frac{1}{n},1-\frac{2}{n}\right)+B\left(\frac{1}{n},\frac{1}{n}\right)\right)&\text{if
$n$ is odd,}\\\
\frac{2}{n|ab|^{1/n}}B\left(\frac{1}{n},\frac{1}{n}\right)&\text{if $ab$ is
positive and $n$ is even,}\\\
\frac{4}{n|ab|^{1/n}}B\left(\frac{1}{n},1-\frac{2}{n}\right)&\text{if $ab$ is
negative and $n$ is even.}\end{cases}$ (3)

In Section 3 we prove that $\lim\limits_{n\rightarrow\infty}A_{F_{n}}=4$.

In 2019, Fouvry and Waldschmidt [4, Théorème 1.5] estimated $A_{\Phi_{n}}$ for
cyclotomic binary forms $\Phi_{n}$ (note that a positive integer $n$ no longer
refers to the degree of the form). It is a consequence of their result that
for any $\varepsilon>0$ there exists $n_{0}=n_{0}(\varepsilon)$ such that for
all $n\geq n_{0}$ the inequalities

$\left(2-n^{-1+\varepsilon}\right)^{2}<A_{\Phi_{n}}<\left(2+n^{-1+\varepsilon}\right)^{2}$

are satisfied. Consequently, $\lim\limits_{n\rightarrow\infty}A_{\Phi_{n}}=4$.

In this article we estimate $A_{F}$ for four families of binary forms. Let
$\Psi_{n}(x)$ and $\Pi_{n}(x)$ denote the minimal polynomials of
$2\cos\left(\frac{2\pi}{n}\right)$ and $2\sin\left(\frac{2\pi}{n}\right)$,
respectively. The first two families that we are interested in are
$\Psi_{n}(x,y)$ and $\Pi_{n}(x,y)$, which are homogenizations of $\Psi_{n}(x)$
and $\Pi_{n}(x)$, respectively. By [15, Lemma],

$\Psi_{n}(x,y)=\prod\limits_{\begin{subarray}{c}1\leq k<\frac{n}{2}\\\
\gcd(k,n)=1\end{subarray}}\left(x-2\cos\left(\frac{2\pi k}{n}\right)y\right).$
(4)

Further, since
$\sin\left(\frac{2\pi}{n}\right)=\cos\left(\frac{2\pi(n-4)}{4n}\right)$, it
follows from (4) that $\sin\left(\frac{2\pi}{n}\right)$ is an algebraic
conjugate of $\cos\left(\frac{2\pi}{c(n)}\right)$, where $c(n)$ is the
denominator of $\frac{n-4}{4n}$ (in lowest terms). Consequently,

$\Pi_{n}(x,y)=\Psi_{c(n)}(x,y).$

See Figure 1 for the graphs of $|\Psi_{13}(x,y)|=1$ and $|\Pi_{9}(x,y)|=1$.
Note that in both cases the fundamental regions are _not_ compact.

Figure 1: Graphs of $|\Psi_{13}(x,y)|=1$ (left) and $|\Pi_{9}(x,y)|=1$ (right)

Next, let $T_{n}(x)$ and $U_{n}(x)$ denote Chebyshev polynomials of the first
and second kinds, respectively. The other two families that we are interested
in are $T_{n}(x,y)$ and $U_{n}(x,y)$, which are homogenizations of $T_{n}(x)$
and $U_{n}(x)$, respectively. It is known [8] that

$T_{n}(x,y)=2^{n-1}\prod\limits_{k=0}^{n-1}\left(x-\cos\left(\frac{(2k+1)\pi}{2n}\right)y\right)$

and

$U_{n}(x,y)=2^{n}\prod\limits_{k=1}^{n}\left(x-\cos\left(\frac{k\pi}{n+1}\right)y\right).$

See Figure 2 for the graphs of $|T_{6}(x,y)|=1$ and $|U_{6}(x,y)|=1$. Note
that in both cases the fundamental regions are _not_ compact. It is also known
that the binary forms $T_{n}(x,y)$ and $U_{n}(x,y)$ both have integer
coefficients [8].

For a positive integer $n$, let $d(n)$ denote the number of its positive
divisors. Let $\varphi(n)$ denote the Euler’s totient function. Our results
are summarized in Theorem 1.1, Corollary 1.2, Theorem 1.3 and Theorem 1.4.

######  Theorem 1.1.

Let $n$ be a positive integer such that $\varphi(n)\geq 6$. Let
$\Psi_{n}(x,y)$ denote the homogenization of the minimal polynomial of
$2\cos\left(\frac{2\pi}{n}\right)$. Then

$\displaystyle A_{\Psi_{n}}$
$\displaystyle>\frac{16}{3}\exp\left(-\frac{2d(n)\log n}{\varphi(n)}\right)$
(5) $\displaystyle A_{\Psi_{n}}$
$\displaystyle<2^{\frac{4}{\varphi(n)}}\exp\left(\frac{2d(n)^{2}\log
n}{\varphi(n)}\right)\times$
$\displaystyle\times\left[\frac{16}{3}+2^{1-\frac{4}{\varphi(n)}}B\left(\frac{1}{2},\frac{1}{2}-\frac{2}{\varphi(n)}\right)-2\pi+\left(\frac{2}{n}B\left(\frac{1}{n},1-\frac{4}{\varphi(n)}\right)-2\right)-\left(\frac{2}{n}B\left(\frac{3}{n},1-\frac{4}{\varphi(n)}\right)-\frac{2}{3}\right)\right].$

Furthermore, $\lim\limits_{n\rightarrow\infty}A_{\Psi_{n}}=\frac{16}{3}$.

Figure 2: Graphs of $|T_{6}(x,y)|=1$ (left) and $|U_{6}(x,y)|=1$ (right)

######  Corollary 1.2.

Let $n$ be a positive integer. Let $\Pi_{n}(x,y)$ denote the homogenization of
the minimal polynomial of $2\sin\left(\frac{2\pi}{n}\right)$. Then
$\Pi_{n}(x,y)=\Psi_{c(n)}(x,y)$, where

$c(n)=\begin{cases}4n&\text{if $n$ is odd,}\\\ 2n&\text{if $n\equiv
2\pmod{4}$,}\\\ n&\text{if $n\equiv 0\pmod{8}$,}\\\ n/2&\text{if $n\equiv
12\pmod{16}$,}\\\ n/4&\text{if $n\equiv 4\pmod{16}$.}\end{cases}$

Furthermore, $\lim\limits_{n\rightarrow\infty}A_{\Pi_{n}}=\frac{16}{3}$.

######  Theorem 1.3.

Let $n$ be a positive integer such that $n\geq 3$. Let $T_{n}(x,y)$ denote the
homogenization of the Chebyshev polynomial of the first kind $T_{n}(x)$. Then

$\frac{8}{3}<A_{T_{n}}<\frac{8}{3}+\frac{2}{3}\left(\sqrt[n]{4}-1\right)+B\left(\frac{1}{2}-\frac{1}{n},\frac{1}{2}\right)-\pi$
(6)

Furthermore, $\lim\limits_{n\rightarrow\infty}A_{T_{n}}=\frac{8}{3}$.

######  Theorem 1.4.

Let $n$ be a positive integer such that $n\geq 3$. Let $U_{n}(x,y)$ denote the
homogenization of the Chebyshev polynomial of the second kind $U_{n}(x)$. Then

$\displaystyle A_{U_{n}}$
$\displaystyle>\frac{8}{3}+\left(B\left(1+\frac{1}{n},\frac{1}{2}\right)-2\right)+\frac{2}{3}\left((n+1)^{-2/n}-1\right)$
(7) $\displaystyle A_{U_{n}}$
$\displaystyle<\frac{8}{3}+\left(B\left(1+\frac{1}{n},\frac{1}{2}\right)-2\right)+B\left(\frac{1}{2}-\frac{1}{n},\frac{1}{2}\right)-\pi$

Furthermore, $\lim\limits_{n\rightarrow\infty}A_{U_{n}}=\frac{8}{3}$.

Theorem 1.1 follows from Lemma 4.1, which we establish in Section 4. The proof
of Corollary 1.2 is outlined in Section 5. Theorem 1.3 follows from Lemma 6.1,
which we establish in Section 6. Theorem 1.4 follows from Lemma 7.1, which we
establish in Section 7.

Our last result, which we present in Section 8, concerns the quantity

$Q(F)=|D_{F}|^{1/n(n-1)}A_{F}$

associated to a binary form $F$ of degree $n\geq 3$ and non-zero discriminant
$D_{F}$. It was demonstrated by Bean [1] that the quantities $A_{F}$ and
$Q(F)$ remain invariant for all binary forms that are equivalent under
$\operatorname{GL}_{2}(\mathbb{Z})$ to $F$. Unlike $A_{F}$, the quantity
$Q(F)$ also remains invariant for all binary forms that, up to multiplication
by a complex number, are equivalent under $\operatorname{GL}_{2}(\mathbb{R})$
to $F$. An important conjecture about $Q(F)$ was formulated by Bean [2,
Conjecture 1]. Let

$F_{n}^{*}(x,y)=\prod\limits_{k=1}^{n}\left(\sin\left(\frac{k\pi}{n}\right)x-\cos\left(\frac{k\pi}{n}\right)y\right).$

######  Conjecture 1.5.

The maximum value $M_{n}$ of $Q(F)$ over the forms $F$ with complex
coefficients of degree $n$ with discriminant $D_{F}\neq 0$ is attained
precisely when $F$ is a form which, up to multiplication by a complex number,
is equivalent under $\operatorname{GL}_{2}(\mathbb{R})$ to the form
$F_{n}^{*}$. That is, $M_{n}=Q(F_{n}^{*})$. Moreover, the limit of the
sequence $\\{M_{n}\\}$ is $2\pi$.

It was proved by Bean and Laugesen [3, Theorem 1] that

$Q(F_{n}^{*})=2^{1-2/n}n^{1/(n-1)}B\left(\frac{1}{2}-\frac{1}{n},\frac{1}{2}\right)$
(8)

and that $\lim\limits_{n\rightarrow\infty}Q(F_{n}^{*})=2\pi$. Let $\nu_{2}(n)$
denote the $2$-adic order of $n$. In [9] it was proved by the author that the
binary form

$S_{n}(x,y)=2^{n-1-\nu_{2}(n)}F_{n}^{*}(x,y)$

has integer coefficients and that the greatest common divisor of its
coefficients is equal to one.111Note that in [9, Proposition 1] where the
coefficients of $F_{n}^{*}$ and $S_{n}$ are determined there should be no $Y$
in front of summation symbols. See Figure 3 for the graph of $|S_{6}(x,y)|=1$.
Just like the graph of $|F_{n}^{*}(x,y)|=1$, the graph of $|S_{n}(x,y)|=1$ is
invariant under rotation by any integer multiple of $\frac{\pi}{n}$. In [9] it
was also demonstrated that

$A_{S_{n}}=4^{\nu_{2}(n)/n}B\left(\frac{1}{2}-\frac{1}{n},\frac{1}{2}\right),$

and as a consequence that $\lim\limits_{n\rightarrow\infty}A_{S_{n}}=\pi$.
Since $Q(F_{n}^{*})$ remains invariant under multiplication by a non-zero
complex number, we see that
$Q(S_{n})=Q(2^{n-1-\nu_{2}(n)}F_{n}^{*})=Q(F_{n}^{*})$. In Section 8 we will
prove that $Q(T_{n})\leq Q(S_{n})$ and $Q(U_{n})\leq Q(S_{n})$ for every
integer $n\geq 3$. We will also prove that

$\lim\limits_{n\rightarrow\infty}Q(\Psi_{n})=\lim\limits_{n\rightarrow\infty}Q(\Pi_{n})=\lim\limits_{n\rightarrow\infty}Q(T_{n})=\lim\limits_{n\rightarrow\infty}Q(U_{n})=\frac{16}{3}.$

Figure 3: Graph of $|S_{6}(x,y)|=1$ $n$ | $Q(\Psi_{n})$ | $Q(\Pi_{n})$ | $Q(S_{n})$ | $Q(T_{n})$ | $Q(U_{n})$
---|---|---|---|---|---
$3$ | $\infty$ | $\infty$ | $15.8997$ | $15.8997$ | $15.8997$
$4$ | $\infty$ | $\infty$ | $11.7726$ | $11.4798$ | $11.5438$
$5$ | $\infty$ | $10.8953$ | $10.3228$ | $9.85622$ | $9.94897$
$6$ | $\infty$ | $\infty$ | $9.55526$ | $8.97702$ | $9.08227$
$7$ | $15.8997$ | $8.5577$ | $9.06881$ | $8.41412$ | $8.52393$
$8$ | $\infty$ | $\infty$ | $8.72772$ | $8.01814$ | $8.12848$
$9$ | $15.8997$ | $9.00056$ | $8.47265$ | $7.72218$ | $7.83097$
Table 1: Values of $Q(\Psi_{n})$, $Q(\Pi_{n})$, $Q(S_{n})$, $Q(T_{n})$ and
$Q(U_{n})$ for $n=3,4,\ldots,9$.

##  2 Beta Function and Its Properties

For a complex number $z$ with positive real part, let

$\Gamma(z)=\int\limits_{0}^{\infty}t^{z-1}e^{-t}dt$ (9)

denote the _gamma function_. This function satisfies the functional equation

$\Gamma(z+1)=z\Gamma(z).$ (10)

The gamma function is analytic, with simple pole at $x=0$. Its residue at
$x=0$ is equal to $1$, i.e.,

$\operatorname{Res}(\Gamma(x),0)=\lim\limits_{x\rightarrow 0}x\Gamma(x)=1.$
(11)

The beta function (2) can be expressed in terms of the gamma function by means
of the relation

$B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}.$ (12)

If in (2) we make the change of variables $t=\sin^{2}\theta$, then we obtain a
trigonometric form of the beta function:

$B(x,y)=2\int\limits_{0}^{\pi/2}(\sin\theta)^{2x-1}(\cos\theta)^{2y-1}d\theta.$
(13)

This representation of $B(x,y)$ is used in the derivation of the formula for
$A_{F_{n}^{*}}$ [9]. In Lemmas 4.1, 6.1 and 7.1 we will use it to obtain upper
and lower bounds on $A_{\Psi_{n}}$, $A_{T_{n}}$ and $A_{U_{n}}$.

Finally, we would like to remark that the values of beta function can be
easily computed numerically, as this function is present in various
mathematical software, such as Maple (Beta(x, y)), Mathematica (Beta(x, y))
and MATLAB (beta(Z, W)).

##  3 The Limit of $A_{F_{n}}$ for $F_{n}(x,y)=ax^{n}+by^{n}$

In this section we prove the following result.

######  Proposition 3.1.

Let $a$, $b$ and $n$ be non-zero integers with $n\geq 3$ and let
$F_{n}(x,y)=ax^{n}+by^{n}$. Then
$\lim\limits_{n\rightarrow\infty}A_{F_{n}}=4$.

###### Proof.

In view of (3) and $\lim\limits_{n\rightarrow\infty}|ab|^{1/n}=1$, the result
will follow once we show that

$\lim\limits_{n\rightarrow\infty}\frac{1}{n}B\left(\frac{1}{n},1-\frac{2}{n}\right)=1\quad\text{and}\quad\lim\limits_{n\rightarrow\infty}\frac{1}{n}B\left(\frac{1}{n},\frac{1}{n}\right)=2.$

To see that this is the case, first note that the identities

$B\left(\frac{1}{n},1-\frac{2}{n}\right)=\frac{\Gamma(1/n)\Gamma(1-2/n)}{\Gamma(1-1/n)}\quad\text{and}\quad
B\left(\frac{1}{n},\frac{1}{n}\right)=\frac{\Gamma(1/n)^{2}}{\Gamma(2/n)}$

follow directly from (12). It follows from (11) that

$\lim\limits_{n\rightarrow\infty}\frac{\Gamma(1/n)}{n}=1\quad\text{and}\quad\lim\limits_{n\rightarrow\infty}\frac{\Gamma(2/n)}{n}=\frac{1}{2}.$

Since $\Gamma(1)=1$, we see that

$\displaystyle\lim\limits_{n\rightarrow\infty}\frac{1}{n}B\left(\frac{1}{n},1-\frac{2}{n}\right)=\lim\limits_{n\rightarrow\infty}\frac{\Gamma(1/n)\Gamma(1-2/n)}{n\Gamma(1-1/n)}=\lim\limits_{n\rightarrow\infty}\frac{\Gamma(1/n)}{n}=1$

and

$\displaystyle\lim\limits_{n\rightarrow\infty}\frac{1}{n}B\left(\frac{1}{n},\frac{1}{n}\right)=\lim\limits_{n\rightarrow\infty}\frac{\Gamma(1/n)^{2}}{n\Gamma(2/n)}=\frac{\lim\limits_{n\rightarrow\infty}\left(\frac{\Gamma(1/n)}{n}\right)^{2}}{\lim\limits_{n\rightarrow\infty}\frac{\Gamma(2/n)}{n}}=2.$

∎

##  4 Upper and Lower Bounds on $A_{\Psi_{n}}$

In this section we prove the following result.

######  Lemma 4.1.

For any $n\in\mathbb{N}$ and $\alpha\in\mathbb{R}$ such that $\varphi(n)\geq
4$ and $2/\varphi(n)<\alpha<1$,

$\displaystyle\int\limits_{-\infty}^{+\infty}\frac{dx}{|\Psi_{n}(x)|^{\alpha}}$
$\displaystyle>\left(4+\frac{16}{\varphi(n)^{2}\alpha^{2}-4}\right)\exp\left(-\frac{d(n)\log
n}{2}\alpha\right)$
$\displaystyle\int\limits_{-\infty}^{+\infty}\frac{dx}{|\Psi_{n}(x)|^{\alpha}}$
$\displaystyle<2^{\alpha}\exp\left(\frac{d(n)^{2}\log
n}{2}\alpha\right),\times$
$\displaystyle\times\left[4+2^{1-\alpha}B\left(\frac{1-\alpha}{2},\frac{1}{2}\right)-2\pi+\frac{2}{n}B\left(\frac{\varphi(n)\alpha-2}{2n},1-\alpha\right)-\frac{2}{n}B\left(\frac{\varphi(n)\alpha+2}{2n},1-\alpha\right)\right].$

Let us see why Theorem 1.1 directly follows from Lemma 4.1.

###### Proof of Theorem 1.1.

It follows from (4) that the degree of $\Psi_{n}$ is equal to $\varphi(n)/2$.
Since the area $A_{\Psi_{n}}$ bounded by the curve $|\Psi_{n}(x,y)|=1$ can be
computed via (1), the inequalities (5) follow from Lemma 4.1 once we take
$\alpha=4/\varphi(n)$.

The fact that $\lim\limits_{n\rightarrow\infty}A_{\Psi_{n}}=\frac{16}{3}$ is a
consequence of the following:

$\lim\limits_{n\rightarrow\infty}B\left(\frac{1}{2},\frac{1}{2}-\frac{2}{\varphi(n)}\right)=B\left(\frac{1}{2},\frac{1}{2}\right)=\frac{\Gamma(1/2)^{2}}{\Gamma(1)}=\pi,$
$\lim\limits_{n\rightarrow\infty}\frac{2}{n}B\left(\frac{1}{n},1-\frac{4}{\varphi(n)}\right)=\lim\limits_{n\rightarrow\infty}\frac{2}{n}\Gamma\left(\frac{1}{n}\right)=2,$
$\lim\limits_{n\rightarrow\infty}\frac{2}{n}B\left(\frac{3}{n},1-\frac{4}{\varphi(n)}\right)=\lim\limits_{n\rightarrow\infty}\frac{2}{n}\Gamma\left(\frac{3}{n}\right)=\frac{2}{3}.$

Further, we have

$\lim\limits_{n\rightarrow\infty}\frac{2d(n)\log
n}{\varphi(n)}=0\quad\text{and}\quad\lim\limits_{n\rightarrow\infty}\frac{2d(n)^{2}\log
n}{\varphi(n)}=0.$

These two limits follow from the fact that, for every integer $n>2$,

$d(n)\leq n^{\frac{1.067}{\log\log n}}$ (14)

and

$\varphi(n)>\frac{n}{e^{\gamma}\log\log n+\frac{3}{\log\log n}}.$ (15)

Here $\gamma\approx 0.5772$ denotes the Euler-Mascheroni constant. The upper
bound on $d(n)$ was derived by Nicolas and Robin [10], while the lower bound
on $\varphi(n)$ can be found in the paper of Rosser and Schoenfeld [12,
Theorem 15]. ∎

Before proving Lemma 4.1, we need to establish three lemmas.

######  Lemma 4.2.

Let $n$ be an integer such that $n\geq 3$. Then, for every real number
$\theta$,

$\Psi_{n}(\pm 2\cos\theta)=(\pm
1)^{\frac{\varphi(n)}{2}}e^{-\frac{\varphi(n)}{2}\theta i}\Phi_{n}\left(\pm
e^{\theta i}\right),$ (16) $\Psi_{n}(\pm 2\cosh\theta)=(\pm
1)^{\frac{\varphi(n)}{2}}e^{\frac{\varphi(n)}{2}\theta}\Phi_{n}\left(\pm
e^{-\theta}\right).$ (17)

###### Proof.

According to Lehmer [5],

$\Psi_{n}(x+x^{-1})=x^{-\varphi(n)/2}\Phi_{n}(x).$

Putting $x=e^{\theta i}$, we obtain the identity (16), where the plus signs
are chosen everywhere. Replacing $\theta$ with $\pi+\theta$, $\theta i$ and
$\pi+\theta i$ we obtain the identity (16) where the minus signs are chosen
everywhere, as well as the two identities in (17). ∎

For a polynomial $f(x)$, let $L(f)$ denote the sum of the absolute values of
coefficients of $f(x)$. The following lemma can be found in [4, Lemme 4.1].

######  Lemma 4.3.

For every integer $n>1$, $L(\Phi_{n})\leq n^{\frac{d(n)}{2}}$.

######  Lemma 4.4.

For every positive integer $n$,

$\prod\limits_{\begin{subarray}{c}m\mid n\\\ m\neq
1\end{subarray}}L(\Phi_{m})\leq\exp\left(\frac{d(n)^{2}\log n}{2}\right).$

###### Proof.

By Lemma 4.3,

$\displaystyle\prod\limits_{\begin{subarray}{c}m\mid n\\\ m\neq
1\end{subarray}}L(\Phi_{m})$ $\displaystyle\leq\prod\limits_{m\mid
n}m^{\frac{d(m)}{2}}$ $\displaystyle=\exp\left(\frac{1}{2}\sum\limits_{m\mid
n}d(m)\log m\right)$ $\displaystyle\leq\exp\left(\frac{d(n)\log
n}{2}\sum\limits_{m\mid n}1\right)$
$\displaystyle=\exp\left(\frac{d(n)^{2}\log n}{2}\right).$

∎

###### Proof of Lemma 4.1.

Note that

$\displaystyle\int\limits_{-\infty}^{+\infty}\frac{dx}{|\Psi_{n}(x)|^{\alpha}}$
$\displaystyle=\int\limits_{-2}^{2}\frac{dx}{|\Psi_{n}(x)|^{\alpha}}+\int\limits_{2}^{+\infty}\frac{dx}{|\Psi_{n}(x)|^{\alpha}}+\int\limits_{-\infty}^{-2}\frac{dx}{|\Psi_{n}(x)|^{\alpha}}$
$\displaystyle=-\int\limits_{0}^{\pi}\frac{d(2\cos\theta)}{|\Psi_{n}(2\cos\theta)|^{\alpha}}+\int\limits_{0}^{+\infty}\frac{d(2\cosh\theta)}{|\Psi_{n}(2\cosh\theta)|^{\alpha}}+\int\limits_{-\infty}^{0}\frac{d(-2\cosh\theta)}{|\Psi_{n}(-2\cosh\theta)|^{\alpha}}$
$\displaystyle=2\int\limits_{0}^{\pi}\frac{\sin\theta
d\theta}{|\Psi_{n}(2\cos\theta)|^{\alpha}}+2\int\limits_{0}^{+\infty}\frac{\sinh\theta
d\theta}{|\Psi_{n}(2\cosh\theta)|^{\alpha}}+2\int\limits_{0}^{+\infty}\frac{\sinh\theta
d\theta}{|\Psi_{n}(-2\cosh\theta)|^{\alpha}}$
$\displaystyle=2\int\limits_{0}^{\pi}\frac{\sin\theta
d\theta}{|\Phi_{n}(e^{\theta
i})|^{\alpha}}+2\int\limits_{0}^{+\infty}\frac{\sinh\theta
d\theta}{e^{(\varphi(n)/2)\alpha\theta}\Phi_{n}(e^{-\theta})^{\alpha}}+2\int\limits_{0}^{+\infty}\frac{\sinh\theta
d\theta}{e^{(\varphi(n)/2)\alpha\theta}\Phi_{n}(-e^{-\theta})^{\alpha}},$

where the last equality follows from Lemma 4.2. Let

$C_{1}=2\int\limits_{0}^{\pi}\frac{\sin\theta d\theta}{|\Phi_{n}(e^{\theta
i})|^{\alpha}},\quad C_{2}=2\int\limits_{0}^{+\infty}\frac{\sinh\theta
d\theta}{e^{(\varphi(n)/2)\alpha\theta}\Phi_{n}(e^{-\theta})^{\alpha}},\quad
C_{3}=2\int\limits_{0}^{+\infty}\frac{\sinh\theta
d\theta}{e^{(\varphi(n)/2)\alpha\theta}\Phi_{n}(-e^{-\theta})^{\alpha}}.$

We will now derive upper and lower bounds on $C_{1}$, $C_{2}$ and $C_{3}$.

We begin with the derivation of a lower bound. By Lemma 4.3,

$|\Phi_{n}(z)|\leq L(\Phi_{n})\max\\{1,|z|\\}^{\varphi(n)}\leq
n^{\frac{d(n)}{2}}$

for all $z\in\mathbb{C}$ such that $|z|\leq 1$. In view of this,

$\displaystyle C_{1}$ $\displaystyle=2\int\limits_{0}^{\pi}\frac{\sin\theta
d\theta}{|\Phi_{n}(e^{\theta i})|^{\alpha}}$ $\displaystyle\geq
2n^{-\frac{d(n)}{2}\alpha}\int\limits_{0}^{\pi}\sin\theta d\theta$
$\displaystyle=4\exp\left(-\frac{d(n)\log n}{2}\alpha\right)$

and

$\displaystyle C_{2}$
$\displaystyle=2\int\limits_{0}^{+\infty}\frac{\sinh\theta
d\theta}{e^{(\varphi(n)/2)\alpha\theta}\Phi_{n}\left(e^{-\theta}\right)^{\alpha}}$
$\displaystyle\geq
2n^{-\frac{d(n)}{2}\alpha}\int\limits_{0}^{+\infty}e^{-\frac{\varphi(n)}{2}\alpha\theta}\sinh\theta
d\theta$
$\displaystyle=2n^{-\frac{d(n)}{2}\alpha}\frac{1}{\left(\frac{\varphi(n)}{2}\alpha\right)^{2}-1}$
$\displaystyle=\frac{8}{\varphi(n)^{2}\alpha^{2}-4}\exp\left(-\frac{d(n)\log
n}{2}\alpha\right).$

Similarly,

$C_{3}\geq\frac{8}{\varphi(n)^{2}\alpha^{2}-4}\exp\left(-\frac{d(n)\log
n}{2}\alpha\right).$

We conclude that

$\int\limits_{-\infty}^{+\infty}\frac{dx}{|\Psi_{n}(x)|^{\alpha}}=C_{1}+C_{2}+C_{3}\geq\left(4+\frac{16}{\varphi(n)^{2}\alpha^{2}-4}\right)\exp\left(-\frac{d(n)\log
n}{2}\alpha\right).$

To derive an upper bound, we will make use of the identity

$z^{n}-1=\prod\limits_{m\mid n}\Phi_{m}(z).$

Notice that

$\Phi_{n}(z)=(z^{n}-1)\prod\limits_{\begin{subarray}{c}m\mid n\\\ m\neq
n\end{subarray}}\Phi_{m}(z)^{-1}.$

Thus for every $z\in\mathbb{C}$ such that $|z|<1$ we have

$\displaystyle|\Phi_{n}(z)|^{-1}$
$\displaystyle=\frac{1}{|z^{n}-1|}\prod\limits_{\begin{subarray}{c}m\mid n\\\
m\neq n\end{subarray}}|\Phi_{m}(z)|$
$\displaystyle=\frac{|z-1|}{|z^{n}-1|}\prod\limits_{\begin{subarray}{c}m\mid
n\\\ 1<m<n\end{subarray}}|\Phi_{m}(z)|$
$\displaystyle\leq\frac{|z-1|}{|z^{n}-1|}\prod\limits_{\begin{subarray}{c}m\mid
n\\\ m\neq 1\end{subarray}}L(\Phi_{m})$
$\displaystyle<\frac{2}{|z^{n}-1|}\exp\left(\frac{d(n)^{2}\log n}{2}\right),$

where the last inequality follows from Lemma 4.4. In view of this,

$\displaystyle C_{1}$ $\displaystyle=2\int\limits_{0}^{\pi}\frac{\sin\theta
d\theta}{|\Phi_{n}(e^{\theta i})|^{\alpha}}$ $\displaystyle\leq
2^{1+\alpha}\exp\left(\frac{d(n)^{2}\log
n}{2}\alpha\right)\int\limits_{0}^{\pi}\frac{\sin\theta d\theta}{|e^{n\theta
i}-1|^{\alpha}}$ $\displaystyle=2^{1+\alpha}\exp\left(\frac{d(n)^{2}\log
n}{2}\alpha\right)\left(\int\limits_{0}^{\pi}\sin\theta
d\theta+\int\limits_{0}^{\pi}\sin\theta\left(|e^{n\theta
i}-1|^{-\alpha}-1\right)d\theta\right)$ $\displaystyle\leq
2^{1+\alpha}\exp\left(\frac{d(n)^{2}\log
n}{2}\alpha\right)\left(2+\int\limits_{0}^{\pi}\left(\frac{1}{|e^{n\theta
i}-1|^{\alpha}}-1\right)d\theta\right)$
$\displaystyle=2^{1+\alpha}\exp\left(\frac{d(n)^{2}\log
n}{2}\alpha\right)\left(2+\int\limits_{0}^{\pi}\frac{d\theta}{|e^{n\theta
i}-1|^{\alpha}}-\pi\right).$

It remains to estimate the integral $\int_{0}^{\pi}\frac{d\theta}{|e^{n\theta
i}-1|^{\alpha}}$. Note that

$|e^{n\theta
i}-1|^{2}=\left(\cos(n\theta)-1\right)^{2}+\sin^{2}(n\theta)=2-2\cos(n\theta)=4\sin^{2}\left(\frac{n\theta}{2}\right).$

Therefore,

$\displaystyle\int_{0}^{\pi}\frac{d\theta}{|e^{n\theta i}-1|^{\alpha}}$
$\displaystyle=2^{-\alpha}\int\limits_{0}^{\pi}\left|\sin\left(\frac{n\theta}{2}\right)\right|^{-\alpha}d\theta$
$\displaystyle=\frac{2^{1-\alpha}}{n}\int\limits_{0}^{n\pi/2}\left|\sin\theta\right|^{-\alpha}d\theta$
$\displaystyle=2^{-\alpha}\cdot
2\int\limits_{0}^{\pi/2}\left(\sin\theta\right)^{-\alpha}d\theta$
$\displaystyle=2^{-\alpha}B\left(\frac{1-\alpha}{2},\frac{1}{2}\right).$

We conclude that

$\displaystyle C_{1}$ $\displaystyle=2\int\limits_{0}^{\pi}\frac{\sin\theta
d\theta}{|\Phi_{n}(e^{\theta i})|^{\alpha}}$ $\displaystyle\leq
2^{\alpha}\exp\left(\frac{d(n)^{2}\log
n}{2}\alpha\right)\left(4+2^{1-\alpha}B\left(\frac{1-\alpha}{2},\frac{1}{2}\right)-2\pi\right).$

Next, we derive an upper bound on $C_{2}$:

$\displaystyle C_{2}$
$\displaystyle=2\int\limits_{0}^{+\infty}\frac{\sinh\theta
d\theta}{e^{(\varphi(n)/2)\alpha\theta}\Phi_{n}(e^{-\theta})^{\alpha}}$
$\displaystyle\leq 2^{1+\alpha}\exp\left(\frac{d(n)^{2}\log
n}{2}\alpha\right)\int\limits_{0}^{+\infty}\frac{\sinh\theta
d\theta}{e^{(\varphi(n)/2)\alpha\theta}(1-e^{-n\theta})^{\alpha}}$
$\displaystyle=2^{1+\alpha}\exp\left(\frac{d(n)^{2}\log
n}{2}\alpha\right)\frac{1}{2n}\int\limits_{0}^{1}\left(t^{\frac{\varphi(n)\alpha-2}{2n}-1}-t^{\frac{\varphi(n)\alpha+2}{2n}-1}\right)(1-t)^{-\alpha}dt$
$\displaystyle=2^{\alpha}\exp\left(\frac{d(n)^{2}\log
n}{2}\alpha\right)\left(\frac{1}{n}B\left(\frac{\varphi(n)\alpha-2}{2n},1-\alpha\right)-\frac{1}{n}B\left(\frac{\varphi(n)\alpha+2}{2n},1-\alpha\right)\right),$

where the second-to-last equality follows from the change of variables
$t=e^{-n\theta}$.

Finally, we derive an upper bound on $C_{3}$. We consider two cases.

_Case 1._ Suppose that $n$ is even. Then

$\displaystyle C_{3}$
$\displaystyle=2\int\limits_{0}^{+\infty}\frac{\sinh\theta
d\theta}{e^{(\varphi(n)/2)\alpha\theta}\Phi_{n}(-e^{-\theta})^{\alpha}}$
$\displaystyle\leq 2^{1+\alpha}\exp\left(\frac{d(n)^{2}\log
n}{2}\alpha\right)\int\limits_{0}^{+\infty}\frac{\sinh\theta
d\theta}{e^{(\varphi(n)/2)\alpha\theta}\left(1-(-e^{-\theta})^{n}\right)^{\alpha}}$
$\displaystyle=2^{1+\alpha}\exp\left(\frac{d(n)^{2}\log
n}{2}\alpha\right)\int\limits_{0}^{+\infty}\frac{\sinh\theta
d\theta}{e^{(\varphi(n)/2)\alpha\theta}\left(1-e^{-n\theta}\right)^{\alpha}}$
$\displaystyle=2^{\alpha}\exp\left(\frac{d(n)^{2}\log
n}{2}\alpha\right)\left(\frac{1}{n}B\left(\frac{\varphi(n)\alpha-2}{2n},1-\alpha\right)-\frac{1}{n}B\left(\frac{\varphi(n)\alpha+2}{2n},1-\alpha\right)\right),$

where the last equality was derived from the estimate for $C_{2}$.

_Case 2._ Suppose that $n$ is odd. Then for any real number $x$ we have
$\Phi_{n}(-x)=\Phi_{2n}(x)$, so

$\displaystyle C_{3}$
$\displaystyle=2\int\limits_{0}^{+\infty}\frac{\sinh\theta
d\theta}{e^{(\varphi(n)/2)\alpha\theta}\Phi_{n}(-e^{-\theta})^{\alpha}}$
$\displaystyle=2\int\limits_{0}^{+\infty}\frac{\sinh\theta
d\theta}{e^{(\varphi(n)/2)\alpha\theta}\Phi_{2n}(e^{-\theta})^{\alpha}}$
$\displaystyle\leq 2^{1+\alpha}\exp\left(\frac{d(n)^{2}\log
n}{2}\alpha\right)\int\limits_{0}^{+\infty}\frac{\sinh\theta
d\theta}{e^{(\varphi(n)/2)\alpha\theta}\left(1-e^{-2n\theta}\right)^{\alpha}}$
$\displaystyle\leq 2^{1+\alpha}\exp\left(\frac{d(n)^{2}\log
n}{2}\alpha\right)\int\limits_{0}^{+\infty}\frac{\sinh\theta
d\theta}{e^{(\varphi(n)/2)\alpha\theta}\left(1-e^{-n\theta}\right)^{\alpha}}$
$\displaystyle=2^{\alpha}\exp\left(\frac{d(n)^{2}\log
n}{2}\alpha\right)\left(\frac{1}{n}B\left(\frac{\varphi(n)\alpha-2}{2n},1-\alpha\right)-\frac{1}{n}B\left(\frac{\varphi(n)\alpha+2}{2n},1-\alpha\right)\right),$

where the last equality was derived from the estimate for $C_{2}$.

We conclude that

$\displaystyle\int\limits_{-\infty}^{+\infty}\frac{dx}{|\Psi_{n}(x)|^{\alpha}}$
$\displaystyle=C_{1}+C_{2}+C_{3}$ $\displaystyle\leq
2^{\alpha}\exp\left(\frac{d(n)^{2}\log n}{2}\alpha\right)\times$
$\displaystyle\times\left(4+2^{1-\alpha}B\left(\frac{1-\alpha}{2},\frac{1}{2}\right)-2\pi+\frac{2}{n}B\left(\frac{\varphi(n)\alpha-2}{2n},1-\alpha\right)-\frac{2}{n}B\left(\frac{\varphi(n)\alpha+2}{2n},1-\alpha\right)\right).$

∎

##  5 Proof of Corollary 1.2

In view of the identity
$\sin\left(\frac{2\pi}{n}\right)=\cos\left(\frac{2\pi(n-4)}{4n}\right)$, we
can follow the argument of Niven [11, III.4] and consider the following five
cases.

  1. (1)

If $n$ is odd, then the fraction $\frac{n-4}{4n}$ is in lowest terms. Thus
$2\cos\left(\frac{2\pi(n-4)}{4n}\right)$ is conjugate to
$2\cos\left(\frac{2\pi}{4n}\right)$ and $c(n)=4n$.

  2. (2)

If $n\equiv 2\pmod{4}$, then $\frac{n-4}{4n}$ reduces to a fraction with
denominator $2n$. Thus $2\cos\left(\frac{2\pi(n-4)}{4n}\right)$ is conjugate
to $2\cos\left(\frac{2\pi}{2n}\right)$ and $c(n)=2n$.

  3. (3)

If $n\equiv 0\pmod{8}$, then $\frac{n-4}{4n}$ reduces to a fraction with
denominator $n$. Thus $2\cos\left(\frac{2\pi(n-4)}{4n}\right)$ is conjugate to
$2\cos\left(\frac{2\pi}{n}\right)$ and $c(n)=n$.

  4. (4)

If $n\equiv 12\pmod{16}$, then $\frac{n-4}{4n}$ reduces to a fraction with
denominator $n/2$. Thus $2\cos\left(\frac{2\pi(n-4)}{4n}\right)$ is conjugate
to $2\cos\left(\frac{2\pi}{n/2}\right)$ and $c(n)=n/2$.

  5. (5)

If $n\equiv 4\pmod{16}$, then $\frac{n-4}{4n}$ reduces to a fraction with
denominator $n/4$. Thus $2\cos\left(\frac{2\pi(n-4)}{4n}\right)$ is conjugate
to $2\cos\left(\frac{2\pi}{n/4}\right)$ and $c(n)=n/4$.

A combination of this result with Theorem 1.1 yields
$\lim\limits_{n\rightarrow\infty}A_{\Pi_{n}}=\lim\limits_{n\rightarrow\infty}A_{\Psi_{n}}=\frac{16}{3}$.

##  6 Upper and Lower Bounds on $A_{T_{n}}$

In this section we prove the following result.

######  Lemma 6.1.

For any $n\in\mathbb{N}$ and $\alpha\in\mathbb{R}$ such that $n\geq 3$ and
$2/n\leq\alpha<1$,

$2+\frac{2}{n^{2}\alpha^{2}-1}<\int_{-\infty}^{+\infty}\frac{dx}{|T_{n}(x)|^{\alpha}}<2+\frac{2^{1+\alpha}}{n^{2}\alpha^{2}-1}+B\left(\frac{1-\alpha}{2},\frac{1}{2}\right)-\pi.$

Let us see why Theorem 1.3 directly follows from Lemma 6.1.

###### Proof of Theorem 1.3.

Since for an arbitrary binary form $F$ of positive degree $n$ the area bounded
by the curve $|F(x,y)|=1$ can be computed via (1), the inequalities (6) follow
from Lemma 6.1 if we take $\alpha=2/n$.

To see that $\lim\limits_{n\rightarrow\infty}A_{T_{n}}=\frac{8}{3}$, it is
sufficient to demonstrate that the limit of
$B\left(\frac{1}{2}-\frac{1}{n},\frac{1}{2}\right)$ as $n$ approaches infinity
is equal to $\pi$. Recall that the gamma function (9) satisfies $\Gamma(1)=1$
and $\Gamma(1/2)=\sqrt{\pi}$. In view of (12), we see that

$\lim\limits_{n\rightarrow\infty}B\left(\frac{1}{2}-\frac{1}{n},\frac{1}{2}\right)=B\left(\frac{1}{2},\frac{1}{2}\right)=\frac{\Gamma(1/2)^{2}}{\Gamma(1)}=\pi.$

∎

We will now turn our attention to the proof of Lemma 6.1.

###### Proof of Lemma 6.1.

Since the function $|T_{n}(x)|$ is even, we find that

$\int_{-\infty}^{+\infty}\frac{dx}{|T_{n}(x)|^{\alpha}}=C_{1}+C_{2},$

where

$C_{1}=2\int\limits_{1}^{+\infty}\frac{dx}{T_{n}(x)^{\alpha}},\,\,\,C_{2}=2\int\limits_{0}^{1}\frac{dx}{|T_{n}(x)|^{\alpha}},$

We will now derive upper and lower bounds on $C_{1}$ and $C_{2}$.

First, we determine upper and lower bounds on $C_{1}$. Recall that

$T_{n}(x)=\frac{(x+\sqrt{x^{2}-1})^{n}+(x-\sqrt{x^{2}-1})^{n}}{2}$

for all $x$ such that $|x|\geq 1$. Hence

$\frac{(x+\sqrt{x^{2}-1})^{n}}{2}<T_{n}(x)<(x+\sqrt{x^{2}-1})^{n}$ (18)

for all $x>1$. Using the lower bound in (18), as well as the change of
variables $x=\csc\theta$, we find that

$\displaystyle C_{1}$
$\displaystyle=2\int\limits_{1}^{+\infty}\frac{dx}{T_{n}(x)^{\alpha}}$
$\displaystyle<2^{\alpha+1}\int\limits_{1}^{+\infty}\frac{dx}{(x+\sqrt{x^{2}-1})^{n\alpha}}$
$\displaystyle=2^{\alpha+1}\int\limits_{0}^{\pi/2}\frac{(\cos\theta)(\sin\theta)^{n\alpha-2}d\theta}{(1+\cos\theta)^{n\alpha}}$
$\displaystyle=2^{\alpha+1}\left.\frac{(\sin\theta)^{n\alpha-1}(n\alpha\cos\theta+1)}{(n^{2}\alpha^{2}-1)(1+\cos\theta)^{n\alpha}}\right|_{0}^{\pi/2}$
$\displaystyle=\frac{2^{\alpha+1}}{n^{2}\alpha^{2}-1}$

A lower bound on $C_{1}$ can be obtained analogously. As a result, we get

$\frac{2}{n^{2}\alpha^{2}-1}<C_{1}<\frac{2^{1+\alpha}}{n^{2}\alpha^{2}-1}.$

Next, we determine upper and lower bounds on $C_{2}$. We apply the
substitution $x=\cos(\theta/n)$ and make use of the identity
$\cos\theta=T_{n}(\cos(\theta/n))$ as follows:222The author is grateful to
Fedor Petrov for recognizing the third equality. See
https://mathoverflow.net/questions/345820/an-integral-of-sinx-cosnx-2-n-from-
pi-to-pi.

$\displaystyle C_{2}$
$\displaystyle=2\int\limits_{0}^{1}\frac{dx}{|T_{n}(x)|^{\alpha}}$
$\displaystyle=\frac{2}{n}\int\limits_{0}^{\pi
n/2}\frac{\sin(\theta/n)d\theta}{|\cos\theta|^{\alpha}}$
$\displaystyle=\frac{2}{n}\int\limits_{0}^{\pi
n/2}\sin(\theta/n)d\theta+\varepsilon_{n},$

where

$\varepsilon_{n}=\frac{2}{n}\int\limits_{0}^{\pi
n/2}\sin(\theta/n)\left(|\cos\theta|^{-\alpha}-1\right)d\theta.$

Note that $\varepsilon_{n}>0$. Since

$\frac{2}{n}\int\limits_{0}^{\pi
n/2}\sin(\theta/n)d\theta=2\int\limits_{0}^{\pi/2}\sin\theta d\theta=2,$

we have

$C_{2}=\frac{2}{n}\int\limits_{0}^{\pi
n/2}\sin(\theta/n)d\theta+\varepsilon_{n}>2.$

It remains to find an upper bound on $C_{2}$. Note that

$\displaystyle\varepsilon_{n}$ $\displaystyle=\frac{2}{n}\int\limits_{0}^{\pi
n/2}\sin(\theta/n)\left(|\cos\theta|^{-\alpha}-1\right)d\theta$
$\displaystyle\leq\frac{2}{n}\int\limits_{0}^{\pi
n/2}|\cos\theta|^{-\alpha}d\theta-\pi$
$\displaystyle=2\int\limits_{0}^{\pi/2}(\cos\theta)^{-\alpha}d\theta-\pi$
$\displaystyle=B\left(\frac{1-\alpha}{2},\frac{1}{2}\right)-\pi,$

where the last equality follows from (13). We conclude that

$2<C_{2}\leq 2+B\left(\frac{1-\alpha}{2},\frac{1}{2}\right)-\pi.$

∎

##  7 Upper and Lower Bounds on $A_{U_{n}}$

In this section we prove the following result.

######  Lemma 7.1.

For any $n\in\mathbb{N}$ and $\alpha\in\mathbb{R}$ such that $n\geq 3$ and
$2/n\leq\alpha<1$,

$\displaystyle\int\limits_{-\infty}^{+\infty}\frac{dx}{|U_{n}(x)|^{\alpha}}$
$\displaystyle>B\left(\frac{2+\alpha}{2},\frac{1}{2}\right)+\frac{2}{n^{2}\alpha^{2}-1}(n+1)^{-\alpha}$
$\displaystyle\int\limits_{-\infty}^{+\infty}\frac{dx}{|U_{n}(x)|^{\alpha}}$
$\displaystyle<B\left(\frac{2+\alpha}{2},\frac{1}{2}\right)+\frac{2}{n^{2}\alpha^{2}-1}+B\left(\frac{1-\alpha}{2},\frac{1}{2}\right)-\pi$

Let us see why Theorem 1.4 directly follows from Lemma 7.1.

###### Proof of Theorem 1.4.

Since for an arbitrary binary form $F$ of positive degree $n$ the area bounded
by the curve $|F(x,y)|=1$ can be computed via (1), the inequalities (7) follow
from Lemma 7.1 if we take $\alpha=2/n$.

To see that $\lim\limits_{n\rightarrow\infty}A_{U_{n}}=\frac{8}{3}$, it is
sufficient to demonstrate that the limits of
$B\left(\frac{1}{2}-\frac{1}{n},\frac{1}{2}\right)$ and
$B\left(1+\frac{1}{n},\frac{1}{2}\right)$ as $n$ approaches infinity are equal
to $\pi$ and $2$, respectively. Recall that the gamma function (9) satisfies
$\Gamma(1)=1$ and $\Gamma(1/2)=\sqrt{\pi}$. In view of (12), we see that

$\lim\limits_{n\rightarrow\infty}B\left(\frac{1}{2}-\frac{1}{n},\frac{1}{2}\right)=B\left(\frac{1}{2},\frac{1}{2}\right)=\frac{\Gamma(1/2)^{2}}{\Gamma(1)}=\pi$

and

$\lim\limits_{n\rightarrow\infty}B\left(1+\frac{1}{n},\frac{1}{2}\right)=B\left(1,\frac{1}{2}\right)=\frac{\Gamma(1)\Gamma(1/2)}{\Gamma(3/2)}=\frac{\Gamma(1)\Gamma(1/2)}{(1/2)\Gamma(1/2)}=2,$

where the second-to-last equality follows from the functional equation (10). ∎

We will now turn our attention to the proof of Lemma 7.1.

###### Proof of Lemma 7.1.

Since the function $|U_{n}(x)|$ is even, we find that

$\int_{-\infty}^{+\infty}\frac{dx}{|U_{n}(x)|^{\alpha}}=C_{1}+C_{2},$

where

$C_{1}=2\int\limits_{1}^{+\infty}\frac{dx}{U_{n}(x)^{\alpha}},\,\,\,C_{2}=2\int\limits_{0}^{1}\frac{dx}{|U_{n}(x)|^{\alpha}}.$

We will now derive upper and lower bounds on $C_{1}$ and $C_{2}$.

Put
$C_{1}=2\int\limits_{1}^{+\infty}\frac{dx}{U_{n}(x)^{\alpha}},\,\,\,C_{2}=2\int\limits_{0}^{1}\frac{dx}{|U_{n}(x)|^{\alpha}}$,
so that
$\int\limits_{-\infty}^{+\infty}\frac{dx}{|U_{n}(x)|^{\alpha}}=C_{1}+C_{2}$.

First, we determine upper and lower bounds on $C_{1}$. Recall that

$U_{n}(x)=\frac{\left(x+\sqrt{x^{2}-1}\right)^{n+1}-\left(x-\sqrt{x^{2}-1}\right)^{n+1}}{2\sqrt{x^{2}-1}}$

for all $x$ such that $|x|\geq 1$. Thus for every $\theta\in[0,\pi/2]$ we have

$\displaystyle(\sin\theta)^{n}U_{n}(\csc\theta)$
$\displaystyle=\frac{(1+\cos\theta)^{n+1}-(1-\cos\theta)^{n+1}}{2\cos\theta}$
$\displaystyle=\frac{\left(2\cos^{2}\frac{\theta}{2}\right)^{n+1}-\left(2\sin^{2}\frac{\theta}{2}\right)^{n+1}}{2\cos\theta}$
$\displaystyle=2^{n}\sum\limits_{k=0}^{n}\left(\sin^{2}\frac{\theta}{2}\right)^{k}\left(\cos^{2}\frac{\theta}{2}\right)^{n-k}.$

Since

$\left(\cos^{2}\frac{\theta}{2}\right)^{n}<\sum\limits_{k=0}^{n}\left(\sin^{2}\frac{\theta}{2}\right)^{k}\left(\cos^{2}\frac{\theta}{2}\right)^{n-k}<(n+1)\left(\cos^{2}\frac{\theta}{2}\right)^{n}$

for every $\theta\in(0,\pi/2]$, we find that

$\left(1+\cos\theta\right)^{n}<(\sin\theta)^{n}U_{n}(\csc\theta)<(n+1)\left(1+\cos\theta\right)^{n}$
(19)

for every $\theta\in(0,\pi/2]$.

Using the change of variables $x=\csc\theta$, as well as the lower bound in
(19), we find that

$\displaystyle C_{1}$
$\displaystyle=2\int\limits_{1}^{+\infty}\frac{dx}{U_{n}(x)^{\alpha}}$
$\displaystyle=2\int\limits_{0}^{\pi/2}\frac{\cos\theta
d\theta}{(\sin\theta)^{2}U_{n}(\csc\theta)^{\alpha}}$
$\displaystyle<2\int\limits_{0}^{\pi/2}\frac{\cos\theta(\sin\theta)^{n\alpha-2}d\theta}{(1+\cos\theta)^{n\alpha}}$
$\displaystyle=2\left.\frac{(\sin\theta)^{n\alpha-1}(n\alpha\cos\theta+1)}{(n^{2}\alpha^{2}-1)(1+\cos\theta)^{n\alpha}}\right|_{0}^{\pi/2}$
$\displaystyle=\frac{2}{n^{2}\alpha^{2}-1}.$

A lower bound on $C_{1}$ can be obtained analogously. As a result, we get

$\frac{2}{n^{2}\alpha^{2}-1}(n+1)^{-\alpha}<C_{1}<\frac{2}{n^{2}\alpha^{2}-1}.$

It remains to determine upper and lower bounds on $C_{2}$. We apply the
substitution $x=\cos\left(\theta/(n+1)\right)$ and make use of the identity

$U_{n}\left(\cos\left(\frac{\theta}{n+1}\right)\right)=\frac{\sin\theta}{\sin\left(\frac{\theta}{n+1}\right)}$

as follows:

$\displaystyle C_{2}$
$\displaystyle=2\int\limits_{0}^{1}\frac{dx}{|U_{n}(x)|^{\alpha}}$
$\displaystyle=\frac{2}{n+1}\int\limits_{0}^{\pi(n+1)/2}\sin\left(\frac{\theta}{n+1}\right)^{1+\alpha}|\sin\theta|^{-\alpha}d\theta$
$\displaystyle=\frac{2}{n+1}\int\limits_{0}^{\pi(n+1)/2}\sin\left(\frac{\theta}{n+1}\right)^{1+\alpha}d\theta+\varepsilon_{n}$
$\displaystyle=2\int\limits_{0}^{\pi/2}(\sin\theta)^{1+\alpha}d\theta+\varepsilon_{n}$
$\displaystyle=B\left(\frac{2+\alpha}{2},\frac{1}{2}\right)+\varepsilon_{n},$

where the last equality follows from (13) and

$\varepsilon_{n}=\frac{2}{n+1}\int\limits_{0}^{\pi(n+1)/2}\sin\left(\frac{\theta}{n+1}\right)^{1+\alpha}\left(|\sin\theta|^{-\alpha}-1\right)d\theta.$

Since $\varepsilon_{n}>0$, we obtain the lower bound
$C_{2}>B\left(\frac{2+\alpha}{2},\frac{1}{2}\right)$.

To find an upper bound on $C_{2}$, note that

$\displaystyle\varepsilon_{n}$
$\displaystyle\leq\frac{2}{n+1}\int\limits_{0}^{\pi(n+1)/2}\left(|\sin\theta|^{-\alpha}-1\right)d\theta$
$\displaystyle=2\int\limits_{0}^{\pi/2}(\sin\theta)^{-\alpha}-\pi$
$\displaystyle=B\left(\frac{1-\alpha}{2},\frac{1}{2}\right)-\pi,$

where the last equality follows from (13). We conclude that

$B\left(\frac{2+\alpha}{2},\frac{1}{2}\right)<C_{2}\leq
B\left(\frac{2+\alpha}{2},\frac{1}{2}\right)+B\left(\frac{1-\alpha}{2},\frac{1}{2}\right)-\pi.$

∎

##  8 Bean’s Conjecture

We conclude this article with the proof of Proposition 8.1, which provides
further theoretical evidence in support of Conjecture 1.5.

######  Proposition 8.1.

For any integer $n\geq 3$, $Q(T_{n})\leq Q(S_{n})$ and $Q(U_{n})\leq
Q(S_{n})$. Furthermore,

$\lim\limits_{n\rightarrow\infty}Q(\Psi_{n})=\lim\limits_{n\rightarrow\infty}Q(\Pi_{n})=\lim\limits_{n\rightarrow\infty}Q(T_{n})=\lim\limits_{n\rightarrow\infty}Q(U_{n})=\frac{16}{3}.$

$n$ | $D_{\Psi_{n}}$ | $A_{\Psi_{n}}$ | $D_{\Pi_{n}}$ | $A_{\Pi_{n}}$ | $D_{S_{n}}$ | $A_{S_{n}}$ | $D_{T_{n}}$ | $A_{T_{n}}$ | $D_{U_{n}}$ | $A_{U_{n}}$
---|---|---|---|---|---|---|---|---|---|---
$3$ | 1 | $\infty$ | $2^{2}3$ | $\infty$ | $2^{2}3^{3}$ | $7.28585$ | $2^{4}3^{3}$ | $5.78286$ | $2^{11}$ | $4.46217$
$4$ | $1$ | $\infty$ | $1$ | $\infty$ | $2^{2}$ | $10.4882$ | $2^{17}$ | $4.30008$ | $2^{16}5^{2}$ | $3.50332$
$5$ | $5$ | $\infty$ | $2^{4}5^{3}$ | $5.78302$ | $2^{12}5^{5}$ | $4.55444$ | $2^{16}5^{5}$ | $3.78568$ | $2^{28}3^{3}$ | $3.19719$
$6$ | $1$ | $\infty$ | $2^{2}3$ | $\infty$ | $2^{16}3^{6}$ | $5.29992$ | $2^{31}3^{6}$ | $3.52082$ | $2^{36}7^{4}$ | $3.04985$
$7$ | $7^{2}$ | $8.31171$ | $2^{6}7^{5}$ | $5.38644$ | $2^{30}7^{7}$ | $3.99650$ | $2^{36}7^{7}$ | $3.35841$ | $2^{64}$ | $2.96434$
$8$ | $2^{3}$ | $\infty$ | $2^{3}$ | $\infty$ | $2^{24}$ | $6.48467$ | $2^{73}$ | $3.24832$ | $2^{64}3^{12}$ | $2.90894$
$9$ | $3^{4}$ | $7.64379$ | $2^{6}3^{9}$ | $5.63543$ | $2^{56}3^{18}$ | $3.75495$ | $2^{64}3^{18}$ | $3.16867$ | $2^{88}5^{7}$ | $2.87035$
Table 2: Invariants associated with $\Psi_{n}$, $\Pi_{n}$, $S_{n}$, $T_{n}$
and $U_{n}$ for $n=3,4,\ldots,9$.

###### Proof.

Since $Q(F_{n}^{*})=Q(S_{n})$, we prove our statement with $F_{n}^{*}$ in
place of $S_{n}$.

Let $n$ be an integer such that $n\geq 3$. According to Tran [14], the
discriminants of $T_{n}$ and $U_{n}$ are given by

$D_{T_{n}}=2^{(n-1)^{2}}n^{n}\quad\text{and}\quad
D_{U_{n}}=2^{n^{2}}(n+1)^{n-2}.$ (20)

Along with Theorems 1.3 and 1.4, these equalities yield the upper bounds

$2^{-\frac{n-1}{n}}n^{-\frac{1}{n-1}}Q(T_{n})<\frac{8}{3}+\frac{2}{3}\left(\sqrt[n]{4}-1\right)+B\left(\frac{1}{2}-\frac{1}{n},\frac{1}{2}\right)-\pi$
(21)

and

$2^{-\frac{n}{n-1}}(n+1)^{-\frac{n-2}{n(n-1)}}Q(U_{n})<\frac{8}{3}+\left(B\left(1+\frac{1}{n},\frac{1}{2}\right)-2\right)+B\left(\frac{1}{2}-\frac{1}{n},\frac{1}{2}\right)-\pi$
(22)

Now, the values of $Q(F_{n}^{*})=Q(S_{n})$, $Q(T_{n})$ and $Q(U_{n})$ for
$n=3,4,\ldots,9$ are given in Table 1. The integrals associated with
$A_{S_{n}}$, $A_{T_{n}}$ and $A_{U_{n}}$ were approximated with Mathematica.
Notice that
$Q(T_{3})=Q(U_{3})=Q(S_{3})=3B\left(\frac{1}{3},\frac{1}{3}\right)$, because
$S_{3}$, $T_{3}$ and $U_{3}$ are all equivalent under
$\operatorname{GL}_{2}(\mathbb{R})$ to $xy(x-y)$. From Table 2 it is clear
that the statement of Corollary 8.1 holds for the aforementioned values of
$n$.

For $n\geq 10$, the inequalities $Q(T_{n})\leq Q(S_{n})$ and $Q(U_{n})\leq
Q(S_{n})$ can be easily verified by combining the formula (8) with the upper
bounds (21) and (22).

The formulas for $D_{T_{n}}$ and $D_{U_{n}}$ given in (20) imply

$\lim\limits_{n\rightarrow\infty}D_{T_{n}}^{1/n(n-1)}=\lim\limits_{n\rightarrow\infty}D_{U_{n}}^{1/n(n-1)}=2.$

Combining this with the fact that
$\lim\limits_{n\rightarrow\infty}A_{T_{n}}=\lim\limits_{n\rightarrow\infty}A_{U_{n}}=\frac{8}{3}$,
we find that

$\lim\limits_{n\rightarrow\infty}Q(T_{n})=\lim\limits_{n\rightarrow\infty}D_{T_{n}}^{\frac{1}{n(n-1)}}A_{T_{n}}=\left(\lim\limits_{n\rightarrow\infty}D_{T_{n}}^{\frac{1}{n(n-1)}}\right)\cdot\left(\lim\limits_{n\rightarrow\infty}A_{T_{n}}\right)=\frac{16}{3}.$

The same calculation applies with $Q(U_{n})$ in place of $Q(T_{n})$.

Next, we prove that
$\lim\limits_{n\rightarrow\infty}Q(\Psi_{n})=\frac{16}{3}$. For a positive
integer $n$, let $\omega(n)$ denote the number of distinct prime divisors of
$n$, and let $n=p_{1}^{e_{1}}\cdots p_{\omega(n)}^{e_{\omega(n)}}$ be the
prime factorization of $n$. According to Liang [6], the discriminant
$D_{K_{n}}$ of the number field
$K_{n}=\mathbb{Q}\left(2\cos\left(\frac{2\pi}{n}\right)\right)$ can be
computed according to the formula

$D_{K_{n}}=\begin{cases}2^{(m-1)2^{m-2}-1}&\textrm{if $n=2^{m}$, $m>2$,}\\\
p^{(mp^{m}-(m+1)p^{m-1}-1)/2}&\textrm{if $n=p^{m}$ or $2p^{m}$, $p>2$
prime,}\\\
\left(\prod_{i=1}^{\omega(n)}p_{i}^{e_{i}-1/(p_{i}-1)}\right)^{\frac{\varphi(n)}{2}}&\textrm{if
$\omega(n)>1,n\neq 2p^{m}$.}\end{cases}$ (23)

It was also established by Liang that the ring of integers
$\mathcal{O}_{K_{n}}$ of $K_{n}$ has a power integral basis, i.e.,
$\mathcal{O}_{K_{n}}=\mathbb{Z}\left[2\cos\left(\frac{2\pi}{n}\right)\right]$.
Consequently, the discriminant $D_{\Psi_{n}}$ of $\Psi_{n}$ is equal to
$D_{K_{n}}$. Since $1\leq D_{\Psi_{n}}\leq n^{\varphi(n)/2}$, it follows from
(15) and Theorem 1.1 that

$\lim\limits_{n\rightarrow\infty}Q(\Psi_{n})=\frac{16}{3}.$

In view of Corollary 1.2, this result also implies
$\lim\limits_{n\rightarrow\infty}Q(\Pi_{n})=\lim\limits_{n\rightarrow\infty}Q(\Psi_{n})=\frac{16}{3}$.
∎

## Acknowledgements

The author is grateful to Prof. Cameron L. Stewart for his numerous
suggestions, to Patrick Naylor for many productive conversations, and to the
anonymous reviewer for their valuable recommendations.

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