††

# Generalized Mahler measure of Laurent polynomials

Subham Roy  Département de mathématiques et de statistique, Université de
Montréal, CP 6128, succ. Centre-Ville. Montréal, QC H3C 3J7, Canada
<EMAIL_ADDRESS>
###### Abstract.

Following the work of Lalín and Mittal on the Mahler measure over arbitrary
tori, we investigate the definition of the generalized Mahler measure for all
Laurent polynomials in two variables when they do not vanish on the
integration torus. We establish certain relations between the standard Mahler
measure and the generalized Mahler measure of such polynomials. Later we focus
our investigation on a tempered family of polynomials originally studied by
Boyd, namely $Q_{r}(x,y)=x+\frac{1}{x}+y+\frac{1}{y}+r$ with $r\in\mathbb{C},$
and apply our results to this family. For the $r=4$ case, we explicitly
calculate the generalized Mahler measure of $Q_{4}$ over any arbitrary torus
in terms of special values of the Bloch–Wigner dilogarithm. Finally, we extend
our results to the several variable setting.

###### Key words and phrases:

Mahler measure; elliptic curve; special values of $L$-functions; dilogarithm.

###### 2020 Mathematics Subject Classification:

Primary 11R06; Secondary 11G05, 14H52, 31A05.

##  1\. Introduction

The (logarithmic) Mahler measure of a non-zero rational function
$P\in\mathbb{C}\left(x_{1},\dots,x_{n}\right)^{*}$ is defined by

(1)
$\mathrm{m}\left(P\right)=\mathrm{m}(P(x_{1},\dots,x_{n})):=\frac{1}{\left(2\pi
i\right)^{n}}\int_{\mathbb{T}^{n}}\log|P\left(x_{1},\dots,x_{n}\right)|\frac{dx_{1}}{x_{1}}\cdots\frac{dx_{n}}{x_{n}},$

where
$\mathbb{T}^{n}=\\{\left(x_{1},\dots,x_{n}\right)\in\mathbb{C}^{*}\times\mathbb{C}^{*}\times\cdots\times\mathbb{C}^{*}:|x_{1}|=\cdots=|x_{n}|=1\\}$.

The first appearance of this quantity (for one variable polynomials) can be
traced back to Lehmer’s work [1] on Mersenne numbers, and its several variable
form first appeared in the work of Mahler [2] regarding a simpler proof of the
Gel’fond-Mahler inequality, and it was later named after him.

In the early $80$’s, Smyth [3] discovered the following remarkable identities:

$\displaystyle\mathrm{m}(x+y+1)=$
$\displaystyle\frac{3\sqrt{3}}{4\pi}L(\chi_{-3},2),$
$\displaystyle\mathrm{m}(1+x+y+z)=$ $\displaystyle\frac{7}{2\pi^{2}}\zeta(3),$

where $L(\chi_{-3},2)$ is the Dirichlet $L$-function of the quadratic
character $\chi_{-3}$ of conductor $3,$ and $\zeta(s)$ is the Riemann zeta
function (for more details see [4]). These are two of the initial formulas for
several variable cases.

Later the work of Boyd [5], Deninger [6], Rodriguez-Villeags [7] and others
provided us with interesting connections among Mahler measure, higher
regulators, and Beĭlinson’s conjectures. The conjectural formulas to support
their work, such as

$\mathrm{m}(P_{k}(x,y))\stackrel{{\scriptstyle?}}{{=}}r_{k}L^{\prime}(E_{N(k)},0),\quad\quad
r_{k}\in\mathbb{Q},$

were eventually proved for certain polynomials, due to Rodriguez-Villegas [7],
Rogers and Zudilin [8, 9] et al. Here $E_{N(k)}$ is an elliptic curve of
conductor $N(k)$ associated to $P_{k},$ and the question mark stands for a
numerical formula that is true for at least 20 decimal places. (See the book
of Brunault and Zudilin [10] for more details.)

In a different direction, Cassaigne and Maillot [11] generalized the formula
found by Smyth to $\mathrm{m}(a+bx+cy)$ for arbitrary complex constants $a,b,\
\mbox{and}\ c:$

(2)
$\pi\mathrm{m}(ax+by+c)=\left\\{\begin{array}[]{ll}\alpha\log|a|+\beta\log|b|+\gamma\log|c|+D\left(\frac{|a|}{|b|}e^{i\gamma}\right)&\mbox{if}\
\Delta\ \mbox{holds},\\\ \log\max\\{|a|,|b|,|c|\\}&\mbox{if}\ \Delta\
\mbox{does not hold},\end{array}\right.$

where $\Delta$ stands for the statement that $|a|,|b|,\ \mbox{and}\ |c|$ are
the lengths of the sides of a planar triangle, and in that case,
$\alpha,\beta,\ \mbox{and}\ \gamma$ are the angles opposite to the sides of
the lengths $|a|,|b|$ and $|c|$ respectively (see Figure 1).

$\beta$$\alpha$$\gamma$$|b|$$|a|$$|c|$ Figure 1. Condition $\Delta$ in
Cassaigne and Maillot’s formula

We also remark that the constant coefficient can be replaced by a variable
without changing the Mahler measure, in the sense that
$\mathrm{m}(ax+by+c)=\mathrm{m}(ax+by+cz).$ Additionally, it is immediate to
see that Cassaigne and Maillot’s result can also be interpreted as

$\displaystyle\mathrm{m}(ax+by+cz)$ $\displaystyle=\frac{1}{\left(2\pi
i\right)^{3}}\int_{\mathbb{T}_{|a|,|b|,|c|}^{3}}\log|x+y+z|\frac{dx}{x}\frac{dy}{y}\frac{dz}{z}$
$\displaystyle=\mathrm{m}\left(|a|x+|b|y+|c|z\right)=\mathrm{m}_{|a|,|b|,|c|}(x+y+z),$

i.e. the standard Mahler measure of $ax+by+cz$ is same as the Mahler measure
of $x+y+z$ over the integration torus $\mathbb{T}_{|a|,|b|,|c|}^{3},$ where

$\mathbb{T}_{|a|,|b|,|c|}^{3}=\\{(x,y,z)\in\mathbb{C}^{*}\times\mathbb{C}^{*}\times\mathbb{C}^{*}:|x|=|a|,|y|=|b|,|z|=|c|\\}.$

This different representation of $\mathrm{m}(ax+by+cz)$ makes (2) a
generalization of Smyth’s result. This leads to the following definition.

######  Definition 1.1.

The generalized Mahler measure of a non-zero rational function
$P\in\mathbb{C}(x_{1},\dots,x_{n})^{*}$ is defined as

$\mathrm{m}_{\mathfrak{a}}(P)=\mathrm{m}_{a_{1},\dots,a_{n}}(P(x_{1},\dots,x_{n})):=\frac{1}{\left(2\pi
i\right)^{n}}\int_{\mathbb{T}_{\mathfrak{a}}^{n}}\log|P\left(x_{1},\dots,x_{n}\right)|\frac{dx_{1}}{x_{1}}\cdots\frac{dx_{n}}{x_{n}},$

where $\mathfrak{a}=(a_{1},\dots,a_{n})\in(\mathbb{R}_{>0})^{n}$ and

$\mathbb{T}_{\mathfrak{a}}^{n}:=\\{(x_{1},\dots,x_{n})\in\mathbb{C}^{*}\times\mathbb{C}^{*}\times\cdots\times\mathbb{C}^{*}:|x_{1}|=a_{1},\dots,|x_{n}|=a_{n}\\}.$

Lalín and Mittal [12] explored this definition over $\mathbb{T}_{q^{2},q}^{2}$
and $\mathbb{T}_{q,q}^{2}$ to obtain relations between some polynomials
mentioned in Boyd’s paper [5], namely

$\begin{array}[]{ll}R_{-2}(x,y)&:=\displaystyle(1+x)(1+y)(x+y)+2xy,\\\
S_{2,-1}(x,y)&:=\displaystyle y^{2}+2xy-x^{3}+x,\end{array}$

for some values of $q\in\mathbb{R}_{>0}.$ They have simultaneously evaluated
$\mathrm{m}_{q^{2},q}(R_{-2})$ and $\mathrm{m}_{q,q}(S_{2,-1})$ in terms of
$\log q$ and special values of $L$-functions when each of them does not vanish
on the respective integration torus. In particular, they established a
relation between the standard Mahler measure and the generalized Mahler
measure. In this article, we provide a way to obtain such relations for a
variety of Laurent polynomials of the form

$Q_{r}(x,y)=r-Q(x,y)\in\mathbb{C}[x^{\pm},y^{\pm}],$

where $r\in\mathbb{C},$ and $Q$ has no constant term.

This project started with a specific family of Boyd’s polynomials, namely

(3) $\left\\{x+\frac{1}{x}+y+\frac{1}{y}+t:t\in\mathbb{C}\right\\}.$

An extension of the methods in [7] and [13] led us to an interesting fact: for
an arbitrarily fixed $(a,b)\in\mathbb{R}_{>0}^{2},$ there exists a large set
of $t\in\mathbb{C}$ such that Mahler measure of these polynomials remains the
same irrespective of deforming the integration torus from $\mathbb{T}^{2}$
($=\mathbb{T}_{1,1}^{2}$) to $\mathbb{T}_{a,b}^{2}.$ In fact, we found that
this method can be extended to a larger family of Laurent polynomials when
they do not vanish on the integration torus.

Let $Q(x,y)$ be a Laurent polynomial in $\mathbb{C}[x^{\pm},y^{\pm}]$ with no
constant term, and define the family of polynomials
$\\{Q_{r}(x,y):r\in\mathbb{C}\\}$ associated to $Q$ as

$Q_{r}(x,y)=r-Q(x,y)\in\mathbb{C}[x^{\pm},y^{\pm}].$

For $a,b>0,$ let $\mathcal{R}_{a,b}$ be the image of the map

(4) $q:\mathbb{T}_{a,b}^{2}\longrightarrow\mathbb{C},\quad\mbox{defined
by}\quad(x,y)\mapsto Q(x,y).$

For $u\in\mathbb{T}_{b}^{1},$ let $Z_{a,u,r}^{1}$ denote the number of zeros
(counting multiplicities) of $Q_{r}(x,u)$ inside the circle
$\mathbb{T}_{a}^{1},$ and let $P_{a,u,r}^{1}$ denote the order of the pole of
$Q_{r}(x,u)$ at $x=0.$ Similarly, for $w\in\mathbb{T}_{a}^{1},$ the number of
zeros (counting multiplicities) of $Q_{r}(w,y)$ inside the circle
$\mathbb{T}_{b}^{1}$ is denoted by $Z_{w,b,r}^{2},$ and the order of the pole
of $Q_{r}(w,y)$ at $y=0$ is denoted by $P_{w,b,r}^{2}.$ Define
$\nu_{a,u,r}^{1}$ and $\nu_{w,b,r}^{2}$ as

(5)
$\nu_{a,u,r}^{1}:=Z_{a,u,r}^{1}-P_{a,u,r}^{1},\quad\mbox{and}\quad\nu_{w,b,r}^{2}:=Z_{w,b,r}^{2}-P_{w,b,r}^{2}.$

Then, for $u=b$ and $w=a,$ we have the following theorem.

######  Theorem 1.2.

Let $a$ and $b$ be positive real numbers, and denote by $U_{a,b}$ the
unbounded open connected component of $\mathbb{C}\setminus\mathcal{R}_{a,b}$
containing some neighbourhood of $r=\infty.$ Then, for $r\in U_{a,b}\cap
U_{1,1},$

$\mathrm{m}_{a,b}(Q_{r})=\mathrm{m}(Q_{r})+\nu_{a,b,r}^{1}\log
a+\nu_{a,b,r}^{2}\log b,$

where $\nu_{a,b,r}^{1}$ and $\nu_{a,b,r}^{2}$ are defined as above, and
$\mathrm{m}_{1,1}(Q_{r})=\mathrm{m}(Q_{r}).$ Moreover, when $r\in U_{a,b}\cap
U_{1,1},$ the quantities $\nu_{a,b,r}^{1}$ and $\nu_{a,b,r}^{2}$ only depend
on $(a,b).$

For $|r|$ large enough, the above relation between the standard Mahler measure
and the generalized Mahler measure of $Q_{r}$ can be obtained by first
expanding $\log\left(1-\frac{Q}{r}\right)$ in a convergent series, and then
integrating each term individually. We should mention here that, in order to
obtain a convergent series expansion of $\log,$ the above procedure is
restricted to a smaller subregion contained in the unbounded region of
$\mathbb{C}\setminus\mathcal{R}_{a,b}.$ Theorem 1.2 establishes this equality
for a larger set, and harmonic properties of Mahler measure then imply that
the equality holds for all $r$ in the unbounded open connected component of
$\mathbb{C}\setminus\mathcal{R}_{a,b}.$

A follow up question can be posed regarding the values of
$\mathrm{m}_{a,b}(Q_{r})$ when $r$ belongs to one of the bounded open
connected components of $\mathbb{C}\setminus\mathcal{R}_{a,b}.$ The next
theorem answers this question when $\nu_{a,b,r}^{1}$ (or $\nu_{a,b,r}^{2}$)
satisfies a particular condition.

We introduce some necessary notation to state the next theorem. $Q_{r}(x,y),$
when considered as a polynomial in $y$ (resp. $x$) of degree $d_{y}$ (resp.
$d_{x}$) with coefficients in $\overline{\mathbb{C}(x)}$ (resp.
$\overline{\mathbb{C}(y)}$), can be expressed as

$\displaystyle Q_{r}(x,y)=$
$\displaystyle(y)^{-v_{2}}\left(Q_{F,r}^{y}(x)(y)^{d_{y}}+Q_{f,r}^{y}(x)+\sum_{j=1}^{d_{y}-1}a_{j,r}^{y}(x)(y)^{j}\right)$
$\displaystyle=$
$\displaystyle(x)^{-v_{1}}\left(Q_{F,r}^{x}(y)(x)^{d_{x}}+Q_{f,r}^{x}(y)+\sum_{j=1}^{d_{x}-1}a_{j,r}^{x}(y)(x)^{j}\right),$

where $v_{1}$ and $v_{2}$ denote the largest powers of $x^{-1}$ and $y^{-1}$
in $Q_{r}(x,y),$ respectively, and $Q_{F,r}^{u}$ and $Q_{f,r}^{u}$ are the
respective leading and "constant" coefficient with respect to the variable
$u,$ for $u=x\ \mbox{or}\ y.$ Then we have the following theorem.

######  Theorem 1.3.

Let $a$ and $b$ be positive real numbers. Let
$r_{0}\in\mathbb{C}\setminus\mathcal{R}_{a,b}$ such that $r_{0}$ belongs to
one of the bounded open connected components of
$\mathbb{C}\setminus\mathcal{R}_{a,b}.$ We denote by $V_{a,b,r_{0}}$ the
bounded open connected component containing $r_{0}.$

  1. slbx()

If all the roots of $Q_{r_{0}}(a,y)$ either lie entirely inside the circle
$\mathbb{T}_{b}^{1}$ or lie entirely outside the circle $\mathbb{T}_{b}^{1},$
then, for all $r\in V_{a,b,r_{0}},$

$\mathrm{m}_{a,b}(Q_{r})-\nu_{a,b,r}^{2}\log
b=\left\\{\begin{array}[]{ll}\mathrm{m}_{a}(Q_{F,r}^{y}(x))&\mbox{when all
roots of}\ Q_{r_{0}}(a,y)\ \mbox{lie inside}\ \mathbb{T}_{b}^{1},\\\
\mathrm{m}_{a}(Q_{f,r}^{y}(x))&\mbox{when all roots of}\ Q_{r_{0}}(a,y)\
\mbox{lie outside}\ \mathbb{T}_{b}^{1}.\end{array}\right.$

  2. slbx()

If all the roots of $Q_{r_{0}}(x,b)$ either lie entirely inside the circle
$\mathbb{T}_{a}^{1}$ or lie entirely outside the circle $\mathbb{T}_{a}^{1}$,
then, for all $r\in V_{a,b,r_{0}},$

$\mathrm{m}_{a,b}(Q_{r})-\nu_{a,b,r}^{1}\log
a=\left\\{\begin{array}[]{ll}\mathrm{m}_{b}(Q_{F,r}^{x}(y))&\mbox{when all
roots of}\ Q_{r_{0}}(x,b)\ \mbox{lie inside}\ \mathbb{T}_{a}^{1},\\\
\mathrm{m}_{b}(Q_{f,r}^{x}(y))&\mbox{when all roots of}\ Q_{r_{0}}(x,b)\
\mbox{lie outside}\ \mathbb{T}_{a}^{1}.\end{array}\right.$

Using Theorems 1.2 and 1.3, Cassaigne and Maillot’s result in (2) follows
immediately when the "triangle condition" does not hold. In this case,
$Q_{c}(x,y)=c-x-y$ for $c\in\mathbb{C}.$ For $a,b\in\mathbb{C}^{*},$
$\mathcal{R}_{|a|,|b|}$ for this family of polynomials is the "closed" annulus
$\left\\{z\in\mathbb{C}:|z|\in\left[\left||a|-|b|\right|,|a|+|b|\right]\right\\}.$
Note that, when $c$ belongs to the unbounded component of
$\mathbb{C}\setminus\mathcal{R}_{|a|,|b|},$ we have $\nu_{|a|,|b|,c}^{j}=0.$
Then, Theorem 1.2 and harmonic properties of Mahler measure imply that, when
$|c|>|a|+|b|,$

$\mathrm{m}_{|a|,|b|}(Q_{c})=\mathrm{m}(|a|x+|b|y+c)=\log|c|.$

On the other hand, Theorem 1.3 implies that, for $|c|<\left||a|-|b|\right|,$

$\mathrm{m}_{|a|,|b|}(Q_{c})=\log\max\\{|a|,|b|\\},$

since $\nu_{|a|,|b|,c}^{1}=1$ (resp. $\nu_{|a|,|b|,c}^{2}=1$) when $|a|>|b|$
(resp. $|b|>|a|$). The combination of both equalities leads to a restatement
of (2) when $\Delta$ does not hold. We should remark that the condition
$\Delta$ in their result is equivalent to the condition
$c\in\mathcal{R}_{|a|,|b|},$ i.e. $Q_{c}$ vanishes on the integration torus. A
more involved approach, using Theorems 1.2 and 1.3 on the family of
polynomials

$\begin{array}[]{ll}R_{\alpha}^{*}(x,y)&:=\displaystyle\alpha-x^{-1}-y^{-1}-xy^{-1}-yx^{-1}-x-y,\quad\quad\alpha\in\mathbb{C},\\\
S_{\beta,-1}^{*}(x,y)&:=\displaystyle\beta-x^{-1}y+x^{2}y^{-1}-y^{-1},\quad\quad\beta\in\mathbb{C},\end{array}$

re-establishes the identities obtained in [12] for $\alpha=-4$ and $\beta=2.$
Note that the aforementioned result(s) involving the generalized Mahler
measure of $R_{-2}^{*}$ (resp. $S_{2,-1}^{*}$) on the torus
$\mathbb{T}_{a,b}^{2}$ only depends on $b$ since the integration torus is
$\mathbb{T}_{b^{2},b}^{2}$ (resp. $\mathbb{T}_{b,b}^{2}$), i.e. $a$ is a
function of $b$ here. Our theorems, along with the method of the Lagrange
multiplier, provide a larger set of $2$-tuples $(a,b)\in\mathbb{R}_{>0}^{2},$
such that a similar type of identities obtained in [12] hold even when $a$ is
not a function of $b.$ An analogous result is exhibited in Section $5$ with a
different Boyd’s family of polynomials given in (3).

Due to the technical difficulties involving the study of the integration path
in the definition of Mahler measure, it is challenging to evaluate
$\mathrm{m}_{a,b}(Q_{r})$ explicitly for all $a,b>0.$ In this regard, Theorems
1.2 and 1.3 have a common feature: the polynomial in consideration does not
vanish on the integration torus.

The next result considers a particular polynomial from our initial family of
polynomials, namely

$Q_{4}(x,y)=x+\frac{1}{x}+y+\frac{1}{y}+4.$

It removes the constraint of being non-zero on the integration torus, and
evaluates the generalized Mahler measure of $Q_{4}(x,y)$ for all $a,b>0.$

######  Theorem 1.4.

Let $a,b\in\mathbb{R}_{>0},$ and define

$c=\sqrt{ab},\quad d=\sqrt{\frac{b}{a}},\quad\mbox{and}\
\mathcal{A}_{c,d}=\frac{1-d^{2}}{1+d^{2}}\cdot\frac{1+c^{2}}{2c},$

such that $c$ and $d$ are both positive real numbers. Then,

$\mathrm{m}_{a,b}(Q_{4}(x,y))=\left\\{\begin{array}[]{ll}\max\\{\log c,-\log
c\\}+\max\\{\log d,-\log d\\}&\mbox{if}\ \left|\mathcal{A}_{c,d}\right|\geq
1,\\\ \\\ \frac{2}{\pi}\left[D(ice^{-i\mu})+D(ice^{i\mu})-\mu\log d+(\log
c)\tan^{-1}\left(\frac{c-c^{-1}}{2\cos\mu}\right)\right]&\mbox{if}\
\left|\mathcal{A}_{c,d}\right|<1,\end{array}\right.$

where
$\mu=\sin^{-1}\left(\mathcal{A}_{c,d}\right)\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right),$
and $D$ is the Bloch–Wigner dilogarithm defined, for $z\in\mathbb{C},$ by

$D\left(z\right)=\mathop{\mathrm{Im}}\left(\mathrm{Li}_{2}\left(z\right)+i\arg\left(1-z\right)\log|z|\right),\quad\quad\mathrm{Li}_{2}\left(z\right)=-\int_{0}^{z}\frac{\log\left(1-v\right)}{v}dv.$

Under a certain change of variables, the polynomial above can be factored into
two linear polynomials [5]. This simplification along with a direct approach
involving a particular differential form and the Bloch–Wigner dilogarithm lead
us to the explicit formula in the statement of Theorem 1.4.

This article is organised as follows. In Section $2,$ we recall some relations
between Mahler measure, the Bloch–Wigner dilogarithm and a particular
differential form appearing after a simplification of the definition in (1).
We end the section with a discussion on Mahler measure over an arbitrary
torus. In Section $3,$ we discuss the proof of Theorem 1.2 and some auxiliary
results required to complete the proof. A brief discussion regarding the
relations between Mahler measure and periods of the associated varieties (such
as, algebraic curves) is also included at the end of this section for
completion. Section $4$ is completely dedicated to the proof of Theorems 1.3.
In Section $5,$ we discuss some applications of Theorems 1.2 and 1.3 to the
family of polynomials in (3). We also derive an explicit expression for the
region $\mathcal{R}_{a,b}$ with some conditions on $a,b.$ We then prove
Theorem 1.4 in Section $6,$ where we use properties of the differential form
and Bloch–Wigner dilogarithm mentioned in Section $2.$ Later, in Section $7,$
we sketch a brief proof of an extension of Theorem 1.2 to the several
variables setting. After concluding remarks on possible directions to pursue
going forward, we end this article with an Appendix containing explicit
calculations involving the region $\mathcal{R}_{a,b}$ mentioned in Section
$5.$

## Acknowledgements

I express my deepest gratitude to my Ph.D. supervisor, Matilde Lalín, for her
invaluable assistance and support, and for sharing several ideas that have
enriched this work. I am grateful to Andrew Granville for his useful comments
and enlightening discussions. I am also thankful to Ali Zahabi for helpful
discussions on Ronkin functions and generalized Mahler measure. Finally, I
would like to express my gratitude to the Faculté des études supérieurs et
postdoctorales (bourses d’excellence) of the Université de Montréal, the
Institute des sciences mathématiques and the Centre de recherches
mathématiques for their financial support.

##  2\. Mahler measure and a differential form

In this section, we briefly review some necessary background prior to proving
the theorems in the following sections.

###  2.1. Jensen’s Formula

We recall a special case of Jensen’s formula. Let $z_{0}\in\mathbb{C}.$ Then

$\displaystyle\frac{1}{2\pi
i}\int_{\mathbb{T}^{1}}\log|z-z_{0}|\frac{dz}{z}=\left\\{\begin{array}[]{lr}\log|z_{0}|&\quad|z_{0}|\geq
1,\\\ \\\ \quad 0&\quad|z_{0}|\leq 1.\end{array}\right.$

###  2.2. Bloch–Wigner Dilogarithm

For $z\in\mathbb{C},$ the Bloch–Wigner dilogarithm $D(z)$ is defined as

(6)
$D\left(z\right)=\mathop{\mathrm{Im}}\left(\mathrm{Li}_{2}\left(z\right)+i\arg\left(1-z\right)\log|z|\right),$

where
$\mathrm{Li}_{2}\left(z\right)=-\int_{0}^{z}\frac{\log\left(1-v\right)}{v}dv.$
In [14], Zagier shows that it can be extended continuously to
$\mathbb{C}\cup\\{\infty\\},$ with $D(\infty)=D(0)=D(1)=0.$ In fact, it is
real-analytic in $\mathbb{C}\setminus\\{0,1\\}.$

For $z\in\mathbb{C}$,

(7) $\displaystyle D\left(\bar{z}\right)=-D\left(z\right).$

This also implies that $D(r)=0$ for all $r\in\mathbb{R}$ (for more details see
[14]). This property of the Bloch–Wigner dilogarithm frequently appears in the
proof of Theorem 1.4.

###  2.3. A differential form and its applications

Let $C$ be a curve over $\mathbb{C}$ which defines a compact Riemann surface,
and let $\mathbb{C}(C)$ be its field of fractions. For
$f,g\in\mathbb{C}(C)^{*},$ we define

(8) $\eta\left(f,g\right):=\log|f|d\arg g-\log|g|d\arg f,$

where $d\arg x$ is defined by $\imaginary(\frac{dx}{x})$. Note that, $\eta$ is
a real $C^{\infty}$ differential $1$-form on $C\setminus S$, where $S$
contains all the zeroes and poles of $f$ and $g$. The following lemma consists
of some useful properties of $\eta$ which are frequently used in later
sections (see [15, 16], [17] for more details).

######  Lemma 2.1.

Let $f,g,h,v\in\mathbb{C}(C)^{*}$ and $a,b\in\mathbb{C}^{*}.$ Then we have

  1. (1)

$\eta(f,g)=-\eta(g,f),$ i.e. $\eta$ is anti-symmetric,

  2. (2)

$\eta(fg,hv)=\eta(f,h)+\eta(g,h)+\eta(f,v)+\eta(g,v),$

  3. (3)

$\eta(a,b)=0,$

  4. (4)

$\eta$ is a closed differential form,

  5. (5)

for $x,1-x\in\mathbb{C}(C)^{*},$

(9) $\eta\left(x,1-x\right)=dD\left(x\right).$

Let $P(x,y)$ be a Laurent polynomial in two variables. Multiplying $P(x,y)$ by
a suitable power of $y,$ we can always assume that
$P(x,y)\in\mathbb{C}[x^{\pm},y]$ is a polynomial of degree $d$ in $y,$ where
$d>0$. Then $P(x,y)$ has the following factorization over
$\overline{\mathbb{C}(x)}:$

$P\left(x,y\right)=P^{*}\left(x\right)\left(y-y_{1}\left(x\right)\right)\left(y-y_{2}\left(x\right)\right)\cdots\left(y-y_{d}\left(x\right)\right),$

where $P^{*}\left(x\right)\in\mathbb{C}[x]$ and $y_{j}:=y_{j}\left(x\right)$
are algebraic functions of $x$ for $j=1,2,\dots,d.$

Applying Jensen’s formula with respect to the variable $y$ in the standard
Mahler measure formula for $P(x,y),$ we obtain

$\displaystyle\mathrm{m}\left(P(x,y)\right)-\mathrm{m}\left(P^{*}(x)\right)=$
$\displaystyle\frac{1}{\left(2\pi
i\right)^{2}}\int_{\mathbb{T}^{2}}\log|P\left(x,y\right)|\frac{dx}{x}\frac{dy}{y}-\mathrm{m}\left(P^{*}(x)\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\left(\sum_{j=1}^{d}\int_{|x|=1,|y_{j}\left(x\right)|\geq
1}\log|y_{j}\left(x\right)|\frac{dx}{x}\right)$ (10) $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\sum_{j=1}^{d}\int_{|x|=1,|y_{j}\left(x\right)|\geq
1}\eta\left(x,y_{j}\right),$

where $\eta$ is defined by (8), and
$\eta(x,y_{j})=i\log|y_{j}(x)|\frac{dx}{x},$ which immediately follows from
the facts that $\log|x|=\log 1=0$ and $\frac{dx}{x}=d(\log|x|+i\arg x).$ Here
we consider $\arg(x)\in[-\pi,\pi).$

Therefore, if $\eta$ can be decomposed as

(11)
$\eta\left(x,y_{j}\right)=\sum_{k}a_{j_{k}}\eta\left(z_{j_{k}},1-z_{j_{k}}\right)=\sum_{k}a_{j_{k}}dD(z_{j_{k}}),$

where $z_{j_{k}},\left(1-z_{j_{k}}\right)\in\mathbb{C}(C)^{*}$ are algebraic
functions of $x,$ and the sum is finite, then (10) can be restated in terms of
Bloch–Wigner dilogarithm:

$\mathrm{m}\left(P\left(x,y\right)\right)-\mathrm{m}\left(P^{*}\left(x\right)\right)=-\frac{1}{2\pi}\sum_{j=1}^{d}\sum_{k}a_{j_{k}}D\left(z_{j_{k}}\right)|_{\partial\\{|x|=1,|y_{j}\left(x\right)\geq
1\\}},$

where $\partial\\{|x|=1,|y_{j}|\geq 1\\}$ is the set of boundary points of
$\\{|x|=1,|y_{j}|\geq 1\\}.$

######  Remark 2.2.

As mentioned in [18], we may have some extra terms of the form
$\eta\left(c,z\right)$ in (11), where $c$ is a constant complex number and $z$
is some algebraic function. In that case, we can still reach a closed formula
by integrating $\eta\left(c,z\right)$ directly (i.e. by integrating
$\log|c|d\arg z$). Also, if $\nu$ is a constant such that $|\nu|=1,$ then
$\eta\left(\nu,z\right)=\log|\nu|d\arg z=0$.

###  2.4. Arbitrary Tori, Mahler measure, and $\eta$

For a Laurent polynomial $P(x,y),$ we analyse the Mahler measure of $P$ over
an arbitrary torus $\mathbb{T}_{a,b}^{2}.$ The following brief description is
essentially reproducing the analysis in Section $3$ of [12]. For simplicity,
we take $d=2$, where $d$ is the degree of $y$ in $P(x,y)$ once $P$ is
multiplied by a suitable power of $y$ to remove any negative power of $y.$

Let $x=ax^{\prime}$ and $y=by^{\prime}.$ Then we have, for
$P^{*}(x)\in\mathbb{C}[x],$

$\displaystyle\mathrm{m}_{a,b}\left(P(x,y)\right)-\mathrm{m}_{a,b}\left(P^{*}(x)\right)=$
$\displaystyle\frac{1}{\left(2\pi
i\right)^{2}}\iint_{|x^{\prime}|=|y^{\prime}|=1}\log|P\left(ax^{\prime},by^{\prime}\right)|\frac{dx^{\prime}}{x^{\prime}}\frac{dy^{\prime}}{y^{\prime}}-\mathrm{m}_{a,b}\left(P^{*}(x)\right)$
$\displaystyle=$ $\displaystyle 2\log b+\frac{1}{2\pi
i}\left(\sum_{j=1}^{2}\int_{|x^{\prime}|=1,|y^{\prime}_{j}|\geq
1}\log|y^{\prime}_{j}|\frac{dx^{\prime}}{x^{\prime}}\right),$ (12)
$\displaystyle=$ $\displaystyle 2\log
b-\frac{1}{2\pi}\sum_{j=1}^{2}\int_{|x|=a,|y_{j}|\geq
b}\eta\left(x/a,y_{j}/b\right).$

where $y_{j}=y_{j}(x)=by^{\prime}_{j}$ are algebraic functions of $x$ for
$j=1,2,$ and

$\eta\left(x/a,y_{j}/b\right)=\eta(x^{\prime},y^{\prime}_{j})=i\log|y^{\prime}_{j}|\frac{dx^{\prime}}{x^{\prime}},$

for $j=1,2,$ and the penultimate equality follows from Jensen’s formula.
Further simplification of the terms involving $y_{j}$’s using (2) of Lemma 2.1
implies

$\mathrm{m}_{a,b}\left(P(x,y)\right)-\mathrm{m}_{a,b}\left(P^{*}(x)\right)=2\log
b-\frac{1}{2\pi}\sum_{j=1}^{2}\int_{|x|=a,|y_{j}|\geq
b}\left[\eta\left(x,y_{j}\right)-\eta\left(a,y_{i}\right)-\eta\left(x,b\right)\right].$

If $\\{|x|=a,|y_{j}|\geq b\\}$ is a closed path, then the integral

$\frac{1}{2\pi}\sum_{j=1}^{2}\int_{|x|=a,|y_{j}|\geq
b}\eta\left(x/a,y_{j}/b\right)$

can be evaluated using Stokes’ theorem (see Deninger [6]). In addition, if
$\\{|x|=a,|y_{i}|\geq b\\}$ is a closed path, the term

$\frac{1}{2\pi}\int_{|x|=a,|y_{j}|\geq b}\eta\left(a,y_{i}\right)=\frac{\log
a}{2\pi}\int_{|x|=a,|y_{j}|\geq b}d\arg y_{j}$

becomes a multiple of $\log a.$ If we have a genus $0$ curve (such as
$C_{4}:Q_{4}(x,y)=0$) then, instead of proceeding as in the direction above,
we may be able to use (9) to relate the Bloch–Wigner dilogarithm and $\eta$
for evaluating the Mahler measure. The evaluation is much simpler in this case
as we will see in the proof of Theorem 1.4.

##  3\. Proof of Theorem 1.2

We are now ready to prove Theorem 1.2 for
$Q_{r}(x,y)=r-Q(x,y)\in\mathbb{C}[x^{\pm},y^{\pm}],$ where $Q(x,y)$ has no
constant term. In this section, we use the notation

$\mathrm{m}_{a,b}(Q_{r}(x,y))=\mathrm{m}_{a,b}(Q_{r})=\mathrm{m}_{a,b}(r)\quad\quad\quad\mbox{for}\
r\in\mathbb{C},$

for simplicity. Our approach is inspired from the methods of Rodriguez-
Villegas [7] and Bertin [13]. We first show that the required equality between
Mahler measures holds for a smaller unbounded region of
$\mathbb{C}\setminus\mathcal{R}_{a,b},$ and then we argue using properties of
harmonic functions that it can be extended to the desired region stated in
Theorem 1.2.

The following lemma formulates the invariance of $\mathrm{m}_{a,b}(r)$ under
certain changes of variables.

######  Lemma 3.1.

Let $a,b$ be positive real numbers. Define $f_{r}(a,b):=\mathrm{m}_{a,b}(r).$
Then $f_{r}$ satisfies the following identities:

$f_{r}(a,b)=f_{r}(b,a)=f_{r}\left(\dfrac{1}{a},b\right)=f_{r}\left(\dfrac{1}{a},\dfrac{1}{b}\right).$

By Lemma 3.1, we can restrict ourselves to the case $a>b>1.$

###### Proof of Lemma 3.1.

Let $a,b>0.$ For $\tilde{Q}_{r}(x,y)=Q_{r}(ax,by),$ the generalized Mahler
measure of $Q_{r}$ satisfies the identity

$\mathrm{m}_{a,b}(Q_{r}(x,y))=\mathrm{m}(\tilde{Q}_{r}(x,y))=\mathrm{m}(\tilde{Q}_{r}).$

The change of variables

$(x,y)\rightarrow(y,x),\quad(x,y)\rightarrow(x^{-1},y),\quad(x,y)\rightarrow(x^{-1},y^{-1}),$

fix $\mathrm{m}(\tilde{Q}_{r}).$ Since
$\mathrm{m}_{a,b}(r)=\mathrm{m}(\tilde{Q}_{r}),$ we have the required
identities involving $f_{r}(a,b)=\mathrm{m}_{a,b}(r).$ ∎

Our main aim is to study $\mathrm{m}_{a,b}(r)$ in terms of the complex
parameter $r.$ Recall that $\mathcal{R}_{a,b}$ is the set of all
$r\in\mathbb{C}$ such that $Q_{r}(x,y)$ vanishes on $\mathbb{T}_{a,b}^{2}.$
Before proceeding to prove the theorem, we state a proposition explaining the
following:

  * •

the behaviour of the roots of $Q_{r}(x,y)$ for each $x\in\mathbb{T}_{a}^{1};$
in particular, the number of roots inside the unit circle
$\mathbb{T}_{b}^{1},$

  * •

the behaviour of the roots of $Q_{r}(x,y)$ for each $y\in\mathbb{T}_{b}^{1};$
in particular, the number of roots inside the unit circle
$\mathbb{T}_{a}^{1}.$

This proposition, in particular, provides us with the quantities
$\nu_{a,b,r}^{2}$ and $\nu_{a,b,r}^{1}$ in the statement of Theorem 1.2. Since
the above two cases are analogous, we consider just the first case.

For $w\in\mathbb{T}_{a}^{1},$ let $\varrho_{a,b,r}^{2}(w)$ denote the number
of roots of $Q_{r}(w,y)$ lying inside the circle $\mathbb{T}_{b}^{1}.$ In
particular, following the discussion preceding the definition in (5), we have,
for $w\in\mathbb{T}_{a}^{1},$

(13)
$\varrho_{a,b,r}^{2}(w)=Z_{w,b,r}^{2}\quad\mbox{and}\quad\varrho_{a,b,r}^{2}(a)=Z_{a,b,r}^{2}=\nu_{a,b,r}^{2}+P_{a,b,r}^{2}=\nu_{a,b,r}^{2}+v_{2},$

where $Z_{w,b,r}^{2}$ is the number of zeros (counting multiplicities) of
$Q_{r}(w,y)$ inside the circle $\mathbb{T}_{b}^{1}$, $P_{w,b,r}^{2}$ is the
order of the pole of $Q_{r}(w,y)$ at $y=0,$ and $v_{2}$ is the largest power
of $y^{-1}$ in $Q_{r}(x,y).$ Then we have the following proposition.

######  Proposition 3.2.

Let $r\in\mathbb{C}\setminus\mathcal{R}_{a,b}.$ Then $\varrho_{a,b,r}^{2}(x)$
is constant for all $x\in\mathbb{T}_{a}^{1}.$

The above discussion along with Proposition 3.2 implies that, for all
$x\in\mathbb{T}_{a}^{1},$ $\varrho_{a,b,r}^{2}(x)=\nu_{a,b,r}^{2}+v_{2}.$ Next
we derive Theorem 1.2 assuming Proposition 3.2.

###### Proof of Theorem 1.2.

For $a$ and $b$ positive real numbers, the torus $\mathbb{T}_{a,b}^{2}$ is
defined as the set
$\\{(x,y)\in\left(\mathbb{C}^{*}\right)^{2}:|x|=a,|y|=b\\}.$ By construction,
$\mathbb{T}_{a,b}^{2}$ is compact. Since the map in (4), namely

$q:\mathbb{T}_{a,b}^{2}\longrightarrow\mathbb{C},\quad\mbox{defined
by}\quad(x,y)\mapsto Q(x,y),$

is continuous, the image of $q$ is compact. That is,
$q(\mathbb{T}_{a,b}^{2})=\mathcal{R}_{a,b}$ is compact, and therefore closed
and bounded in $\mathbb{C}.$ In other words, $\max_{r\in\mathcal{R}_{a,b}}|r|$
exists. We denote

$R_{a,b}:=\max_{r\in\mathcal{R}_{a,b}}|r|,\quad\mbox{and}\quad
R_{a,b,1,1}:=\max\\{R_{a,b},R_{1,1}\\}.$

Following a construction in [7], we define

$\tilde{\mathrm{m}}_{a,b}(r)=\log r-\sum_{n\geq
0}\frac{a_{n,a,b}}{n}r^{-n},\quad|r|>R_{a,b,1,1},r\notin(-\infty,0],$

where $\log$ denotes the principal branch of the logarithm, and $a_{n,a,b}$ is
defined as follows:

$\displaystyle a_{n,a,b}=\left[\frac{1}{(2\pi
i)^{2}}\int_{\mathbb{T}_{a,b}^{2}}\frac{dxdy}{xy(1-r^{-1}Q(x,y))}\right]_{n}=\frac{1}{(2\pi
i)^{2}}\int_{\mathbb{T}_{a,b}^{2}}Q(x,y)^{n}\frac{dx}{x}\frac{dy}{y},$

Here $[T(s)]_{n}$ denotes the coefficient of $s^{-n}$ in the series $T(s).$ It
is immediate to see that $\tilde{\mathrm{m}}_{a,b}$ is holomorphic in the
region defined by $|r|>R_{a,b,1,1}$ and $r\notin(-\infty,0]$. Also,

$\real(\tilde{\mathrm{m}}_{a,b}(r))=\mathrm{m}_{a,b}(r),\quad|r|>R_{a,b,1,1}.$

######  Lemma 3.3.

For $|r|>R_{a,b,1,1},$

$\frac{d\tilde{\mathrm{m}}_{a,b}}{dr}=\frac{d\tilde{\mathrm{m}}_{1,1}}{dr}.$

###### Proof of Lemma 3.3.

In order to prove the statement, it is enough to show that
$a_{n,a,b}=a_{n,1,1}$ for all $n.$ The above construction of the coefficients
and the integral expression of these terms in [7] yield that

$\displaystyle a_{n,a,b}$ $\displaystyle=\frac{1}{(2\pi
i)^{2}}\int_{\mathbb{T}_{a,b}^{2}}Q(x,y)^{n}\frac{dx}{x}\frac{dy}{y}=\frac{1}{(2\pi
i)^{2}}\int_{\mathbb{T}^{2}}Q(ax^{\prime},by^{\prime})^{n}\frac{dx^{\prime}}{x^{\prime}}\frac{dy^{\prime}}{y^{\prime}}$
$\displaystyle=\left[Q(ax^{\prime},by^{\prime})^{n}\right]_{0}=\left[Q(x^{\prime},y^{\prime})^{n}\right]_{0}$
$\displaystyle=\frac{1}{(2\pi
i)^{2}}\int_{\mathbb{T}^{2}}Q(x^{\prime},y^{\prime})^{n}\frac{dx^{\prime}}{x^{\prime}}\frac{dy^{\prime}}{y^{\prime}}$
$\displaystyle=a_{n,1,1}.$

The equality
$\left[Q(ax^{\prime},by^{\prime})^{n}\right]_{0}=\left[Q(x^{\prime},y^{\prime})^{n}\right]_{0}$
follows from the fact that the constant term gathers the terms with degree
$0,$ which are invariant under the multiplications of $x$ and $y$ by $a$ and
$b,$ respectively. This concludes the proof. ∎

Due to the above identity, we can denote the coefficients as
$a_{n}:=a_{n,a,b}=a_{n,1,1}$ for the rest of the argument. From the definition
of $\tilde{\mathrm{m}}_{a,b},$ it follows that

(14) $\frac{d\tilde{\mathrm{m}}_{a,b}}{dr}=\frac{1}{(2\pi
i)^{2}}\int_{\mathbb{T}_{a,b}^{2}}\frac{1}{r-Q(x,y)}\frac{dx}{x}\frac{dy}{y},\quad\quad|r|>R_{a,b,1,1},$

where we include the region $r\in(-\infty,0]\cap\\{|r|>R_{a,b,1,1}\\}$ by
continuity. We need to show that $\frac{d\tilde{\mathrm{m}}_{a,b}}{dr}$ is in
fact holomorphic in $|r|>R_{a,b,1,1}.$

For $r\in\mathbb{C},$ define

(15) $\mathcal{F}_{a,b}(r):=\frac{1}{(2\pi
i)^{2}}\int_{\mathbb{T}_{a,b}^{2}}\frac{1}{r-Q(x,y)}\frac{dx}{x}\frac{dy}{y}.$

Note that the integrand

$\left.\frac{1}{r-Q(x,y)}\right|_{(x,y)\in\mathbb{T}_{a,b}^{2}}$

is holomorphic in $r$ when $|r|>R_{a,b,1,1}.$ In fact, we will now show that
$\mathcal{F}_{a,b}(r)$ is holomorphic as well on $|r|>R_{a,b,1,1}.$ The
integrand, as well as the integral in (14), are bounded on
$\mathbb{T}_{a,b}^{2}$. This implies that
$\frac{d^{j}\mathcal{F}_{a,b}}{dr^{j}}$ exists and is holomorphic for $j=1$
(and therefore for all $j\geq 1$). Hence, $\mathcal{F}_{a,b}(r)$ is
holomorphic in $|r|>R_{a,b,1,1}.$

From Lemma 3.3 we have, for $|r|>R_{a,b,1,1},$

$\frac{d\tilde{\mathrm{m}}_{a,b}(r)}{dr}=\frac{d\tilde{\mathrm{m}}_{1,1}(r)}{dr},$

and all the quantities are holomorphic in the mentioned region. Integrating
both sides with respect to $r,$ we get

$\tilde{\mathrm{m}}_{a,b}(r)=\tilde{\mathrm{m}}_{1,1}(r)+\tilde{f}(a,b),\quad\quad\mbox{for}\
|r|>R_{a,b,1,1},$

where $\tilde{f}(a,b)$ is the integration constant which only depends on $a$
and $b.$ Taking the real part of both sides yields

(16) $\mathrm{m}_{a,b}(r)=\mathrm{m}_{1,1}(r)+f(a,b),\quad\quad\mbox{for}\
|r|>R_{a,b,1,1},$

where $\mathop{\mathrm{Re}}(\tilde{f}(a,b))=f(a,b).$

Notice that $\mathrm{m}_{a,b}(r)$ is harmonic on $U_{a,b},$ the unbounded
component of $\mathbb{C}\setminus\mathcal{R}_{a,b}$ which contains
$\\{|r|>R_{a,b}\\},$ and $\mathrm{m}_{1,1}(r)+f(a,b)$ is also harmonic on
$U_{1,1},$ since $f(a,b)$ is constant for $a,b$ fixed. The equality (16)
implies that $\mathrm{m}_{a,b}(r)$ and $\mathrm{m}(r)+f(a,b)$ coincide in the
open neighbourhood $|r|>R_{a,b,1,1}.$ Therefore, they must be equal in
$U_{a,b}\cap U_{1,1},$ that is

(17) $\displaystyle\real(\tilde{\mathrm{m}}_{a,b}(r))$
$\displaystyle=\mathrm{m}_{a,b}(r)=\mathrm{m}(r)+f(a,b),\quad\quad\quad\mbox{for}\
r\in O_{a,b}:=U_{a,b}\cap U_{1,1}$

We now proceed to evaluate $f(a,b)$ in terms of $a,b.$ Since
$\mathcal{R}_{a,b}$ is compact for $a,b>0,$ it is bounded for such $a,b.$ Let
$0<\delta<1$ such that $a,b>\delta.$ Let $\mathcal{M}_{a,b}$ be the subset of
$\mathbb{R}_{>0}^{2}$ defined by

$\mathcal{M}_{a,b}=[a-\delta,a+\delta]\times[b-\delta,b+\delta].$

Note that $(a,b)\in\mathcal{M}_{a,b}.$ Since $\mathcal{M}_{a,b}$ is compact,
and the map $(\alpha,\beta)\mapsto R_{\alpha,\beta}$ is continuous for all
$(\alpha,\beta)$ in $\mathcal{M}_{a,b},$ we conclude that the subset
$\\{R_{\alpha,\beta}:(\alpha,\beta)\in\mathcal{M}_{a,b}\\}$ is compact in
$\mathbb{R}_{>0}.$ Then
$\tilde{R}_{a,b}:=\max_{(\alpha,\beta)\in\mathcal{M}_{a,b}}R_{\alpha,\beta}$
exists, and is finite. Now choose an $R\in\mathbb{R}_{>0}$ such that

$R>\tilde{R}_{a,b}+R_{1,1}.$

The choice of $R$ implies that, for $(\alpha,\beta)\in\mathcal{M}_{a,b},$
$\tilde{\mathrm{m}}_{\alpha,\beta}(R)$ is holomorphic, and (16) yields

(18) $\mathrm{m}_{\alpha,\beta}(R)=\mathrm{m}_{1,1}(R)+f(\alpha,\beta).$

Let $A_{a,b,\delta}\subset\mathbb{C}^{2}$ be the poly-annulus
$A_{a,b,\delta}=A_{a,\delta}\times A_{b,\delta},$ where
$A_{a,\delta}=\\{z\in\mathbb{C}:a-\delta<|z|<a+\delta\\}$ and
$A_{b,\delta}=\\{z\in\mathbb{C}:b-\delta<|z|<b+\delta\\}.$ Note that
$\mathbb{T}_{a,b}^{2}\subset A_{a,b,\delta}.$ Since

$Q_{R}(x,y)\in\mathbb{C}\setminus(-\infty,0]\quad\quad\mbox{for}\ (x,y)\in
A_{a,b,\delta},$

$\log(Q_{R}(x,y))$ is holomorphic in $A_{a,b,\delta},$ where $\log$ is the
principal branch of logarithm. Let $\tilde{W}_{a,b}$ denote the set of all
$(\alpha,\beta)\in\mathcal{M}_{a,b}$ such that

$\tilde{\mathrm{m}}_{\alpha,\beta}(R)=\frac{1}{(2\pi
i)^{2}}\int_{\mathbb{T}_{\alpha,\beta}^{2}}\log(Q_{R}(x,y))\frac{dx}{x}\frac{dy}{y}.$

Note that $\tilde{W}_{a,b}$ is an open subset of $\mathcal{M}_{a,b},$ and it
also contains $(a,b).$

Next we compute the functions
$\alpha\frac{\partial\tilde{\mathrm{m}}_{\alpha,\beta}(R)}{\partial\alpha}$
and $\beta\frac{\partial\tilde{\mathrm{m}}_{\alpha,\beta}(R)}{\partial\beta}.$
We only show here the computation of
$\alpha\frac{\partial\tilde{\mathrm{m}}_{\alpha,\beta}(R)}{\partial\alpha}$
when $\alpha$ belongs to an open subinterval of $(a-\delta,a+\delta)$
containing $a,$ since the other case is analogous.

Note that $\tilde{\mathrm{m}}_{\alpha,\beta}(R)$ and $\log(Q_{R})$ are well-
defined and finite-valued on $\tilde{W}_{a,b}$ and $A_{a,b,\delta},$
respectively. Therefore, we can consider their partial derivatives with
respect to $\alpha,$ and obtain

$\displaystyle\alpha\frac{\partial\tilde{\mathrm{m}}_{\alpha,\beta}(R)}{\partial\alpha}$
$\displaystyle=\alpha\frac{\partial}{\partial\alpha}\left(\frac{1}{(2\pi
i)^{2}}\int_{\mathbb{T}_{\alpha,\beta}^{2}}\log(Q_{R}(x,y))\frac{dx}{x}\frac{dy}{y}\right)$
$\displaystyle=\frac{1}{(2\pi
i)^{2}}\int_{\mathbb{T}_{\alpha,\beta}^{2}}\alpha\frac{\partial\log(Q_{R}(x,y))}{\partial
x}\frac{\partial x}{\partial\alpha}\frac{dx}{x}\frac{dy}{y}$
$\displaystyle=\frac{1}{(2\pi
i)^{2}}\int_{\mathbb{T}_{\alpha,\beta}^{2}}x\frac{\partial_{x}Q_{R}(x,y)}{Q_{R}(x,y)}\frac{dx}{x}\frac{dy}{y}$
(19) $\displaystyle=\frac{1}{(2\pi
i)^{2}}\int_{|y|=\beta}\left(\int_{|x|=\alpha}\frac{\partial_{x}Q_{R}(x,y)}{Q_{R}(x,y)}dx\right)\frac{dy}{y},$

where $\partial_{x}=\frac{\partial}{\partial x},$ and the penultimate equality
follows from the facts that $x=\alpha e^{i\theta}$ and $\theta$ does not
depend on $\alpha.$ For a fixed $y_{0}$ such that $|y_{0}|=\beta,$ the
integrand

$\int_{|x|=\alpha}\frac{\partial_{x}Q_{R}(x,y_{0})}{Q_{R}(x,y_{0})}dx=Z_{\alpha,y_{0},R}^{1}-P_{\alpha,y_{0},R}^{1}$

is an integer, where $Z_{\alpha,y_{0},R}^{1}$ denotes the number of zeros
(counting multiplicity) of the Laurent polynomial $Q_{R}(x,y_{0})$ inside the
circle $\mathbb{T}_{\alpha}^{1},$ and $P_{\alpha,y_{0},R}^{1}$ denote the
order of pole of $Q_{R}(x,y_{0})$ at $x=0.$ Let
$\nu_{\alpha,R}^{1}(y_{0}):=Z_{\alpha,y_{0},R}^{1}-P_{\alpha,y_{0},R}^{1}.$
From Proposition 3.2 (when applied to the torus
$\mathbb{T}_{\alpha,\beta}^{2}$), it follows that $\nu_{\alpha,R}^{1}(y)$ is
constant for all $y$ in $\mathbb{T}_{\beta}^{1}.$ We define
$\nu_{\alpha,\beta,R}^{1}:=\nu_{\alpha,R}^{1}(y)\in\mathbb{Z},$ for all
$y\in\mathbb{T}_{\beta}^{1}.$ Therefore, (19) can be simplified to

(20)
$\frac{\partial\tilde{\mathrm{m}}_{\alpha,\beta}(R)}{\partial\alpha}=\frac{\nu_{\alpha,\beta,R}^{1}}{\alpha}.$

Similarly,

(21)
$\frac{\partial\tilde{\mathrm{m}}_{\alpha,\beta}(R)}{\partial\beta}=\frac{\nu_{\alpha,\beta,R}^{2}}{\beta},$

where
$\nu_{\alpha,\beta,R}^{2}=Z_{\alpha,\beta,R}^{2}-P_{\alpha,\beta,R}^{2}.$ Here
$Z_{\alpha,\beta,R}^{2}$ and $P_{\alpha,\beta,R}^{2}$ are similarly defined.

Since the integer-valued functions $\nu_{\alpha,\beta,R}^{1}$ and
$\nu_{\alpha,\beta,R}^{2}$ depend on $\alpha$ and $\beta$ continuously, they
are constant on $\tilde{W}_{a,b}\subset\ \mbox{int}(\mathcal{M}_{a,b}).$ In
other words,

$\nu_{a,b,R}^{1}=\nu_{\alpha,\beta,R}^{1},\quad\mbox{and}\quad\nu_{a,b,R}^{2}=\nu_{\alpha,\beta,R}^{2},\quad\quad\mbox{for
all}\ (\alpha,\beta)\in\tilde{W}_{a,b}.$

Integrating (20) with respect to $\alpha$ and then taking the real part yields

$\mathrm{m}_{\alpha,\beta}(R)=\mathrm{m}_{1,1}(R)+\nu_{a,b,R}^{1}\log\alpha+F(\beta),$

where $F$ is a function of $\beta$ which does not depend on $\alpha$ and $R.$
A similar process when applied to (21) implies that

$\mathrm{m}_{\alpha,\beta}(R)=\mathrm{m}_{1,1}(R)+\nu_{a,b,R}^{2}\log\beta+G(\alpha),$

where $G$ is independent of $\beta$ and $R.$ From the above equalities and
(18), we conclude that

(22)
$\mathrm{m}_{\alpha,\beta}(R)=\mathrm{m}_{1,1}(R)+\nu_{a,b,R}^{1}\log\alpha+\nu_{a,b,R}^{2}\log\beta+c,$

for all $(\alpha,\beta)\in\mathcal{M}_{a,b},$ and some constant $c$
independent of $\alpha,\beta,R.$ As $|R|>R_{1,1},$ evaluating (22) at
$\alpha=1,\beta=1$ we obtain $c=0.$ Then, combining (17) and (22) together, we
derive that

(23) $f(a,b)=\nu_{a,b,R}^{1}\log a+\nu_{a,b,R}^{2}\log
b,\quad\quad\quad\mbox{for}\ r\in O_{a,b}.$

Since $f(a,b)$ in (17) is independent of $r,$ comparing (23) with (17) we
obtain that, for $j=1,2,$ $\nu_{a,b,R}^{j}$ is constant in $O_{a,b},$ i.e.

$\nu_{a,b,R}^{j}=\nu_{a,b,r}^{j},\quad\quad\mbox{when}\ r\in
O_{a,b},j\in\\{1,2\\}.$

This concludes the proof of Theorem 1.2, namely

$\mathrm{m}_{a,b}(r)=\mathrm{m}_{1,1}(r)+\nu_{a,b,r}^{1}\log
a+\nu_{a,b,r}^{2}\log b,\quad\quad\quad\mbox{for}\ r\in O_{a,b}=U_{a,b}\cap
U_{1,1}.$

∎

###  3.1. Invariance of $\nu_{w,b,r}^{2}$

It now remains to prove Proposition 3.2, which tells us that, for all
$w\in\mathbb{T}_{a}^{1},$ $\varrho_{a,b,r}^{2}(w)$ is constant. Moreover, (13)
implies that the constant is $\nu_{a,b,r}^{2}+v_{2},$ where $v_{2}$ is the
largest power of $y^{-1}$ in $Q_{r}(x,y).$ In particular, we will show that
$\nu_{w,b,r}^{2}=\nu_{a,b,r}^{2}$ for all $w\in\mathbb{T}_{a}^{1},$ where
$\nu_{w,b,r}^{2}$ is given in (5). Before proceeding with the proof, we first
consider the resultant of the polynomial $Q_{r}$ with respect to $y.$

Recall that

$Q_{r}(x,y)=y^{-v_{2}}Q_{F,r}^{y}(x)\prod_{j=1}^{d_{y}}(y-y_{j,r}(x)),$

where $y_{j,r}(x)$ are algebraic functions in $x,$ and $v_{2}$ is as defined
above.

Here and in what follows for the rest of this section, we denote
$Q_{F,r}(x):=Q_{F,r}^{y}(x),d:=d_{y}.$

Let $D_{r}(x)$ denote the resultant of $Q_{r}(x,y)$ and
$\frac{\partial}{\partial y}Q_{r}(x,y)$ with respect to $y.$

Then the algebraic solutions $y_{j,r}$ are holomorphic in some neighbourhood
of $x$ for any $x\in\mathbb{C}\setminus S_{r},$ where

(24) $S_{r}=\\{z\in\mathbb{C}:Q_{F,r}(z)D_{r}(z)=0\\}$

is a finite subset of $\mathbb{C}.$

Let $\textbf{y}_{r}(x)$ be the $d$-valued global analytic function, with
$d$-branches $y_{1,r},\dots y_{d,r},$ such that
$Q_{r}(x,\textbf{y}_{r}(x))=0.$ Then $S_{r}$ is called the set of critical
points of $\textbf{y}_{r}(x).$ If $x^{\prime}$ is a critical point of
$\textbf{y}_{r}(x),$ then $x^{\prime}$ is either an algebraic branch point or
a pole (for more details see [19]).

  1. (bx0)

If $x^{\prime}\in S_{r}$ is an algebraic branch point, i.e. when
$D_{r}(x^{\prime})=0,$ then, in a sufficiently small neighbourhood
$U_{x^{\prime}}$ of $x^{\prime}$ (which does not contain any other critical
points), the multi-set $\\{y_{1,r},\dots y_{d,r}\\}$ can be decomposed into a
number of non-intersecting cycles

$\\{f_{1}(x),\dots,f_{k_{1}}(x)\\},\dots,\\{f_{k_{1}+\dots+k_{t-1}+1}(x),\dots,f_{k_{1}+\dots+k_{t}}(x)\\},$

such that $\sum_{n=1}^{t}k_{n}=d,$ and $f_{j}(x)=y_{l,r}(x)$ for some
$j,l\in\\{1,\dots,d\\}.$ The elements of the first cycle can be represented as
convergent Puiseux series of the local parameter
$\tau=(x-x^{\prime})^{1/k_{1}}$ in a small enough neighbourhood of $\tau=0.$
The elements of the rest of the cycles follow analogous convergent series
representations. Therefore, a single turn around $x^{\prime}$ in a circle
$C^{\prime}\subset U_{x^{\prime}}$ converts the Puiseux series of elements in
one cycle into each other in a cyclic order, i.e. $f_{1}\rightarrow
f_{2}\rightarrow\cdots\rightarrow f_{k_{1}}\rightarrow f_{1}$ etc.

  2. (bx0)

If $x^{\prime}\in S_{r}$ is a pole, that is when $Q_{F,r}(x^{\prime})=0,$
then, substituting $y$ with $yQ_{F,r}(x),$ we return to the first case where
the local parameter of the convergent series is $\tau=1/x.$

Recall that, for $w\in\mathbb{T}_{a}^{1},$ $\varrho_{a,b,r}^{2}(w)$ denote the
number of roots of $Q_{r}(w,y)$ lying inside the circle $\mathbb{T}_{b}^{1}.$
We are now ready to prove Proposition 3.2.

###### Proof of Proposition 3.2.

First fix an arbitrary $r\in\mathbb{C}\setminus\mathcal{R}_{a,b}.$ Note that
$\varrho_{a,b,r}^{2}$ defines a function from $\mathbb{T}_{a}^{1}$ to
$\mathbb{Z}$ via the map $x\mapsto\varrho_{a,b,r}^{2}(x),$ where $\mathbb{Z}$
is equipped with discrete topology.

If $x_{0}\in\mathbb{T}_{a}^{1}$ is not a critical point of $\textbf{y}_{r},$
i.e. $x_{0}\notin S_{r},$ where $S_{r}$ is given in (24), then, for all
$j=1,\dots,d,$ $y_{j,r}$ is holomorphic in a sufficiently small neighbourhood
$U_{x_{0}}$ of $x_{0}$ which does not contain any critical point. Therefore,
$|y_{j,r}(x)|$ is continuous in $U_{x_{0}}.$ Since $Q_{r}$ does not vanish on
$\mathbb{T}_{a,b}^{2},$ we have $|y_{j,r}(x)|\neq b$ for all $x\in
U_{x_{0}}\cap\mathbb{T}_{a}^{1},$ $j.$ Therefore, if, for any $l=1,\dots,d,$
$|y_{l,r}(x_{0})|<1$ (resp. $|y_{l,r}(x_{0})|>1$), then, for all $x\in
U_{x_{0}\cap\mathbb{T}_{a}^{1}},$ $|y_{l,r}(x)|<1$ (resp.
$|y_{l,r}(x_{0})|>1$). In other words, $\varrho_{a,b,r}^{2}(x)$ is constant
for all $x\in U_{x_{0}}\cap\mathbb{T}_{a}^{1}.$ In particular,
$\varrho_{a,b,r}^{2}$ is continuous at $x_{0}.$

If $x_{1}\in\mathbb{T}_{a}^{1}\cap S_{r},$ then there exists a sufficiently
small neighbourhood $U_{x_{1}}$ of $x_{1}$ which does not contain any critical
point except $x_{1}.$ Then the convergent Puiseux series expansions of
$y_{1,r},\dots,y_{d,r}$ in $U_{x_{1}}$ imply that, for all $j,$ $|y_{j,r}|$ is
continuous in $U_{x_{1}},$ and this brings us to the previous case. From
properties (1), (2) and the above discussion, we conclude that, in the
neighbourhood $U_{x_{1}}$ of $x_{1},$ $\varrho_{a,b,r}^{2}$ is constant. This
implies that $\varrho_{a,b,r}^{2}$ is continuous at $x_{1}.$

We now have a continuous function $\varrho_{a,b,r}^{2}$ from a connected set
$\mathbb{T}_{a}^{1}$ to a discrete set $\mathbb{Z}.$ Since only connected
subsets of $\mathbb{Z}$ are singletons, we derive that $\varrho_{a,b,r}^{2}$
is constant in $\mathbb{T}_{a}^{1},$ and thus completing the proof of the
statement. ∎

###  3.2. A brief discussion on generalized Mahler measure in terms of periods

In this section, our aim is to describe the whole phenomena mentioned above in
terms of periods when the curve defined by the polynomial has non-zero genus.
Rodriguez-Villegas [7], Deninger [6] et al. showed that, for a family of non-
zero polynomials $\\{Q_{r}:r\in\mathbb{C}\\},$ the quantity
$\frac{d\tilde{\mathrm{m}}(Q_{r})}{dr}$ is in fact a period of the non-
singular curve $C_{r}$ associated to $Q_{r},$ when the genus of $C_{r}$ is
non-zero for generic $r.$ Note that, for generic $r,$ $C_{r}$ is a non-
singular hypersurface in $\mathbb{P}^{2}.$ Here, using Griffith’s method [20],
we extend their idea to the arbitrary torus case. In other words, we show that
even if we change the integration torus $\mathbb{T}_{a,b}^{2},$ for $r\in
U_{a,b},$

$\frac{d\tilde{\mathrm{m}}_{a,b}(Q_{r})}{dr}\quad\mbox{is a period of}\
C_{r}.$

We use this to derive (17).

A typical holomorphic differential (or holomorphic $1$-form) of $C_{r}$ can be
expressed as the Poincaré residue of

$\psi_{1,r}=\frac{f(x,y)dxdy}{Q_{r}(x,y)},\quad\quad\quad\quad\deg f\leq\deg
Q_{r}-3,$

along $C_{r}.$ In other words,

$\omega_{1,r}=\frac{f(x,y)dx}{\frac{\partial Q_{r}}{\partial
y}}=\frac{f(x,y)dy}{\frac{\partial Q_{r}}{\partial x}},\quad\quad\deg
f\leq\deg Q_{r}-3$

defines a holomorphic $1$-form on $C_{r}.$ Therefore, the Poincaré residue of

$\omega_{r}=\frac{1}{Q_{r}}\frac{dxdy}{xy}=\frac{1}{r-Q(x,y)}\frac{dx}{x}\frac{dy}{y}$

along $C_{r}$ is a holomorphic $1$-form, and we denote it by
$\mathrm{Res}(\omega_{r}).$

Now $\Gamma_{a,b}:=\mathbb{T}_{a,b}^{2}$ defines a $2$-cycle in
$\mathbb{P}^{2}.$ Additionally, for $r\notin\mathcal{R}_{a,b},$ $\Gamma_{a,b}$
defines a $2$-cycle in $\mathbb{P}^{2}\setminus C_{r}.$ Since any $2$-cycle in
$\mathbb{P}^{2}\setminus C_{r}$ is homologous to a tube over an $1$-cycle on
$C_{r}$ [20], we have

$\int_{\gamma_{a,b}}\mathrm{Res}(\omega_{r})=\frac{1}{2\pi
i}\int_{\tau(\gamma_{a,b})}\omega_{r},$

where $\gamma_{a,b}\in H_{1}(C_{r},\mathbb{Z})$ is a $1$-cycle lying on
$C_{r},$ and $\tau(\gamma_{a,b}),$ a tube over $\gamma_{a,b},$ is a $2$-cycle
homologous to $\Gamma_{a,b}$ in $H_{2}(\mathbb{P}^{2}\setminus
C_{r},\mathbb{Z}).$ We know that $\int_{\gamma}\mathrm{Res}(\omega_{r})$ is a
period of $C_{r}$ for $\gamma\in H_{1}(C_{r},\mathbb{Z}).$ Therefore, we have
shown that, indeed, for $r\in\mathbb{C}\setminus\mathcal{R}_{a,b},$

$\mathcal{F}_{a,b}(r)=\frac{1}{(2\pi
i)^{2}}\int_{\Gamma_{a,b}}\omega_{r}=\frac{1}{(2\pi
i)^{2}}\int_{\tau(\gamma_{a,b})}\omega_{r}=\frac{1}{2\pi
i}\int_{\gamma_{a,b}}\mathrm{Res}(\omega_{r})\quad\mbox{is a period of}\
C_{r},$

where $\mathcal{F}_{a,b}(r)$ is given in (15). As a result, it satisfies one
of the Picard-Fuchs differential equations associated to $C_{r},$ in
particular the Picard-Fuchs differential equation associated to the
holomorphic $1$-form $\mathrm{Res}(\omega_{r}).$ Since, $\mathcal{F}_{a,b}(r)$
is holomorphic for $|r|>R,$ it agrees with a holomorphic solution of the
corresponding differential equation on the unbounded component of
$\mathbb{C}\setminus\mathcal{R}_{a,b}$ containing an open neighbourhood of
$r=\infty,$ namely $\\{r:|r|>R\\}.$

Recall that $U_{a,b}$ denotes the unbounded component of
$\mathbb{C}\setminus\mathcal{R}_{a,b}.$ Following the arguments in [7] and
[13], we conclude that, in fact for $r\in U_{1,1},$ $\mathcal{F}_{1,1}(r)$ is
holomorphic. This produces the equality

(25) $\mathcal{F}_{a,b}(r)=\mathcal{F}_{1,1}(r)\quad\quad\quad\mbox{for}\ r\in
U_{a,b}\cap U_{1,1},$

Integrating (25) with respect to $r,$ and then taking the real part and using
the harmonicity of the Mahler measure we conclude that, for $r\in U_{a,b}\cap
U_{1,1},$

$\mathrm{m}_{a,b}(r)=\mathrm{m}_{1,1}(r)+f(a,b),$

where $f$ is a function of $a$ and $b.$

##  4\. Proof of Theorem 1.3

In this section, our goal is to provide the proof of Theorem 1.3, and
eventually evaluate $\mathrm{m}_{a,b}(Q_{r})$ when
$r\in\mathbb{C}\setminus(\mathcal{R}_{a,b}\cup U_{a,b}).$ Our proof uses
Proposition 3.2 to conclude that, for all $r$ from a small enough
neighbourhood in one of the bounded regions under consideration, certain
properties of the roots of $Q_{r}(a,y)$ or $Q_{r}(x,b)$ remain invariant.
This, combined with the properties of harmonic functions along with Rouché’s
theorem, gives us the desired results.

Recall that $Q_{r}(x,y),$ considered as a polynomial in $y$ of degree $d_{y}$
with coefficients in $\overline{\mathbb{C}(x)},$ can be factored in
$\overline{\mathbb{C}(x)}[y]$ as

(26) $\displaystyle Q_{r}(x,y)=$
$\displaystyle(y)^{-v_{2}}\left(Q_{F,r}^{y}(x)(y)^{d_{y}}+Q_{f,r}^{y}(x)+\sum_{j=1}^{d_{y}-1}a_{j,r}^{y}(x)(y)^{j}\right)$
(27) $\displaystyle=$
$\displaystyle(y)^{-v_{2}}Q_{F,r}^{y}(x)\prod_{j=1}^{d_{y}}(y-y_{j,r}(x)),$

where the $y_{j,r}(x)$ are algebraic functions in $x,$ $v_{2}$ is the order of
the pole of $Q_{r}(a,y)$ at $y=0,$ and $Q_{F,r}^{y}(x)$ and $Q_{f,r}^{y}(x)$
are the respective leading and "constant" coefficient with respect to the
variable $y.$ Similarly, we can factor $Q_{r},$ considered as a polynomial in
$x$ of degree $d_{x}$ with coefficients in $\overline{\mathbb{C}(y)},$ as

$\displaystyle Q_{r}(x,y)=$
$\displaystyle(x)^{-v_{1}}\left(Q_{F,r}^{x}(y)(x)^{d_{x}}+Q_{f,r}^{x}(y)+\sum_{j=1}^{d_{x}-1}a_{j,r}^{x}(y)(x)^{j}\right)$
$\displaystyle=$
$\displaystyle(x)^{-v_{1}}Q_{F,r}^{x}(y)\prod_{j=1}^{d_{x}}(x-x_{j,r}(y)),$

where the $x_{j,r}(y)$ are algebraic functions in $y,$ $v_{1}$ is the order of
the pole of $Q_{r}(x,b)$ at $x=0,$ and $Q_{F,r}^{x}(y)$ and $Q_{f,r}^{x}(y)$
are the respective leading and "constant" coefficient with respect to the
variable $x.$

Let $Z_{F,r}^{u}=\\{z\in\mathbb{C}:Q_{F,r}^{u}(z)=0\\},$
$Z_{f,r}^{u}=\\{z\in\mathbb{C}:Q_{f,r}^{u}(z)=0\\},$ where $u=x\ \mbox{or}\
y,$ and $V_{a,b,r_{0}}$ denotes the bounded open connected component of
$\mathbb{C}\setminus\mathcal{R}_{a,b}$ containing $r_{0}.$

Since the proofs of the statements in (i) and (ii) of Theorem 1.3 are similar,
here we restrict ourselves in proving the statement (i)

###### Proof of Theorem 1.3.

In (26), we see that the polynomial $Q_{r}(x,y)$ can be expressed in terms of
$y_{j,r}(x)$ (algebraic functions in $x$), $v_{2},Q_{F,r}^{y}(x)$ and
$Q_{f,r}^{y}(x).$ For simplicity we denote

$Q_{F,r}(x):=Q_{F,r}^{y}(x),Q_{f,r}(x):=Q_{f,r}^{y}(x),\ \mbox{and}\
d:=d_{y}.$

Proposition 3.2 and the assumption in (i) in the statement of Theorem 1.3
yield that $\varrho_{a,b,r_{0}}^{2}(x)=d\ \mbox{or}\ 0$ for all
$x\in\mathbb{T}_{a}^{1}.$ In particular, $\varrho_{a,b,r_{0}}^{2}(a)=d\
\mbox{or}\ 0,$ depending on whether all the roots of $Q_{r}(a,y)$ lie entirely
inside or entirely outside the circle $\mathbb{T}_{b}^{1}.$

The following three cases can occur when $\varrho_{a,b,r_{0}}^{2}(a)=d\
\mbox{or}\ 0.$

  1. bx0:

For all $x\in\mathbb{T}_{a}^{1},$

$Q_{F,r_{0}}(x)\cdot Q_{f,r_{0}}(x)\neq 0.$

  2. bx0:

$Q_{F,r_{0}}^{y}$ vanishes on $\mathbb{T}_{a}^{1},$ but $Q_{f,r_{0}}^{y}$ does
not, i.e.

$Z_{F,r_{0}}^{y}\cap\mathbb{T}_{a}^{1}\neq\varnothing,\quad\mbox{and}\quad
Z_{f,r_{0}}^{y}\cap\mathbb{T}_{a}^{1}=\varnothing,$

  3. bx0:

$Q_{f,r_{0}}^{y}$ vanishes on $\mathbb{T}_{a}^{1},$ but $Q_{F,r_{0}}^{y}$ does
not, i.e.

$Z_{f,r_{0}}^{y}\cap\mathbb{T}_{a}^{1}\neq\varnothing,\quad\mbox{and}\quad
Z_{F,r_{0}}^{y}\cap\mathbb{T}_{a}^{1}=\varnothing.$

Case 1: Since

$Q_{F,r_{0}}(x)\cdot Q_{f,r_{0}}(x)\neq 0\quad\quad\mbox{for all}\
x\in\mathbb{T}_{a}^{1},$

the discussion preceding the proof of Proposition 3.2 implies that the
algebraic functions $y_{j,r_{0}}(x)$ may have only an algebraic branch point
at $x=a\in\mathbb{T}_{a}^{1}.$ From Proposition 3.2 we know that
$\nu_{a,b,r_{0}}^{2}$ is constant in $\mathbb{T}_{a}^{1}.$ Therefore, we can
in fact assume that $x=a$ is not a branch point of $y_{j,r_{0}}(x)$ for all
$j.$ Indeed, if $x=a$ is branch point, then there exists an
$x_{0}\in\mathbb{T}_{a}^{1}$ close enough to $a$ such that $x_{0}\notin
S_{r},$ where $S_{r}$ is given in (24). We replace $a$ with $x_{0}$ in the
statement, and proceed. Here we provide a proof of Case 1 when
$\varrho_{a,b,r_{0}}^{2}(a)=d,$ since the case when
$\varrho_{a,b,r_{0}}^{2}(a)=0$ is similar. Recall that the condition
$\varrho_{a,b,r_{0}}^{2}(a)=d$ (resp. $\varrho_{a,b,r_{0}}^{2}(a)=0$) is
equivalent to the condition that all the roots of $Q_{r_{0}}(a,y)$ lie inside
(resp. outside) the circle $\mathbb{T}_{b}^{1}.$

The polynomial $Q_{r}(x,y)$ has additional structure: $Q_{r}(x,y)=r-Q(x,y)$
where $Q$ does not contain any constant term, and $r$ is the constant
coefficient in $Q_{r}.$ Therefore, after multiplying $Q_{r}$ by $y^{v_{2}},$
we find from (26) that one, and only one, among the set of the coefficients

$\mathrm{Coeff}_{Q_{r},x}:=\\{Q_{F,r}(x),Q_{f,r}(x),a_{1,r}(x),\dots,a_{d-1,r}(x)\\}\subset\overline{\mathbb{C}(x)}$

contains $r$ as its constant term, namely the coefficient of $y^{v_{2}}$ in
$y^{v_{2}}Q_{r}(x,y).$ Let $b_{v_{2},r}(x)$ denotes the said coefficient. Then
$b_{v_{2},r}(x)\in\mathrm{Coeff}_{Q_{r},x},$ and
$b_{v_{2},r}(x)-b_{v_{2},r_{0}}(x)=r-r_{0}.$ Since all the coefficients,
except $b_{v_{2},r},$ do not depend on $r$ by construction, the above
discussion further implies that

(28)
$\displaystyle\left\\{\left|Q_{F,r}(x)-Q_{F,r_{0}}(x)\right|,\left|Q_{f,r}(x)-Q_{f,r_{0}}(x)\right|\right\\}\cup\left\\{\left|a_{j,r}(x)-a_{j,r_{0}}(x)\right|:1\leq
j\leq d-1\right\\}$ $\displaystyle=$
$\displaystyle\left\\{0,\left|b_{v_{2},r}(x)-b_{v_{2},r_{0}}(x)\right|\right\\}=\left\\{0,\left|r-r_{0}\right|\right\\}.$

In other words, if, for example, $Q_{F,r}(x)=b_{v_{2},r}(x),$ then

$Q_{F,r}(x)-Q_{F,r_{0}}(x)=r-r_{0},\quad
Q_{f,r}(x)=Q_{f,r_{0}}(x),\quad\mbox{and, for all}\ j,\
a_{j,r}(x)=a_{j,r_{0}}(x).$

Next we investigate the relation between $|Q_{r}(a,y)-Q_{r_{0}}(a,y)|$ and
$|Q_{r_{0}}(a,y)|$ when $y$ takes values in certain sufficiently small
circles.

Let

$\epsilon_{ij}=\frac{1}{b}\left|y_{i,r_{0}}(a)-y_{j,r_{0}}(a)\right|\quad\quad\mbox{and}\quad\quad\epsilon_{k}=\frac{1}{b}\min_{t\in\mathbb{T}_{b}^{1}}\left|y_{k,r_{0}}(a)-t\right|.$

Since all the roots of $Q_{r_{0}}(a,y)$ are distinct and lie inside the circle
$\mathbb{T}_{b}^{1}$, the quantities $\epsilon_{ij},\epsilon_{k}$ are non-zero
for any $i,j,k\in\\{1,\dots,d\\}$ such that $i\neq j.$

We denote

$\Upsilon=\min_{\begin{subarray}{c}1\leq i<j\leq d\\\ 1\leq k\leq
d\end{subarray}}\left\\{\epsilon_{ij},\epsilon_{k}\right\\}.$

Note that $\Upsilon>0.$ Let
$\epsilon\in\left(0,\Upsilon\right)\cap\left(0,1\right).$ We define the closed
discs

$D_{j}=\\{z:|z-y_{j,r_{0}}(a)|\leq\epsilon\\},\quad\mbox{for}\ j=1,\dots,d.$

Let $C_{j}=\partial D_{j}$ be the boundary of $D_{j}.$ The choice of
$\epsilon$ then confirms that the discs $D_{j}$ are disjoint and
$Q_{r_{0}}(a,y)$ does not vanish on $C_{j}.$ This implies
$\psi_{j,\epsilon,r_{0}}:=\min_{y\in C_{j}}|Q_{r_{0}}(a,y)|$ is positive for
each $j.$

Let $\delta_{j,r_{0},\epsilon}:=\frac{\psi_{j,\epsilon,r_{0}}}{d+1}.$ Then,
for $y\in C_{j},$ and $r\in V_{a,b,r_{0}}$ such that

$|r-r_{0}|<\delta_{j,r_{0},\epsilon},$

we have

$\displaystyle\left|Q_{r}(a,y)-Q_{r_{0}}(a,y)\right|$ $\displaystyle=$
$\displaystyle\left|\left(Q_{F,r}(a)-Q_{F,r_{0}}(a)\right)(y)^{d}+\left(Q_{f,r}(a)-Q_{f,r_{0}}(a)\right)+\sum_{j=1}^{d-1}\left(a_{j,r}(a)-a_{j,r_{0}}(a)\right)(y)^{j}\right|$
$\displaystyle\leq$
$\displaystyle|r-r_{0}|\left(\sum_{j=0}^{d}|\epsilon|^{j}\right)\leq(d+1)|r-r_{0}|<\psi_{j,\epsilon,r_{0}}\leq\left|Q_{r_{0}}(a,y)\right|,$

where the first inequality follows from (28).

This implies that, for $j=1,\dots,d,$

$\left|Q_{r}(a,y)-Q_{r_{0}}(a,y)\right|<\left|Q_{r_{0}}(a,y)\right|$

on $C_{j}.$ Therefore, it follows from Rouché’s Theorem that $Q_{r}(a,y)$ and
$Q_{r_{0}}(a,y)$ have the same number of root(s) in the interior of $D_{j}$
when $|r-r_{0}|<\delta_{j,r_{0},\epsilon}.$ Moreover, for

$\delta(\epsilon,r_{0})=\min_{1\leq j\leq d}\delta_{j,r_{0},\epsilon}>0,$

the choice of $\epsilon$ implies that, when
$|r-r_{0}|<\delta(\epsilon,r_{0}),$ all the roots of $Q_{r}(a,y)$ lie entirely
inside the circle $\mathbb{T}_{b}^{1}.$

When $|r-r_{0}|<\delta(\epsilon,r_{0}),$ another application of Proposition
3.2 yields that all the roots of $Q_{r}(x,y)$ lie inside $\mathbb{T}_{b}^{1}$
for every $x\in\mathbb{T}_{a}^{1}.$ Following the discussion in Section 2.4
regarding the Mahler measure over arbitrary tori, we conclude that, for
$r\in\\{z:|z-r_{0}|<\delta_{\epsilon,r_{0}}\\}\subset V_{a,b,r_{0}},$

$\displaystyle\mathrm{m}_{a,b}(Q_{r}(x,y))$
$\displaystyle=\mathrm{m}_{a,b}\left((y)^{-v_{2}}Q_{F,r}(x)\prod_{j=1}^{d}(y-y_{j,r}(x))\right)$
$\displaystyle=\mathrm{m}_{a}(Q_{F,r}(x))-v_{2}\log
b+\mathrm{m}_{a,b}\left(\prod_{j=1}^{d}(y-y_{j,r}(x))\right)$ (29)
$\displaystyle=\mathrm{m}_{a}(Q_{F,r}(x))-v_{2}\log b+d\log b.$

Similarly, when all roots of $Q_{r_{0}}(a,y)$ lie outside the circle
$\mathbb{T}_{b}^{1},$ we have for
$r\in\\{z:|z-r_{0}|<\delta_{\epsilon,r_{0}}\\}\subset V_{a,b,r_{0}},$

$\displaystyle\mathrm{m}_{a,b}(Q_{r}(x,y))=$
$\displaystyle\mathrm{m}_{a}(Q_{F,r}(x))-v_{2}\log
b+\mathrm{m}_{a}(Q_{f,r}(x))-\mathrm{m}_{a}(Q_{F,r}(x))$ (30) $\displaystyle=$
$\displaystyle\mathrm{m}_{a}(Q_{f,r}(x))-v_{2}\log b.$

Recall that, $\nu_{a,b,r}^{2}$ denotes the difference between the number of
zeros (counting multiplicity) of $Q_{r}(a,y)$ inside $\mathbb{T}_{b}^{1}$ and
the order of pole of $Q_{r}(a,y)$ at $y=0.$ Then the above discussion implies
that, for $r\in\\{z:|z-r_{0}|<\delta_{\epsilon,r_{0}}\\}\subset V_{a,b},$

$\nu_{a,b,r}^{2}=\nu_{a,b,r_{0}}^{2}=\varrho_{a,b,r_{0}}^{2}(x)-v_{2}=d-v_{2}\
\mbox{or}\ -v_{2}.$

Since $\mathrm{m}_{a}(Q_{F,r}(x))$ and $\mathrm{m}_{a}(Q_{f,r}(x))$ are
harmonic, and $\mathrm{m}_{a,b}(Q_{r}(x,y))$ is harmonic for all $r\in
V_{a,b,r_{0}}\setminus\mathcal{S}_{a,b,r_{0}}$ (where
$\mathcal{S}_{a,b,r_{0}}$ is a finite set containing all the $r\in
V_{a,b,r_{0}}$ such that $Q_{r}(x,y)$ is singular), the equalities in (29) and
(30) can be extended to a larger set
$V_{a,b,r_{0}}\setminus\mathcal{S}_{a,b,r_{0}}.$ using the harmonicity of
Mahler measure. In other words, for $r,r_{0}\in
V_{a,b,r_{0}}\setminus\mathcal{S}_{a,b,r_{0}},$

(31) $\mathrm{m}_{a,b}(Q_{r})-\nu_{a,b,r_{0}}^{2}\log
b=\left\\{\begin{array}[]{ll}\mathrm{m}_{a}(Q_{F,r}(x))&\mbox{all roots of}\
Q_{r_{0}}(a,y)\ \mbox{lie inside}\ \mathbb{T}_{b}^{1},\\\
\mathrm{m}_{a}(Q_{f,r}(x))&\mbox{all roots of}\ Q_{r_{0}}(a,y)\ \mbox{lie
outside}\ \mathbb{T}_{b}^{1}.\end{array}\right.$

By continuity, (31) holds for all $r\in V_{a,b,r_{0}},$ and this concludes the
proof of the Case 1.

Recall that $Z_{F,r}^{y}=\\{z\in\mathbb{C}:Q_{F,r}^{y}(z)=0\\},$ and
$Z_{f,r}^{y}=\\{z\in\mathbb{C}:Q_{f,r}^{y}(z)=0\\}.$
Case 2: If

$Z_{F,r_{0}}^{y}\cap\mathbb{T}_{a}^{1}\neq\varnothing,\quad\mbox{and}\quad
Z_{f,r_{0}}^{y}\cap\mathbb{T}_{a}^{1}=\varnothing,$

then there exists $x^{\prime}\in Z_{F,r_{0}}^{y}\cap\mathbb{T}_{a}^{1},$ and a
$l\in\\{1,\dots,d_{y}\\}$ such that $y_{l,r_{0}}(x)$ has a pole at
$x^{\prime}.$ Then Proposition 3.2 and the conditions in the statement of
Theorem 1.3 imply that all the roots of $Q_{r_{0}}(a,y)$ lie outside the
circle $\mathbb{T}_{b}^{1},$ and we can choose an $x_{0}\in\mathbb{T}_{a}^{1}$
in a sufficiently small neighbourhood of $x^{\prime},$ such that $x_{0}$ is
not a pole of $y_{j,r_{0}}$ for all $j.$ Such choice is possible since the set
of critical points $S_{r_{0}}$ of the global analytic function
$\textbf{y}_{r_{0}}$ is a finite set. Then a similar argument as in Case $1$
implies that, for all $r\in V_{a,b,r_{0}},$

$\mathrm{m}_{a,b}(Q_{r})-\nu_{a,b,r}^{2}\log
b=\mathrm{m}_{a}(Q_{f,r}^{y}(x)).$

Case 3: If

$Z_{f,r_{0}}^{y}\cap\mathbb{T}_{a}^{1}\neq\varnothing,\quad\mbox{and}\quad
Z_{F,r_{0}}^{y}\cap\mathbb{T}_{a}^{1}=\varnothing,$

then there exists $x^{\prime\prime}\in Z_{f,r_{0}}^{y}\cap\mathbb{T}_{a}^{1},$
and a $p\in\\{1,\dots,d_{y}\\}$ such that $y_{p,r_{0}}(x)$ has a zero at
$x^{\prime\prime}.$ Again, Proposition 3.2 and the conditions in the statement
of Theorem 1.3 imply that all the roots of $Q_{r_{0}}(a,y)$ lie inside the
circle $\mathbb{T}_{b}^{1},$ and we can choose an
$x_{1}\in\mathbb{T}_{a}^{1},$ such that $x_{1}\notin S_{r_{0}}\cup
Z_{f,r_{0}}^{y},$ and $Q_{r_{0}}(x_{1},y)$ has all the roots inside
$\mathbb{T}_{b}^{1}.$ With these conditions, we have, for all $r\in
V_{a,b,r_{0}},$

$\mathrm{m}_{a,b}(Q_{r})-\nu_{a,b,r}^{2}\log
b=\mathrm{m}_{a}(Q_{F,r}^{y}(x)).$

This concludes the proof of the statement (i). Statement (ii) follows from an
analogous argument. ∎

##  5\. Generalized Mahler measure of a family of polynomials

In this section, we consider the family of polynomials

$\left\\{Q_{r}(x,y)=x+\frac{1}{x}+y+\frac{1}{y}+r:r\in\mathbb{C}\right\\}.$

Boyd [5], Deninger [6], Rodriguez-Villegas [7], Lalín [9], Rogers and Zudilin
[8] et al have successfully evaluated the (standard) Mahler measure of $Q_{r}$
for different values of $r$ in terms of special values of Dirichlet
$L$-functions, $L$-functions of elliptic curves, special values of the Bloch-
Wigner dilogarithm, etc. Our aim here is to apply Theorems 1.2 and 1.3 to
evaluate the generalized Mahler measure of $Q_{r}.$

Before proceeding with this evaluation we recall some notation associated to
the considered family of polynomials for the reader’s convenience.

  1. 1.

The map in (4) is defined in this case as

$q:\mathbb{T}_{a,b}^{2}\mapsto\mathbb{C},\quad\quad(x,y)\mapsto
x+\frac{1}{x}+y+\frac{1}{y}.$

  2. 2.

The image of $q$ is denoted by $\mathcal{R}_{a,b}.$ The elements of
$\mathcal{R}_{a,b}$ are of the form

$r=\left(a+a^{-1}\right)\cos\alpha+\left(b+b^{-1}\right)\cos\beta+i\left[\left(a-a^{-1}\right)\sin\alpha+\left(b-b^{-1}\right)\sin\beta\right],$

where $\alpha,\beta\in[-\pi,\pi).$

  3. 3.

Since $\mathcal{R}_{a,b}$ is compact,
$R_{a,b}=\max_{r\in\mathcal{R}_{a,b}}|r|$ exists.

  4. 4.

$U_{a,b}$ denotes the unbounded open connected component of
$\mathbb{C}\setminus\mathcal{R}_{a,b}.$ It contains the region
$\\{|r|>R_{a,b}\\};$ since $R_{1,1}=4,$ we have $U_{a,b}\subseteq U_{1,1}.$

Now we are ready to apply our theorems to evaluate the generalized Mahler
measure of $Q_{r}.$

###  5.1. Generalized Mahler measure on the unbounded component of
$\mathbb{C}\setminus\mathcal{R}_{a,b}$

In [7] Rodriguez-Villegas expressed the (standard) Mahler measure of $Q_{r}$
in terms of Eisenstein-Kronecker series for any $r\in\mathbb{C}.$ Combining
his proof and Theorem 1.2, we will show that, for fixed $a,b>0,$ there exists
a large open subset of $\mathbb{C},$ namely $O_{a,b}=U_{a,b}\cap U_{1,1},$
such that if $r\in O_{a,b},$ then the Mahler measure remains unchanged
irrespective of the dependence of the integration torus on $(a,b).$ We will in
fact go further and show that $O_{a,b}$ is the unbounded component of
$\mathbb{C}\setminus\mathcal{R}_{a,b},$ namely $U_{a,b}.$ Later in this
section, we will give an explicit expression of the region $O_{a,b},$ as well
as of the region $\mathcal{R}_{a,b}.$

Recall that, for fixed $a,b>0,$ $Q_{r}$ does not vanish on
$\mathbb{T}^{2}_{a,b}$ if and only if $r\notin\mathcal{R}_{a,b}.$ In order to
show that, for a fixed $a,b>0,$

$\mathrm{m}_{a,b}(Q_{r})=\mathrm{m}(Q_{r}),\quad\quad\quad\mbox{for all}\ r\in
O_{a,b},$

it suffices to evaluate $\nu_{a,b,r}^{j}$ for $j=1,2.$ Since these quantities
are constant in the region $O_{a,b},$ we can choose a suitable $r$ and apply
Theorem 1.2 to evaluate them. Let

$R=R_{a,b}+R_{1,1}=a+\frac{1}{a}+b+\frac{1}{b}+4.$

Note that $R\in O_{a,b}$ and $R\notin(-\infty,0].$

Recall that $\nu_{a,b,r}^{1}$ denotes the difference between the number of
zeros (counting multiplicity), namely $Z_{a,b,r}^{1}$ and the number of poles
(counting multiplicity), namely $P_{a,b,r}^{1},$ of $Q_{r}(x,b)$ inside the
circle $\mathbb{T}_{a}^{1},$ i.e.

$\nu_{a,b,r}^{1}=Z_{a,b,r}^{1}-P_{a,b,r}^{1},$

and that $\nu_{a,b,r}^{2}$ is also defined in a similar way.

Since $Q_{R}(x,b)$ is holomorphic everywhere except for a simple pole at
$x=0,$ we have $P_{a,b,R}^{1}=1.$ Therefore, $xQ_{R}(x,b)$ has no pole in
$\mathbb{C}.$ Now $xQ_{R}(x,b)$ can be factored in $\mathbb{C}[x]$ as

$xQ_{R}(x,b)=\left(x-x_{+}\right)\left(x-x_{-}\right),$

where

$x_{\pm}=\frac{-\left(R+b+\frac{1}{b}\right)\pm\sqrt{\left(R+b+\frac{1}{b}\right)^{2}-4}}{2}.$

Notice that $x_{+}\cdot x_{-}=1,$ and since $R+b+\frac{1}{b}>a+\frac{1}{a},$
we also have

$\displaystyle|x_{-}|$
$\displaystyle=\left|\frac{R+b+\frac{1}{b}+\sqrt{\left(R+b+\frac{1}{b}\right)^{2}-4}}{2}\right|=\frac{R+b+\frac{1}{b}+\sqrt{\left(R+b+\frac{1}{b}\right)^{2}-4}}{2}$
$\displaystyle=\frac{R+b+\frac{1}{b}+\sqrt{\left(a+\frac{1}{a}+b+\frac{1}{b}+b+\frac{1}{b}+6\right)\left(a+\frac{1}{a}+b+\frac{1}{b}+b+\frac{1}{b}+2\right)}}{2}$
$\displaystyle\geq a+\frac{1}{a}.$

Since $a+\frac{1}{a}>\max\left\\{a,\frac{1}{a}\right\\},$ we have
$|x_{+}|\leq\frac{1}{a+\frac{1}{a}}<a,$ and therefore, $Z_{a,b,R}^{1}=1.$ By
the definition of $\nu_{a,b,R}^{1},$ it follows that $\nu_{a,b,R}^{1}=0.$ A
similar argument shows that $\nu_{a,b,R}^{2}=0.$ Combining Theorem 1.2 and the
values obtained above, we derive that, for $r\in U_{a,b}\subset U_{1,1},$

$\mathrm{m}_{a,b}(Q_{r})=\mathrm{m}(Q_{r}),$

and the required $O_{a,b}$ is in fact the region $U_{a,b}.$

Until now we have been fixing $a,b>0$ in our discussion. Next, we want to show
that our theorem can even be applied to a fixed suitable $r$ in order to
obtain certain values of $(a,b)$ such that the equality
$\mathrm{m}_{a,b}(Q_{r})=\mathrm{m}(Q_{r})$ still holds.

For some particular values of $r\in\mathbb{R}\cup i\mathbb{R},$ the standard
Mahler measure of $Q_{r}$ has been proven to be the same as (up to a rational
multiple) a special value of $L$-function of the elliptic curve corresponding
to $Q_{r}$ due to Boyd [5], Rodriguez-Villegas [7], Deninger [6], Rogers and
Zudilin [8], Lalín and Rogers [9] et al. Therefore, an interesting direction
would be to search for values of $(a,b)$ such that changing the integration
torus from $\mathbb{T}^{2}$ ($=\mathbb{T}_{1,1}^{2}$) to
$\mathbb{T}_{a,b}^{2}$ keeps the Mahler measure fixed. In order to do so,
first notice that, for all $r>R_{a,b},$ Theorem 1.2 implies that

$\mathrm{m}_{a,b}(Q_{r})=\mathrm{m}(Q_{r}).$

Since $a$ and $b$ are fixed arbitrarily, we can fix $r=r_{0}>4,$ and conclude
that, for all $2$-tuples $(a,b)$ satisfying

$a+\frac{1}{a}+b+\frac{1}{b}<r_{0},$

we have $\mathrm{m}_{a,b}(Q_{r_{0}})=\mathrm{m}(Q_{r_{0}}).$ Since the change
of variables $r\mapsto-r$ covers the case when $r<-4,$ it is sufficient to
consider the $r>4$ case here.

For $r\in i\mathbb{R},$ it suffices to investigate the imaginary part of
$r\in\mathcal{R}_{a,b}.$ Indeed, once we calculate the
$\max_{r\in\mathcal{R}_{a,b}}\mathop{\mathrm{Im}}(r),$ we can conclude that
all $r^{\prime}\in\mathbb{C},$ such that
$\mathop{\mathrm{Im}}(r^{\prime})>\max_{r\in\mathcal{R}_{a,b}}\mathop{\mathrm{Im}}(r),$
belong to the unbounded component of $\mathbb{C}\setminus\mathcal{R}_{a,b},$
namely $U_{a,b}.$ The following discussion results in gathering the required
$2$-tuples $(a,b)$ such that

$\mathrm{m}_{a,b}(Q_{r_{0}})=\mathrm{m}(Q_{r_{0}})$

for a fixed $r^{\prime}=r_{0}.$

Recall that, any element in $\mathcal{R}_{a,b}$ can be written as

$r=\left(a+a^{-1}\right)\cos\alpha+\left(b+b^{-1}\right)\cos\beta+i\left[\left(a-a^{-1}\right)\sin\alpha+\left(b-b^{-1}\right)\sin\beta\right],$

where $\alpha,\beta\in[-\pi,\pi).$ Notice that,

$\left|\mathop{\mathrm{Im}}(r)\right|=\left|\left(a-a^{-1}\right)\sin\alpha+\left(b-b^{-1}\right)\sin\beta\right|\leq\left|a-a^{-1}\right|+\left|b-b^{-1}\right|,$

and, for $\alpha=\beta\in\\{-\frac{\pi}{2},\frac{\pi}{2}\\},$ we have

$r_{\max,i\mathbb{R}}=i\left[\left|a-a^{-1}\right|+\left|b-b^{-1}\right|\right].$

Therefore, when $a$ and $b$ are fixed, we have
$\mathrm{m}_{a,b}(Q_{r})=\mathrm{m}(Q_{r})$ for all $r\in\left\\{z\in
i\mathbb{R}:|z|>\left|a-a^{-1}\right|+\left|b-b^{-1}\right|\right\\}.$ Then a
similar argument as in the real case shows that, for a fixed $r_{0}\in
i\mathbb{R}_{>0},$ the Mahler measure of $Q_{r_{0}}$ over the integration
torus $\mathbb{T}_{a,b}^{2}$ is same as the standard Mahler measure, i.e.

$\mathrm{m}_{a,b}(Q_{r_{0}})=\mathrm{m}(Q_{r_{0}}),$

for all the $2$-tuples $(a,b)$ satisfying

$\left|a-a^{-1}\right|+\left|b-b^{-1}\right|<|r_{0}|.$

Here we mention two such examples for $r=8$ and $r=2i.$

######  Example 5.1 (r = 8).

We provide two cases: $(I)$ when $b=a,$ and $(II)$ when $b=\sqrt{a}.$ Notice
that, case $(I)$ keeps the symmetry of the polynomial

$Q_{8}(x,y)=x+\frac{1}{x}+y+\frac{1}{y}+8$

in the variables $x$ and $y.$ In other words, under the change of variables
$x\mapsto y$ and $y\mapsto x,$ the polynomial $Q_{8}$ remains unchanged, and
so does the integration torus $\mathbb{T}_{a,a}^{2}$. On the other hand, case
$(II)$ breaks the symmetry as then the above changes of variables change the
integration torus from $\mathbb{T}_{a,\sqrt{a}}^{2}$ to
$\mathbb{T}_{\sqrt{a},a}^{2}.$ In spite of the differences between these two
cases, there are certain values of $a$ such that

$\mathrm{m}_{a,\sqrt{a}}(Q_{8})=\mathrm{m}_{a,a}(Q_{8})=\mathrm{m}(Q_{8})=4L^{\prime}(E_{24},0),$

where $E_{24}$ is an elliptic curve of conductor $24$ associated to $Q_{8}.$
Here the last equality follows from combining the results due to Rogers and
Zudilin [8], and Lalín and Rogers [9], where they showed

$\mathrm{m}(Q_{8}(x,y))=\mathrm{m}(Q_{2}(x,y))=L^{\prime}(E_{24},0).$

From the above discussion, we find that, when $(I)$ $a=b,$ the equality

$\mathrm{m}_{a,a}(Q_{8})=\mathrm{m}(Q_{8})$

holds for all $a$ satisfying

$a+\frac{1}{a}<4\Longleftrightarrow 2-\sqrt{3}<a<2+\sqrt{3}.$

Similarly, when $(II)$ $b=\sqrt{a},$ we find that, for

$\displaystyle a+\frac{1}{a}+\sqrt{a}+\frac{1}{\sqrt{a}}<8$
$\displaystyle\Longleftrightarrow$
$\displaystyle\frac{17-\sqrt{41}-\sqrt{2\left(157-17\sqrt{41}\right)}}{4}<a<\frac{17-\sqrt{41}+\sqrt{2\left(157-17\sqrt{41}\right)}}{4},$

the equality

$\mathrm{m}_{a,\sqrt{a}}(Q_{8})=\mathrm{m}(Q_{8})$

holds. Since,

$\frac{17-\sqrt{41}+\sqrt{2\left(157-17\sqrt{41}\right)}}{4}>2+\sqrt{3}$

and

$\frac{17-\sqrt{41}-\sqrt{2\left(157-17\sqrt{41}\right)}}{4}=\left[\frac{17-\sqrt{41}+\sqrt{2\left(157-17\sqrt{41}\right)}}{4}\right]^{-1},$

we obtain

$\mathrm{m}_{a,\sqrt{a}}(Q_{8})=\mathrm{m}_{a,a}(Q_{8})=\mathrm{m}(Q_{8})=4L^{\prime}(E_{24},0)\quad\quad\mbox{for
all}\ a\in\left(2-\sqrt{3},2+\sqrt{3}\right).$

######  Example 5.2 (r = 2$i$).

In 2011, Mellit [21] showed that

$\mathrm{m}(Q_{2i})=L^{\prime}(E_{40},0),$

where $E_{40}$ is an elliptic curve of conductor $40,$ associated to $Q_{2i}.$

When $b=a,$ Theorem 1.2 implies that
$\mathrm{m}_{a,a}(Q_{2i})=\mathrm{m}(Q_{2i})$ is true for

$\left|a-a^{-1}\right|<1\Longleftrightarrow\frac{\sqrt{5}-1}{2}<a<\frac{\sqrt{5}+1}{2}.$

Similarly, when $b=\sqrt{a},$ the equality

$\mathrm{m}_{a,\sqrt{a}}(Q_{2i})=\mathrm{m}(Q_{2i})$

holds for all $a$ satisfying

$\left|a-\frac{1}{a}\right|+\left|\sqrt{a}-\frac{1}{\sqrt{a}}\right|<2\Leftrightarrow
a_{0}<a<a_{1},$

where $a_{0}\approx 0.530365\dots$ and $a_{1}\approx 1.88549\dots$ satisfy
$X-1/X+\sqrt{X}-1/\sqrt{X}-2=0$ and $X-1/X+\sqrt{X}-1/\sqrt{X}+2=0,$
respectively. Since $a_{0}<\frac{\sqrt{5}-1}{2}$ and
$a_{1}>\frac{\sqrt{5}+1}{2},$ we obtain

$\mathrm{m}_{a,a}(Q_{2i})=\mathrm{m}_{a,\sqrt{a}}(Q_{2i})=\mathrm{m}(Q_{2i})=L^{\prime}(E_{40},0)$

for all $a\in\left(\frac{\sqrt{5}-1}{2},\frac{\sqrt{5}+1}{2}\right).$

###  5.2. Generalized Mahler measure on bounded component(s) of
$\mathbb{C}\setminus\mathcal{R}_{a,b}$

In this section, our goal is to evaluate $\mathrm{m}_{a,b}(Q_{r})$ when $r$
belongs to the bounded connected component(s) of
$\mathbb{C}\setminus\mathcal{R}_{a,b}.$ In particular, we show there can be at
most one such component of $\mathbb{C}\setminus\mathcal{R}_{a,b}.$ Later, we
apply Theorem 1.3 to calculate $\mathrm{m}_{a,b}(Q_{r})$ for all $r$ in said
component.

####  5.2.1. Existence of at most one bounded open connected component of
$\mathbb{C}\setminus\mathcal{R}_{a,b}$

Our aim here is to show that there exists at most one bounded open connected
component of $\mathbb{C}\setminus\mathcal{R}_{a,b}$ for any $a,b>0.$

Recall that the elements of $\mathcal{R}_{a,b}$ are of the form

(32)
$r=\left(a+a^{-1}\right)\cos\alpha+\left(b+b^{-1}\right)\cos\beta+i\left[\left(a-a^{-1}\right)\sin\alpha+\left(b-b^{-1}\right)\sin\beta\right],$

where $\alpha,\beta\in[-\pi,\pi).$ We have

$R_{a,b}=\max_{r\in\mathcal{R}_{a,b}}|r|=a+a^{-1}+b+b^{-1},\ \mbox{and}\
r_{a,b}:=\min_{r\in\mathcal{R}_{a,b}}|r|=\left(a+a^{-1}\right)-\left(b+b^{-1}\right).$

Then

$a=b\Longleftrightarrow r_{a,b}=0\Longleftrightarrow 0\in\mathcal{R}_{a,b}.$

From this point onwards we assume that $a\geq b\geq 1.$ The other cases follow
analogously using Lemma 3.1.

######  Proposition 5.3.

There exists at most one bounded open connected component of
$\mathbb{C}\setminus\mathcal{R}_{a,b},$ and if it exists, then it contains
$0.$

Note that, if $0\in\mathcal{R}_{a,b},$ then Proposition 5.3 implies that there
is no bounded open connected component in
$\mathbb{C}\setminus\mathcal{R}_{a,b}.$ Before we proceed to prove the
proposition, we note some useful properties of $\mathcal{R}_{a,b}.$

  1. (bxsc)

The points $r\in\mathcal{R}_{a,b}$ can be interpreted as points on the
ellipses

(33)
$E_{b,z}:\left|r-(z+2)\right|+\left|r-(z-2)\right|=2\left(b+b^{-1}\right),$

where $z\in\mathbb{C}$ lies on the ellipse

(34) $e_{a}:\left|z-2\right|+\left|z+2\right|=2\left(a+a^{-1}\right).$

In other words, elements of $\mathcal{R}_{a,b}$ can be identified with points
on ellipses $E_{b,z}$ defined by (33) with centres on the ellipse $e_{a}$ in
(34). Note that, the centre ($c$) and the foci ($c_{1}$ and $c_{2}$) of the
ellipse $e_{a}$ are the points $c=0,c_{1}=-2$ and $c_{2}=2.$ The centre
($C_{z}$) and the foci ($C_{1,z}$ and $C_{2,z}$) of the ellipse $E_{b,z}$ are

$C_{z}=z,C_{1,z}=z-2,\ \mbox{and}\ C_{2,z}=z+2.$

Any point $p\in\mathbb{C}$ lying inside (resp. outside) the ellipse $E_{b,z}$
satisfies
$\left|p-C_{1,z}\right|+\left|p-C_{2,z}\right|<2\left(b+b^{-1}\right)$ (resp.
$\left|p-C_{1,z}\right|+\left|p-C_{2,z}\right|>2\left(b+b^{-1}\right)$). Since
the length of the minor axis of $e_{a}$ is $2\left(a-a^{-1}\right),$ we derive
that, for $z\in e_{a},$

$\left|\mathop{\mathrm{Im}}\left(C_{z}\right)\right|\leq\left(a-a^{-1}\right),$

and, for all $h\in\left(-a+a^{-1},a-a^{-1}\right),$ there exists a
$\tilde{z}\in e_{a}$ such that

(35) $\mathop{\mathrm{Im}}\left(C_{\tilde{z}}\right)=h.$

  2. (bxsc)

The region $\mathcal{R}_{a,b}$ is symmetric with respect to the imaginary and
real axes, i.e. if $r\in\mathcal{R}_{a,b},$ then
$-\bar{r},\bar{r}\in\mathcal{R}_{a,b}.$ Let $Q_{\pm,\pm}$ denote the four
quadrants of $\mathbb{C},$ namely $Q_{+,+}=\\{s\in
C:\mathop{\mathrm{Re}}(s)\geq 0,\mathop{\mathrm{Im}}(s)\geq
0\\},Q_{+,-}=\\{s\in C:\mathop{\mathrm{Re}}(s)\geq
0,\mathop{\mathrm{Im}}(s)\leq 0\\}$ and so on. Then, the changes of variable
$\\{\alpha\mapsto\pi-\alpha,\beta\mapsto\pi-\beta\\}$ and
$\\{\alpha\mapsto-\alpha,\beta\mapsto-\beta\\}$ applied to (32) take points
$r\in\mathcal{R}_{a,b}\cap Q_{+,\pm}$ to $-\bar{r}\in\mathcal{R}_{a,b}\cap
Q_{-,\pm},$ and $r\in\mathcal{R}_{a,b}\cap Q_{\pm,+}$ to
$\bar{r}\in\mathcal{R}_{a,b}\cap Q_{\pm,-}.$

For the rest of this section, $\mathcal{R}_{a,b}^{c}$ denotes the region
defined by $\mathbb{C}\setminus\mathcal{R}_{a,b}.$

###### Proof of Proposition 5.3.

Recall that $U_{a,b}$ denotes the connected unbounded component of
$\mathcal{R}_{a,b}^{c}.$ First we consider the case $a>b.$

From the discussion at the beginning of this section, we have
$0\in\mathcal{R}_{a,b}^{c},$ and moreover the open disc
$\\{u\in\mathbb{C}:|u|<r_{a,b}\\}$ is contained in one of the bounded open
connected components of $\mathcal{R}_{a,b}^{c}.$ Let $V_{a,b}$ denote this
component. Note that $0\in V_{a,b}.$

In order to prove the statement, we note that the property (B) above implies
that we can restrict ourselves to the quadrant $Q_{+,+}.$

Since $a>b,$ the ellipse $E_{b,z_{0}}$ lies completely in the interior of
$Q_{+,-},$ where $z_{0}=-i\left(a-a^{-1}\right).$ This implies that, for $P\in
Q_{+,+},$ we have
$\left|P-C_{1,z_{0}}\right|+\left|P-C_{2,z_{0}}\right|>2\left(b+b^{-1}\right),$
where $C_{1,z_{0}}$ and $C_{2,z_{0}}$ are the foci of $E_{b,z_{0}}.$ In other
words, the point $P$ lies completely outside the ellipse $E_{b,z_{0}}$ with
centre at $z_{0}=-i\left(a-a^{-1}\right)\in e_{a}.$ Let $z_{1}$ and $z_{2}$
denote the points $a+a^{-1}$ and $i\left(a-a^{-1}\right),$ respectively. We
now consider the following cases:

  1. (scsc)

$\mathop{\mathrm{Im}}(P)>\mathop{\mathrm{Im}}(z_{2})=\left(a-a^{-1}\right),$

  2. (scsc)

$0\leq\mathop{\mathrm{Im}}(P)\leq\mathop{\mathrm{Im}}(z_{2})=\left(a-a^{-1}\right).$

Let $E^{a,b}$ and $e^{a,b}$ denote the boundaries of $U_{a,b}$ and $V_{a,b}$
respectively, i.e.

$E^{a,b}=\overline{U}_{a,b}\setminus U_{a,b},\quad\mbox{and}\quad
e^{a,b}=\overline{V}_{a,b}\setminus V_{a,b}.$

Therefore, $E^{a,b}\cup e^{a,b}\subset\mathcal{R}_{a,b}.$ If $P\notin
U_{a,b}\cup V_{a,b},$ then $P$ lies in the region bounded by $E^{a,b}$ and
$e^{a,b}.$ Then the line
$L_{\mathop{\mathrm{Re}}(P)}:t=\mathop{\mathrm{Re}}(P)$ intersects $E^{a,b}$
at some point $P^{\prime}\in Q_{+,+}.$ By construction,
$\mathop{\mathrm{Re}}(P)=\mathop{\mathrm{Re}}(P^{\prime}),$ and
$\mathop{\mathrm{Im}}(P)\leq\mathop{\mathrm{Im}}(P^{\prime}),$ where the
equality holds iff $P=P^{\prime}.$ Note that $P=P^{\prime}$ is the trivial
case. Therefore, we assume that $P\neq P^{\prime}.$

Figure 2.
$\mathcal{R}_{a,b},U_{a,b},V_{a,b},e_{a},C_{j,z},P,P^{\prime},L_{\mathop{\mathrm{Re}}(P)}$
when $a>b$ and (I) holds.

Claim: For $P\in Q_{+,+},$ if (I) holds, and $P\notin V_{a,b}\cup U_{a,b},$
then there exists an ellipse $E_{b,z^{\prime}},$ with centre at $z^{\prime}\in
e_{a},$ such that

$\left|P-C_{1,z^{\prime}}\right|+\left|P-C_{2,z^{\prime}}\right|<2\left(b+b^{-1}\right).$

Since $\mathop{\mathrm{Im}}(C_{j,z})\leq a-a^{-1}$ for all $z\in e_{a}$ and
$j=1,2,$ case (I) implies that
$|P^{\prime}-C_{j,z}|^{2}-|P-C_{j,z}|^{2}>|P^{\prime}-P|^{2}>0$ (see Figure
2), and then we have

$\left|P-C_{1,z}\right|+\left|P-C_{2,z}\right|<\left|P^{\prime}-C_{1,z}\right|+\left|P^{\prime}-C_{2,z}\right|.$

On the other hand, $P^{\prime}\in E_{a,b}\subset\mathcal{R}_{a,b},$ i.e. there
exists a $z^{\prime}\in e_{a}$ such that
$\left|P^{\prime}-C_{1,z^{\prime}}\right|+\left|P^{\prime}-C_{2,z^{\prime}}\right|=2\left(b+b^{-1}\right).$
This concludes the proof of the claim.

Now note that $z\in e_{a}$ can also be written as
$z=\left(a+a^{-1}\right)\cos\alpha_{z}+i\left(a-a^{-1}\right)\sin\alpha_{z},$
for $\alpha_{z}\in[-\pi,\pi).$ Recall that $C_{1,z}=z-2$ and $C_{2,z}=z+2.$
Therefore, when $a$ is fixed, $\left|P-C_{j,z}\right|$ is a continuous
function of $\alpha_{z}$ for $j=1,2.$ Let

$\Theta(\alpha_{z}):=\left|P-C_{1,z}\right|+\left|P-C_{2,z}\right|-2\left(b+b^{-1}\right)$

define a function from $[-\pi,\pi)$ to $\mathbb{R}.$ From the claim, we have
already concluded that, for the case (I), either $P\in U_{a,b}\cup V_{a,b},$
or there are $z_{0},z^{\prime}\in e_{b}$ such that

$\left|P-C_{1,z_{0}}\right|+\left|P-C_{2,z_{0}}\right|>2\left(b+b^{-1}\right),\quad\mbox{and}\
\left|P-C_{1,z^{\prime}}\right|+\left|P-C_{2,z^{\prime}}\right|<2\left(b+b^{-1}\right).$

This implies that $\Theta$ is a continuous function which takes both negative
and positive values, and, using Mean Value Theorem (MVT) on $\Theta,$ we
derive that there exists $z_{1}\in e_{a}$ such that
$\Theta(\alpha_{z_{1}})=0.$ In other words, $P\in\mathcal{R}_{a,b},$ which
completes the proof of the proposition for $a>b$ when (I) holds.

For case (II), note that

$0\leq\mathop{\mathrm{Im}}(P),\mathop{\mathrm{Im}}(C_{z})\leq
a-a^{-1},\quad\quad\mbox{for all}\ z\in e_{a},$

where $C_{z}$ $(=z)$ is the centre of $E_{b,z}$ given in property (A). Then,
the continuous property of $\mathop{\mathrm{Im}}\left(C_{z}\right)$ mentioned
in (35) implies that there exists a $z^{\prime\prime}\in e_{a}$ such that
$\mathop{\mathrm{Im}}(P)=\mathop{\mathrm{Im}}(C_{z^{\prime\prime}})$ (see
Figure 3). Let $L_{\mathop{\mathrm{Im}}(P)}:t=\mathop{\mathrm{Im}}(P)$ denote
the line joining $P$ and $C_{z^{\prime\prime}},$ and let
$L_{\mathop{\mathrm{Im}}(P)}$ intersect $E^{a,b}$ at $P_{1}$ in $Q_{+,+}$ such
that any $t\in L_{\mathop{\mathrm{Im}}(P)}$ satisfying
$\mathop{\mathrm{Re}}(t)>\mathop{\mathrm{Re}}(P_{1})$ lies in $U_{a,b}.$ Then
$P_{1}$ has a representation as in (32), namely

$P_{1}=\left(a+a^{-1}\right)\cos\alpha+\left(b+b^{-1}\right)\cos\beta+i\left[\left(a-a^{-1}\right)\sin\alpha+\left(b-b^{-1}\right)\sin\beta\right],$

for some $\alpha,\beta\in[0,\pi/2).$ Moreover, there exist
$\gamma\in[0,\pi/2)$ such that
$\mathop{\mathrm{Im}}(P)=\mathop{\mathrm{Im}}(P^{\prime})=\mathop{\mathrm{Im}}(C_{z^{\prime\prime}})=\left(a-a^{-1}\right)\sin\gamma.$
The case $\mathop{\mathrm{Re}}(P)=\mathop{\mathrm{Re}}(P_{1})$ is trivial as
the above discussion implies that $P=P_{1}.$ If
$\mathop{\mathrm{Re}}(P)>\mathop{\mathrm{Re}}(P_{1}),$ then from the
definition of $P_{1}$ it follows that $P\in U_{a,b}.$ Therefore, it only
remains to investigate the case when
$0\leq\mathop{\mathrm{Re}}(P)<\mathop{\mathrm{Re}}(P_{1}).$

(a) (b)

Figure 3.
$\mathcal{R}_{a,b},U_{a,b},V_{a,b},e_{a},C_{z},P,P_{1},L_{\mathop{\mathrm{Im}}(P)}$
when $a>b$ and (II) holds; left (resp. right) figure shows the case when
$L_{\mathop{\mathrm{Im}}(P)}\cap e^{a,b}\neq\varnothing$ (resp.
$L_{\mathop{\mathrm{Im}}(P)}\cap e^{a,b}=\varnothing$).

Define the function
$f:\left[\alpha,\gamma\right]\rightarrow\left[0,\beta\right],$ which sends
$\psi$ to $f(\psi)$ such that

$\left(a-a^{-1}\right)\sin\gamma=\left(a-a^{-1}\right)\sin\alpha+\left(b-b^{-1}\right)\sin\beta=\left(a-a^{-1}\right)\sin\psi+\left(b-b^{-1}\right)\sin
f(\psi).$

Note that this is a continuous onto function from a connected set. Then the
graph of this function, namely
$\Gamma_{f}:=\\{(\psi,f(\psi)):\psi\in\left[\alpha,\gamma\right]\\},$ is also
a connected set. Consider another function
$g:\Gamma_{f}\rightarrow\mathbb{R},$ defined by

$(\psi,f(\psi))\mapsto\left(a+a^{-1}\right)\cos\psi+\left(b+b^{-1}\right)\cos
f(\psi).$

Note that this function is a continuous function, and there exist
$\chi,\xi\in\Gamma_{f}$ such that
$g(\chi)=\mathop{\mathrm{Re}}(P_{1})=\left(a+a^{-1}\right)\cos\alpha+\left(b+b^{-1}\right)\cos\beta,$
and
$g(\xi)=\mathop{\mathrm{Re}}\left(C_{z^{\prime\prime}}\right)=\left(a+a^{-1}\right)\cos\gamma.$

Now if
$\mathop{\mathrm{Re}}(P)\in\left(\mathop{\mathrm{Re}}\left(C_{z^{\prime\prime}}\right),\mathop{\mathrm{Re}}(P_{1})\right),$
we claim that there exists a $\psi_{0}\in\left[\alpha,\gamma\right]$ such that

$P=\left(a+a^{-1}\right)\cos\psi_{0}+\left(b+b^{-1}\right)\cos
f(\psi_{0})+i\left[\left(a-a^{-1}\right)\sin\psi_{0}+\left(b-b^{-1}\right)\sin
f(\psi_{0})\right],$

which will imply that $P\in\mathcal{R}_{a,b}.$ Indeed, $g$ is a continuous
function on a connected set, and $g(\xi)<g(\chi).$ Therefore, all the values
of the interval $(g(\xi),g(\chi))$ are attained by $g.$ In particular, such
$\psi_{0}$ exists. This proves the statement of the proposition when
$\mathop{\mathrm{Im}}(P)\in\left[0,a-a^{-1}\right]$ and
$\mathop{\mathrm{Re}}(P)\geq\mathop{\mathrm{Re}}\left(C_{z^{\prime\prime}}\right).$

Therefore, it remains to consider the case when (II) holds along with
$\mathop{\mathrm{Re}}(P)\in\left[0,\mathop{\mathrm{Re}}\left(C_{z^{\prime\prime}}\right)\right).$

If the line $L_{\mathop{\mathrm{Im}}(P)}$ intersect $e^{a,b}$ in $Q_{+,+},$
then we consider the intersection point with the smallest non-negative real
part. In other words, if $K_{1},\dots,K_{l}\in Q_{+,+}\cap e^{a,b}\cap
L_{\mathop{\mathrm{Im}}(P)}$ are distinct with
$0\leq\mathop{\mathrm{Re}}(K_{1})<\dots<\mathop{\mathrm{Re}}(K_{l}),$ then
consider $\mathop{\mathrm{Re}}(K_{1}).$ We want to show that, in fact there
can be at most one intersection point of $L_{\mathop{\mathrm{Im}}(P)}$ and
$e^{a,b}$ in $Q_{+,+}.$ That is, if
$\mathop{\mathrm{Re}}(P)\in(\mathop{\mathrm{Re}}(K_{1}),\mathop{\mathrm{Re}}(C_{z^{\prime\prime}})),$
then $P\in\mathcal{R}_{a,b};$ but this follows from a similar argument as
above. Therefore, it only remains to investigate the case when
$L_{\mathop{\mathrm{Im}}(P)}$ does not intersect $e^{a,b}$ in $Q_{+,+}.$ Let
$K_{0}$ be the intersection point of $L_{\mathop{\mathrm{Im}}(P)}$ and the
imaginary axis. Then we have $\mathop{\mathrm{Re}}(K_{0})=0,$ and
$\mathop{\mathrm{Re}}(P)\in(\mathop{\mathrm{Re}}(K_{0}),\mathop{\mathrm{Re}}(C_{z^{\prime\prime}}))\subset
L_{\mathop{\mathrm{Im}}(P)}.$ Again, an analogous argument as above implies
that $P\in\mathcal{R}_{a,b}.$

We collect all the results above, and then using the symmetry of
$\mathcal{R}_{a,b}$ (see property (B)) we conclude that if $P\notin
U_{a,b}\cup V_{a,b},$ then $P\in\mathcal{R}_{a,b},$ which completes the proof
of the proposition for $a>b.$ The case $a=b$ follows from a similar argument.
∎

####  5.2.2. Application of Theorem 1.3 to the bounded component

Now we are ready to apply Theorem 1.3 to $V_{a,b},$ and evaluate
$\mathrm{m}_{a,b}(Q_{r}).$ Firstly, we need to investigate the roots of
$xQ_{r_{0}}(x,b).$ Since, $0\in V_{a,b},$ we can choose $r_{0}=0$ in our
theorem. In particular, we need to count the number of roots of $xQ_{0}(x,b)$
lying inside the circle $|x|=a.$ By Lemma 3.1, we can also assume $a>b>1.$

Factoring $xQ_{0}(x,b)$ in $\mathbb{C}[x],$ we obtain that

$xQ_{0}(x,b)=x^{2}+\left(b+\frac{1}{b}\right)x+1=\left(x+b\right)\left(x+\frac{1}{b}\right).$

Since $a>b>1,$ both roots of $xQ_{0}(x,b))$ lies inside the circle $|x|=a.$
Also note that $Q_{F,0}^{x}(y)$ and $Q_{f,0}^{x}(y)$ in (26) are equal to the
constant function $\textbf{1}.$ Applying Theorem 1.3, we have, for $a>b>1$ and
$r\in V_{a,b},$

$\mathrm{m}_{a,b}(Q_{r})=\nu_{a,b,0}^{1}\log a=\log a,$

where the last equality follows from the fact that

$\nu_{a,b,0}^{1}=Z_{a,b,0}^{1}-P_{a,b,0}^{1}=2-1=1.$

Other cases, such as $b>a>1,$ $a>1>b$ etc, follow from a combination Lemma 3.1
and a similar arguments as above.

###  5.3. Explicit description of $\mathcal{R}_{a,b}$

In this section, our main goal is to describe the region $\mathcal{R}_{a,b}$
explicitly when $a$ and $b$ satisfy some additional conditions. Since $a,b>0,$
let $x=\log a$ and $y=\log b.$ For the rest of this section we consider
$(a,b)$ as exponentials $(e^{x},e^{y}).$

Any $r\in\mathcal{R}_{a,b}$ in (32) can be expressed in terms of of hyperbolic
functions as

$r=2\left[\cosh x\cos\alpha+\cosh y\cos\beta+i\left(\sinh x\sin\alpha+\sinh
y\sin\beta\right)\right]=2\cosh(x+i\alpha)+2\cosh(y+i\beta),$

with $\alpha,\beta\in[-\pi,\pi).$. When considered as a subset of
$\mathbb{R}^{2},$ the points of $\mathcal{R}_{a,b}$ admit a geometric
interpretation: each
$r=\left(\mathop{\mathrm{Re}}(r),\mathop{\mathrm{Im}}(r)\right)\in\mathcal{R}_{a,b}$
corresponds to a point $(u_{r},v_{r})$ on the ellipse

$\frac{\left(u_{r}-s\right)^{2}}{\cosh^{2}y}+\frac{\left(v_{r}-t\right)^{2}}{\sinh^{2}y}=4,$

with

$\frac{s^{2}}{\cosh^{2}x}+\frac{t^{2}}{\sinh^{2}y}=4.$

Though the above geometric interpretation is not very informative, the
boundaries of $\mathcal{R}_{a,b}$ can be expressed explicitly when $x$ and $y$
satisfy certain conditions. This leads to ways of explicitly describing
$O_{a,b}$($=U_{a,b}$) and $V_{a,b}$ in those cases.

######  Lemma 5.4.

Let $a,b>0$ such that $a\neq b,$ and recall that $x:=\log a$ and $y:=\log b.$
If

(36) $\sinh^{2}\left(\frac{x+y}{2}\right)\left[1+\cosh x\cosh
y\right]\geq\max\\{\sinh^{2}x,\sinh^{2}y\\},$

then the outer boundary of $\mathcal{R}_{a,b}$ is given by

$\frac{u^{2}}{\left(\cosh x+\cosh y\right)^{2}}+\frac{v^{2}}{\left(\sinh
x+\sinh y\right)^{2}}=4.$

Figure 4. $\mathcal{R}_{a,b},$ when $a=1.5$ and $b=1.07$ does not satisfy
(36), (38) and (39).

This lemma implies that, when the condition (36) holds, any
$r=(\mathop{\mathrm{Re}}(r),\mathop{\mathrm{Im}}(r))\in\mathcal{R}_{a,b}$
satisfies

(37) $\frac{(\mathop{\mathrm{Re}}(r))^{2}}{\left(\cosh x+\cosh
y\right)^{2}}+\frac{(\mathop{\mathrm{Im}}(r))^{2}}{\left(\sinh x+\sinh
y\right)^{2}}\leq 4.$

The region $O_{a,b}$ is then defined by

$\left(\frac{\mathop{\mathrm{Re}}(r)}{a+a^{-1}+b+b^{-1}}\right)^{2}+\left(\frac{\mathop{\mathrm{Im}}(r)}{a-a^{-1}+b-b^{-1}}\right)^{2}>1.$

The following lemma expresses the innermost boundary of $\mathcal{R}_{a,b}$
explicitly, subject to some conditions on $x$ and $y.$

######  Lemma 5.5.

Let $a,b>0$ such that $a\neq b,$ and define $x:=\log a$ and $y:=\log b.$ If

(38) $\min\\{\left|\tanh y\cosh x\right|,\left|\tanh x\cosh y\right|\\}>1,$

and

(39) $\cosh^{2}\left(\frac{x+y}{2}\right)\left[\cosh x\cosh
y-1\right]\geq\max\\{\sinh^{2}x,\sinh^{2}y\\},$

then the inner boundary of $\mathcal{R}_{a,b}$ is

$\frac{s^{2}}{\left(\cosh x-\cosh y\right)^{2}}+\frac{t^{2}}{\left(\sinh
x-\sinh y\right)^{2}}=4.$

Figure 5. $\mathcal{R}_{a,b},$ when $a=10$ and $b=4$ satisfy (36), (38) and
(39).

From Lemma 5.5, we deduce that, when the above conditions are satisfied, we
have $r=(\mathop{\mathrm{Re}}(r),\mathop{\mathrm{Im}}(r))\in\mathcal{R}_{a,b}$
if and only if

(40) $\frac{(\mathop{\mathrm{Re}}(r))^{2}}{\left(\cosh x-\cosh
y\right)^{2}}+\frac{(\mathop{\mathrm{Im}}(r))^{2}}{\left(\sinh x-\sinh
y\right)^{2}}\geq 4.$

The proofs of the above lemmas are included in the Appendix for completion.

In conclusion, when all the conditions in the statements of Lemmas 5.4 and 5.5
are satisfied, the region $\mathcal{R}_{a,b}$ has a concrete description:
$r=(\mathop{\mathrm{Re}}(r),\mathop{\mathrm{Im}}(r))\in\mathcal{R}_{a,b}$ if
and only if

$\left(\frac{\mathop{\mathrm{Re}}(r)}{a+a^{-1}+b+b^{-1}}\right)^{2}+\left(\frac{\mathop{\mathrm{Im}}(r)}{a-a^{-1}+b-b^{-1}}\right)^{2}\leq
1$

and

$\left(\frac{\mathop{\mathrm{Re}}(r)}{a+a^{-1}-b-b^{-1}}\right)^{2}+\left(\frac{\mathop{\mathrm{Im}}(r)}{a-a^{-1}-b+b^{-1}}\right)^{2}\geq
1.$

##  6\. Generalized Mahler measure of $X+\frac{1}{X}+Y+\frac{1}{Y}+4$

In this section, our goal is to provide a proof of Theorem 1.4, and evaluate

$\mathrm{m}_{a,b}(Q_{4}):=\mathrm{m}_{a,b}(Q_{4}(x,y))=\mathrm{m}_{a,b}\left(x+\frac{1}{x}+y+\frac{1}{y}+4\right)$

for all $a,b>0.$

Our method of proof is mostly inspired from the proof of Theorem $12$ in [22].
We start by factoring $Q_{4}(x,y)$ into linear factors following the changes
of variables considered by Boyd (see Section $2$A in [5]). The change of
variables

$x\mapsto\frac{w}{z}\quad\mbox{and}\quad y\mapsto wz$

applied to $Q_{4}(x,y)$ yields that

(41)
$P\left(w,z\right)=Q_{4}\left(\frac{w}{z},wz\right)=\frac{1}{wz}\left(1+iw+iz+wz\right)\left(1-iw-
iz+wz\right).$

Since
$\mathrm{m}_{a,b}(S(x,y)T(x,y))=\mathrm{m}_{a,b}(S(x,y))+\mathrm{m}_{a,b}(T(x,y)),$
it is sufficient to evaluate the Mahler measures of the linear polynomials
$\left(1\pm iw\pm iz+wz\right)$ over
$\mathbb{T}_{c,d}^{2}=\\{(w,z)\in\mathbb{C}^{*}\times\mathbb{C}^{*}:|w|=c,|z|=d\\},$
where

$c=\sqrt{ab},\ d=\sqrt{\frac{b}{a}}.$

Afterwards, using the changes of variables, we can evaluate
$\mathrm{m}_{a,b}(Q_{4}).$ The changes of variables

$w\mapsto-w\quad\mbox{and}\quad z\mapsto-z$

transform $\left(1+iw+iz+wz\right)$ to $\left(1-iw-iz+wz\right).$ As these
changes of variables preserve the Mahler measure, we find that

$\displaystyle\mathrm{m}_{a,b}(Q_{4})=\mathrm{m}_{c,d}(P(w,z))=$
$\displaystyle\mathrm{m}_{c,d}\left(\frac{1}{wz}\right)+\mathrm{m}_{c,d}\left(1+iw+iz+wz\right)+\mathrm{m}_{c,d}\left(1-iw-
iz+wz\right)$ $\displaystyle=$ $\displaystyle-\log
cd+2\mathrm{m}_{c,d}\left(1+iw+iz+wz\right)$ (42) $\displaystyle=$
$\displaystyle-\log b+2\mathrm{m}_{c,d}\left(1+iw+iz+wz\right),$

where the last equality follows from the fact that
$cd=\sqrt{ab}\cdot\sqrt{\frac{b}{a}}=b.$

Among the terms in (42), it remains to evaluate

$\frac{1}{2}\left(\mathrm{m}_{c,d}(P)+\log
cd\right)=\frac{1}{2}\left(\mathrm{m}_{a,b}(Q_{4})+\log
b\right)=\mathrm{m}_{c,d}(1+iw+iz+wz).$

Note that $z(w)=-\frac{1+iw}{i+w}$ is the only root of $R(w,z)=1+iw+iz+wz,$
when considered as a polynomial in $z.$ Therefore,

$\displaystyle\mathrm{m}_{c,d}(R(w,z))$
$\displaystyle=\mathrm{m}_{c,d}(w+i)+\mathrm{m}_{c,d}\left(z+\frac{1+iw}{i+w}\right)$
$\displaystyle=\frac{1}{(2\pi
i)^{2}}\int_{\mathbb{T}_{c,d}^{2}}\log|w+i|\frac{dw}{w}\frac{dz}{z}+\mathrm{m}_{c,d}\left(z+\frac{1+iw}{i+w}\right)$
(43) $\displaystyle=\frac{1}{2\pi
i}\int_{|w|=c}\log|w+i|\frac{dw}{w}+\mathrm{m}_{c,d}\left(z+\frac{1+iw}{i+w}\right).$

To evaluate the first integral, we apply the change of variables
$w=cw^{\prime}$ and Jensen’s formula (see (2.1)) to obtain

(46) $\displaystyle\frac{1}{2\pi i}\int_{|w|=c}\log|w+i|\frac{dw}{w}=\log
c+\frac{1}{2\pi
i}\int_{|w^{\prime}|=1}\log\left|w^{\prime}+\frac{i}{c}\right|\frac{dw^{\prime}}{w^{\prime}}=$
$\displaystyle\left\\{\begin{array}[]{ll}\log c&\mbox{if}\ c>1,\\\
0&\mbox{if}\ c\leq 1.\end{array}\right.$

It now suffices to evaluate

(47) $\displaystyle\mathrm{m}_{c,d}\left(z+\frac{1+iw}{i+w}\right)$
$\displaystyle=\frac{1}{(2\pi
i)^{2}}\int_{\mathbb{T}_{c,d}^{2}}\log\left|z+\frac{1+iw}{i+w}\right|\frac{dw}{w}\frac{dz}{z}$
$\displaystyle=\frac{1}{2\pi i}\int_{|w|=c}\left(\frac{1}{2\pi
i}\int_{|z|=d}\log\left|z+\frac{1+iw}{i+w}\right|\frac{dz}{z}\right)\frac{dw}{w}$

to complete the proof. Note that $\frac{1}{2\pi
i}\int_{|z|=d}\log\left|z+\frac{1+iw}{i+w}\right|\frac{dz}{z}$ can be
simplified to

(48) $\frac{1}{2\pi
i}\int_{|z|=d}\log\left|z+\frac{1+iw}{i+w}\right|\frac{dz}{z}=\left\\{\begin{array}[]{ll}\log\left|\frac{1+iw}{i+w}\right|&\mbox{if}\
\left|\frac{1+iw}{i+w}\right|>d,\\\ \\\ \log d&\mbox{if}\
\left|\frac{1+iw}{i+w}\right|\leq d\end{array}\right.$

following an application of Jensen’s formula.

Let $\gamma_{>d}$ and $\gamma_{\leq d}$ be the two collections of arcs defined
by

$\gamma_{>d}=\\{w:|w|=c,|z(w)|>d\\},\quad\quad\gamma_{\leq
d}=\\{w:|w|=c,|z(w)|\leq d\\}.$

Then (47) can be expressed as

$\displaystyle\mathrm{m}_{c,d}\left(z+\frac{1+iw}{i+w}\right)$
$\displaystyle=\frac{1}{2\pi i}\int_{|w|=c}\left(\frac{1}{2\pi
i}\int_{|z|=d}\log\left|z+\frac{1+iw}{i+w}\right|\frac{dz}{z}\right)\frac{dw}{w}$
(49) $\displaystyle=\frac{1}{2\pi
i}\int_{\gamma_{>d}}\log\left|\frac{1+iw}{i+w}\right|\frac{dw}{w}+\frac{1}{2\pi
i}\int_{\gamma_{\leq d}}\log d\frac{dw}{w}.$

Since $\mathop{\mathrm{Im}}\left(\frac{dw}{w}\right)=d\arg w,$ the
differential form can be represented in terms of $\eta$ as

$\log\left|\frac{1+iw}{i+w}\right|\frac{dw}{w}=\log|z(w)|\frac{dw}{w}=-i\left(\eta(w,z(w))-\eta(c,z(w))\right).$

The second term above can be further simplified to

$\eta(c,z(w))=\eta(c,iz(w))-\eta(c,i)=\eta(c,iz(w))=(\log
c)d\arg\left(\frac{1+iw}{1-iw}\right),$

where $iz(w)=i\frac{1+iw}{i+w}=\frac{1+iw}{1-iw}.$ Therefore, once we have
determined $\gamma_{>d}$ and $\gamma_{\geq d}$ explicitly, the integrals in
(49) can be evaluated individually using the properties of $\eta$ and the
following two lemmas.

######  Lemma 6.1.

For $w,z(w)$ mentioned above, $\eta\left(w,z(w)\right)$ decomposes as

$\eta(w,z(w))=\eta\left(-iw,1+iw\right)-\eta\left(iw,1-iw\right).$

######  Lemma 6.2 (Lemma 16, [22]).

For $c\in\mathbb{R}_{>0}$ and $\theta\in[-\pi,\pi),$ let $w=ce^{i\theta}$ and
$\psi=\theta+\frac{\pi}{2}.$ Then

$d\arg\left(\frac{1+iw}{1-iw}\right)=\frac{2\left(c^{-1}-c\right)\cos\psi}{\left(c^{-1}-c\right)^{2}+4\sin^{2}\psi}d\psi.$

Using property (9) of $\eta$ we can rewrite $\eta(w,z(w))$ in Lemma 6.1 as

(50) $\eta(w,z(w))=dD\left(-iw\right)-dD\left(iw\right),$

where $D$ is the Bloch–Wigner dilogarithm given in (6).

The evaluation of the remaining integral involving $\eta(c,z(w))$ ($=\log c\
d\arg\left(\frac{1+iw}{1-iw}\right)$) over the integration path $\gamma_{>d}$
follows from the lemma below.

######  Lemma 6.3.

For $c\in\mathbb{R}_{>0}$ and $\theta\in[-\pi,\pi),$ let $w=ce^{i\theta}.$ Let
$\alpha,\beta\in[-\pi,\pi).$ Then

$\int_{w(\alpha)}^{w(\beta)}d\arg\left(\frac{1+iw}{1-iw}\right)=\tan^{-1}\left(\frac{2\cos\alpha}{c-c^{-1}}\right)-\tan^{-1}\left(\frac{2\cos\beta}{c-c^{-1}}\right),$

where $w(\alpha)=ce^{i\alpha}$ and $w(\beta)=ce^{i\beta}.$

We omit the proof of Lemma 6.2 since it is an intermediate step in the Lemma
$16$ of [22]. We should also remark that Lemma 6.3 is a generalized version of
the Lemma 16 in [22], which states the above result for the case $\alpha=-\pi$
and $\beta=0.$ We will see later that the proof of Lemma 6.3 also follows from
an argument similar to the proof in [22]. We now sketch the proofs of Lemma
6.1 and 6.3.

###### Proof of Lemma 6.1.

Using properties of $\eta$ in Lemma 2.1, $\eta(w,z(w))$ decomposes as

$\displaystyle\eta(w,z(w))$
$\displaystyle=\eta\left(w,\frac{1+iw}{i+w}\right)$
$\displaystyle=\eta(w,1+iw)-\eta(w,i+w)$
$\displaystyle=\eta(-iw,1+iw)-\eta(-i,1+iw)-\eta(iw,i+w)+\eta(i,i+w)$
$\displaystyle=\eta(-iw,1+iw)-\eta(iw,1-iw)-\eta(iw,i)$
$\displaystyle=\eta(-iw,1+iw)-\eta(iw,1-iw),$

where we applied Remark 2.2, which implies that
$\eta(\zeta,f(w))=0=\eta(f(w),\zeta)$ for any root of unity $\zeta$ and any
function $f(w)$ of $w.$ ∎

###### Proof of Lemma 6.3:.

We first assume that $c\in\mathbb{R}_{>0}\setminus\\{1\\},$ and the case $c=1$
follows from a continuity argument.

For $\alpha,\beta\in[-\pi,\pi)$ and $w=ce^{i\theta},$ Lemma 6.2 yields that

$\int_{w(\alpha)}^{w(\beta)}d\arg\left(\frac{1+iw}{1-iw}\right)=\int_{\alpha+\frac{\pi}{2}}^{\beta+\frac{\pi}{2}}\frac{2\left(c^{-1}-c\right)\cos\psi}{\left(c^{-1}-c\right)^{2}+4\sin^{2}\psi}d\psi=-\int_{\cos\alpha}^{\cos\beta}\frac{2(c-c^{-1})}{(c-c^{-1})^{2}+4t^{2}}dt,$

where $\psi=\theta+\frac{\pi}{2},$ $w(\phi)=ce^{i\phi},$ and the last equality
follows from the change of variables $\sin\psi\mapsto t.$

Further, the change of variables $\frac{2t}{c-c^{-1}}\mapsto u$ gives that

$\int_{w(\alpha)}^{w(\beta)}d\arg\left(\frac{1+iw}{1-iw}\right)=-\int_{\frac{2\cos\alpha}{c-c^{-1}}}^{\frac{2\cos\beta}{c-c^{-1}}}\frac{du}{1+u^{2}}=\tan^{-1}\left(\frac{2\cos\alpha}{c-c^{-1}}\right)-\tan^{-1}\left(\frac{2\cos\beta}{c-c^{-1}}\right),$

which proves the lemma. ∎

In order to apply Lemma 6.1 and Lemma 6.2 to (49), it is necessary to
explicitly express $\gamma_{\leq d}$ and $\gamma_{>d}.$ Since $\gamma_{>d}$
and $\gamma_{\leq d}$ are disjoint, and

$\\{w:|w|=c\\}=\gamma_{>d}\cup\gamma_{\leq d},$

it suffices to understand $\gamma_{>d}.$

Recall that $z(w)=-\frac{1+iw}{i+w}.$ Then,
$|z(w)|>d\Leftrightarrow\left|\frac{1+iw}{i+w}\right|>d\Leftrightarrow\left|1+iw\right|>d\left|i+w\right|.$
Since both sides of the inequality are non-negative, we can square them and
get

$\displaystyle\left|1+iw\right|>d\left|i+w\right|$
$\displaystyle\Leftrightarrow\left|1+iw\right|^{2}>d^{2}\left|i+w\right|^{2}$
$\displaystyle\Leftrightarrow 2(1+d^{2})\real(iw)>(d^{2}-1)(1+|w|^{2})$
$\displaystyle\Leftrightarrow\real(ie^{i\theta})>\frac{d^{2}-1}{1+d^{2}}\cdot\frac{1+c^{2}}{2c},$

where the last inequality follows from the fact that $w=ce^{i\theta}$ for
$\theta\in[-\pi,\pi).$ In other words, the condition $|z(w)|>d$ is equivalent
to the condition

(51)
$-1\leq-\mathop{\mathrm{Re}}(ie^{i\theta})=\sin\theta<\frac{1-d^{2}}{1+d^{2}}\cdot\frac{1+c^{2}}{2c},$

with $\theta\in[-\pi,\pi).$ For simplicity we denote

$\mathcal{A}_{c,d}:=\frac{1-d^{2}}{1+d^{2}}\cdot\frac{1+c^{2}}{2c}.$

As $|\sin\theta|\leq 1,$ there are three cases to consider.

  * Case 1:

If $\mathcal{A}_{c,d}\leq-1,$ then
$\gamma_{>d}=\varnothing\quad\mbox{and}\quad\gamma_{\leq d}=\\{w:|w|=c\\}.$

  * Case 2:

If $\mathcal{A}_{c,d}\geq 1,$ then
$\gamma_{>d}=\\{w:|w|=c\\}\quad\mbox{and}\quad\gamma_{\leq d}=\varnothing.$

  * Case 3:

If $\left|\mathcal{A}_{c,d}\right|<1,$ then, for $w=ce^{i\theta}$ with
$\theta\in[-\pi,\pi),$
$\gamma_{>d}=\left\\{w:|w|=c,-1\leq\sin\theta<\mathcal{A}_{c,d}\right\\}$ and
$\gamma_{\leq d}=\left\\{w:|w|=c,\mathcal{A}_{c,d}\leq\sin\theta\leq
1\right\\}.$

Now we have everything needed to evaluate (49), namely

$\mathrm{m}_{c,d}\left(z+\frac{1+iw}{i+w}\right)=\frac{1}{2\pi
i}\int_{\gamma_{>d}}\log\left|\frac{1+iw}{i+w}\right|\frac{dw}{w}+\frac{1}{2\pi
i}\int_{\gamma_{\leq d}}\log d\frac{dw}{w}.$

Case 1: Since $\gamma_{>d}=\varnothing,$ the integrals in (49) can be
evaluated individually to obtain

$\frac{1}{2\pi
i}\int_{\gamma_{>d}}\log\left|\frac{1+iw}{i+w}\right|\frac{dw}{w}=0,\quad\mbox{and}\quad\frac{1}{2\pi
i}\int_{\gamma_{\leq d}}\log d\frac{dw}{w}=\frac{1}{2\pi i}\int_{|w|=c}\log
d\frac{dw}{w}=\log d.$

Therefore, in this case we have

(52) $\mathrm{m}_{c,d}\left(z+\frac{1+iw}{i+w}\right)=\log d.$

Case 2: Since $\gamma_{\leq d}=\varnothing,$ the second integral in (49)
contributes nothing. However the first integral can be decomposed into simpler
integrals, i.e.

$\displaystyle\frac{1}{2\pi
i}\int_{\gamma_{>d}}\log\left|\frac{1+iw}{i+w}\right|\frac{dw}{w}$
$\displaystyle=\frac{1}{2\pi
i}\int_{|w|=c}\log\left|\frac{1+iw}{i+w}\right|\frac{dw}{w}$
$\displaystyle=\frac{1}{2\pi
i}\int_{|w|=c}\log\left|1+iw\right|\frac{dw}{w}-\frac{1}{2\pi
i}\int_{|w|=c}\log\left|i+w\right|\frac{dw}{w}$ $\displaystyle=0.$

Therefore, when $\gamma_{>d}=\\{|w|=c\\}$ (and $\gamma_{\leq d}=\varnothing$),
then

(53) $\mathrm{m}_{c,d}\left(z+\frac{1+iw}{i+w}\right)=0.$

Case 3: Since

$\left|\mathcal{A}_{c,d}\right|<1,$

we have two sub-cases to consider.

3a: When

$-1<\mathcal{A}_{c,d}<0,$

then $\sin^{-1}\left(\mathcal{A}_{c,d}\right)\in[-\pi,0).$ For simplicity, we
denote $\tau=\sin^{-1}\left(\mathcal{A}_{c,d}\right)$ such that
$\tau\in\left(-\frac{\pi}{2},0\right).$ Note that $\sin\tau=\sin(-\pi-\tau).$
Then the boundary values of $\gamma_{>d}$ are

$\partial\gamma_{>d}=\\{w(-\pi-\tau),w(\tau)\\}=\\{ce^{i(-\pi-\tau)},ce^{i\tau}\\}=\\{-ce^{-i\tau},ce^{i\tau}\\},$

where $w(\theta)=ce^{i\theta}.$ The integration path $\gamma_{\leq d}$ is then
the union of the arcs joining $w(-\pi)$ and $w(-\pi-\tau),$ and joining
$w(\tau)$ and $w(\pi).$ Therefore

$\partial\gamma_{\leq d}=\\{w(-\pi),w(-\pi-\tau),w(\tau),w(\pi)\\}$

are the boundary values of $\gamma_{\leq d}.$ All the paths are assumed to be
traversed counter-clockwise. Now, we have all the tools to calculate (49) in
this case. Combining Lemma 6.1 and (50) with the above discussion we obtain

$\displaystyle\mathrm{m}_{c,d}\left(z+\frac{1+iw}{i+w}\right)=$
$\displaystyle\frac{1}{2\pi
i}\int_{\gamma_{>d}}\log\left|\frac{1+iw}{i+w}\right|\frac{dw}{w}+\frac{1}{2\pi
i}\int_{\gamma_{\leq d}}\log d\frac{dw}{w}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\int_{\gamma_{>d}}\eta(w,z(w))+\frac{1}{2\pi}\int_{\gamma_{>d}}\eta(c,z(w))+\frac{1}{2\pi
i}\int_{\gamma_{\leq d}}\log d\frac{dw}{w}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\int_{\gamma_{>d}}(dD(-iw)-dD(iw))$ (54)
$\displaystyle+\frac{\log
c}{2\pi}\int_{w(-\pi-\tau)}^{w(\tau)}d\arg\left(\frac{1+iw}{1-iw}\right)+\frac{\log
d}{2\pi}\left(\int_{-\pi}^{-\pi-\tau}+\int_{\tau}^{\pi}\right)d\theta,$

where the simplification of the last integral follows from the above
discussion regarding $\partial\gamma_{\leq d}$ and substituting $w$ with
$w(\theta)=ce^{i\theta}.$ The first integral in (54) can be evaluated using
Stokes’ theorem as

$\displaystyle\frac{1}{2\pi}\int_{\gamma_{>d}}(dD(-iw)-dD(iw))=$
$\displaystyle\frac{1}{2\pi}\left[D(-iw)-D(iw)\right]_{\partial\gamma_{>d}}$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\left[D(-iw)-D(iw)\right]_{w(-\pi-\tau)}^{w(\tau)}$
(55) $\displaystyle=$
$\displaystyle-\frac{1}{\pi}\left(D(ice^{-i\tau})+D(ice^{i\tau})\right),$

where the last equality follows from the property (7) of the Bloch-Wigner
dilogarithm.

Substituting $\alpha=-\pi-\tau$ and $\beta=\tau$ in the statement of Lemma
6.3, we evaluate the second integral in (54):

(56) $\frac{\log
c}{2\pi}\int_{w(-\pi-\tau)}^{w(\tau)}d\arg\left(\frac{1+iw}{1-iw}\right)=-\frac{\log
c}{\pi}\tan^{-1}\left(\frac{2\cos\tau}{c-c^{-1}}\right).$

The remaining integral’s contribution is

(57) $\frac{\log
d}{2\pi}\left(\int_{-\pi}^{-\pi-\tau}+\int_{\tau}^{\pi}\right)d\theta=\frac{\log
d}{2\pi}\left[-\pi-\tau+\pi+\pi-\tau\right]=\frac{\pi-2\tau}{2\pi}\log d.$

Then (55), (56) and (57) together yield that

$\mathrm{m}_{c,d}\left(z+\frac{1+iw}{i+w}\right)=\frac{1}{\pi}\left[D(ice^{-i\tau})+D(ice^{i\tau})-(\log
c)\tan^{-1}\left(\frac{2\cos\tau}{c-c^{-1}}\right)+\left(\frac{\pi}{2}-\tau\right)\log
d\right].$

3b: It remains to evaluate the case when

$0<\mathcal{A}_{c,d}<1.$

This condition is equivalent to

$\sin^{-1}\left(\mathcal{A}_{c,d}\right)\in(0,\pi).$

Again, for simplicity, we denote
$\kappa=\sin^{-1}\left(\mathcal{A}_{c,d}\right)$ such that
$\kappa\in\left(0,\frac{\pi}{2}\right).$ Since $\sin\kappa=\sin(\pi-\kappa)$
and $\sin\pi=0,$ the boundary values in this case are

$\partial\gamma_{>d}=\\{w(-\pi),w(\kappa),w(\pi-\kappa),w(\pi)\\},$

and

$\partial\gamma_{\leq d}=\\{w(\kappa),w(\pi-\kappa)\\}.$

The arcs are considered to be oriented in a counter-clockwise direction. From
a similar argument as before, we deduce that

$\mathrm{m}_{c,d}\left(z+\frac{1+iw}{i+w}\right)=\frac{1}{\pi}\left[D(ice^{-i\kappa})+D(ice^{i\kappa})-(\log
c)\tan^{-1}\left(\frac{2\cos\kappa}{c-c^{-1}}\right)+\left(\frac{\pi}{2}-\kappa\right)\log
d\right].$

We combine the results obtained in 3a and 3b to obtain

(58)
$\mathrm{m}_{c,d}\left(z+\frac{1+iw}{i+w}\right)=\frac{1}{\pi}\left[D(ice^{-i\mu})+D(ice^{i\mu})-(\log
c)\tan^{-1}\left(\frac{2\cos\mu}{c-c^{-1}}\right)+\left(\frac{\pi}{2}-\mu\right)\log
d\right],$

where
$\mu=\sin^{-1}\left(\mathcal{A}_{c,d}\right)\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right).$
This concludes the evaluation of
$\mathrm{m}_{c,d}\left(z+\frac{1+iw}{i+w}\right)$ for the different cases.

Recall that $R(w,z)=1+iw+iz+wz.$ In order to evaluate
$\mathrm{m}_{c,d}(R(w,z)),$ it suffices to collect the equalities in (46),
(52), (53) and (58). We deduce

$\mathrm{m}_{c,d}(R(w,z))=\max\\{\log
c,0\\}+\left\\{\begin{array}[]{ll}\max\\{\log d,0\\}&\mbox{if}\
\left|\mathcal{A}_{c,d}\right|\geq 1,\\\ \\\
\frac{1}{\pi}\left[D(ice^{-i\mu})+D(ice^{i\mu})\right.\\\ \left.-(\log
c)\tan^{-1}\left(\frac{2\cos\mu}{c-c^{-1}}\right)+\left(\frac{\pi}{2}-\mu\right)\log
d\right]&\mbox{if}\ \left|\mathcal{A}_{c,d}\right|<1,\end{array}\right.,$

where
$\mu=\sin^{-1}\left(\mathcal{A}_{c,d}\right)\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right).$
Notice that
$\mathrm{m}_{c,d}(R(w,z))=\frac{1}{2}\left(\mathrm{m}_{a,b}(Q_{4})+\log
b\right),$ where $a=\frac{c}{d}$ and $b=cd.$ This implies that when
$\left|\mathcal{A}_{c,d}\right|\geq 1,$ we have

(59) $\mathrm{m}_{a,b}(Q_{4})=\max\\{\log c,-\log c\\}+\max\\{\log d,-\log
d\\}.$

On the other hand, when $\left|\mathcal{A}_{c,d}\right|<1,$

$\displaystyle\mathrm{m}_{a,b}(Q_{4})=$ $\displaystyle\max\\{\log c,-\log
c\\}-\log d+\frac{2}{\pi}\left[D(ice^{-i\mu})+D(ice^{i\mu})\right]$
$\displaystyle-\frac{2\log
c}{\pi}\tan^{-1}\left(\frac{2\cos\mu}{c-c^{-1}}\right)+\left(1-\frac{2\mu}{\pi}\right)\log
d$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\left[D(ice^{-i\mu})+D(ice^{i\mu})-\mu\log
d\right]+\frac{2\log
c}{\pi}\left[\max\left\\{\frac{\pi}{2},-\frac{\pi}{2}\right\\}-\tan^{-1}\left(\frac{2\cos\mu}{c-c^{-1}}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{2}{\pi}\left[D(ice^{-i\mu})+D(ice^{i\mu})-\mu\log
d\right]+\frac{2\log c}{\pi}\tan^{-1}\left(\frac{c-c^{-1}}{2\cos\mu}\right),$

where
$\mu=\sin^{-1}\left(\mathcal{A}_{c,d}\right)\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right),$
and the simplification of the last term follows from the fact that

$\mbox{if}\ x>0,\quad\pi/2-\tan^{-1}(x)=\tan^{-1}\left(x^{-1}\right),$

and

$\mbox{if}\ x<0,\quad-\pi/2-\tan^{-1}(x)=\tan^{-1}\left(x^{-1}\right).$

Therefore, (59) along with the above discussion implies that

$\mathrm{m}_{a,b}(Q_{4})=\left\\{\begin{array}[]{ll}\max\\{\log c,-\log
c\\}+\max\\{\log d,-\log d\\}&\mbox{if}\ \left|\mathcal{A}_{c,d}\right|\geq
1,\\\ \\\ \frac{2}{\pi}\left[D(ice^{-i\mu})+D(ice^{i\mu})-\mu\log d+(\log
c)\tan^{-1}\left(\frac{c-c^{-1}}{2\cos\mu}\right)\right]&\mbox{if}\
\left|\mathcal{A}_{c,d}\right|<1,\end{array}\right.$

where
$\mu=\sin^{-1}\left(\mathcal{A}_{c,d}\right)\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right),$
which completes the proof of Theorem 1.4.

##  7\. Extension of Theorem 1.2 to several variables

We end this article by extending Theorem 1.2 to $n$-variable Laurent
polynomials for $n\geq 3.$

Let $P_{k}(x_{1},\dots,x_{n})\in\mathbb{C}[x_{1}^{\pm},\dots,x_{n}^{\pm}]$ be
a Laurent polynomial in $n$-variable such that

$P_{k}:=P_{k}(x_{1},\dots,x_{n})=k-P(x_{1},\dots,x_{n}),$

where $P$ has no constant term. Let $\mathbb{T}_{\mathfrak{a}}^{n}$ be the
integration torus in the definition of $\mathrm{m}_{\mathfrak{a}}(P_{k}),$
where $\mathfrak{a}=(a_{1},\dots,a_{n}).$

We consider the image $\mathcal{K}_{\mathfrak{a}}$ of the map from
$\mathbb{T}_{\mathfrak{a}}^{n}$ to $\mathbb{C}$ defined by
$(x_{1},\dots,x_{n})\mapsto P(x_{1},\dots,x_{n}).$ A similar argument as in
the $2$-variable case shows that $\mathcal{K}_{\mathfrak{a}}$ is compact in
$\mathbb{C}.$ Therefore, $\max_{k\in\mathcal{K}_{\mathfrak{a}}}|k|$ exists and
is finite. Let $K_{\mathfrak{a}}$ denote this maximum, and let
$K_{\mathfrak{a},\mathfrak{1}}=\max\\{K_{\mathfrak{a}},K_{\mathfrak{1}}\\},$
where $\mathfrak{1}=(1,\dots,1).$ Then, generalizing the steps in [7] and
[13], we define

(60) $\tilde{\mathrm{m}}_{\mathfrak{a}}(P_{k})=\log k-\sum_{m\geq
0}\frac{a_{m,\mathfrak{a}}}{m}k^{-m},\quad|k|>K_{\mathfrak{a},\mathfrak{1}},k\notin(-\infty,0],$

where $\log$ denotes the principal branch of the logarithm, and
$a_{m,\mathfrak{a}}$ is defined as follows:

$\displaystyle a_{m,\mathfrak{a}}=$ $\displaystyle\left[\frac{1}{(2\pi
i)^{n}}\int_{\mathbb{T}_{\mathfrak{a}}^{2}}\frac{dx_{1}\cdots
dx_{n}}{x_{1}\cdots
x_{n}(1-r^{-1}P(x_{1},\dots,x_{n}))}\right]_{m}=\frac{1}{(2\pi
i)^{n}}\int_{\mathbb{T}_{\mathfrak{a}}^{n}}P(x_{1},\dots,x_{n})^{n}\frac{dx_{1}}{x_{1}}\cdots\frac{dx_{n}}{x_{n}},$

where $[T(s)]_{m}$ denotes the coefficient of $s^{-m}$ in the series $T(s).$

It is immediate to see that $\tilde{\mathrm{m}}_{\mathfrak{a}}(P_{k})$ is
holomorphic in the region defined by the intersection of
$\\{|k|>K_{\mathfrak{a},\mathfrak{1}}\\}$ and
$\mathbb{C}\setminus(-\infty,0]$. Also,

$\real(\tilde{\mathrm{m}}_{\mathfrak{a}}(P_{k}))=\mathrm{m}_{\mathfrak{a}}(P_{k}),\quad|k|>K_{\mathfrak{a},\mathfrak{1}}.$

A similar argument as in the $2$-variable case shows that
$a_{m,\mathfrak{a}}=a_{m,\mathfrak{1}}$ for all $m\geq 0,$ and therefore, we
have the following equality:

(61)
$\frac{d\tilde{\mathrm{m}}_{\mathfrak{a}}(P_{k})}{dk}=\frac{d\tilde{\mathrm{m}}_{\mathfrak{1}}(P_{k})}{dk},\quad\quad\mbox{for}\
|k|>K_{\mathfrak{a},\mathfrak{1}}.$

In addition, similarly as in the $2$-variable case, the integral
representation

$\frac{d\tilde{\mathrm{m}}_{\mathfrak{a}}(P_{k})}{dk}=\frac{1}{(2\pi
i)^{n}}\int_{\mathbb{T}_{\mathfrak{a}}^{n}}\frac{1}{k-P(x_{1},\dots,x_{n})}\frac{dx_{1}}{x_{n}}\cdots\frac{dx_{n}}{x_{n}}$

is in fact holomorphic in $\\{|k|>K_{\mathfrak{a},\mathfrak{1}}\\}.$

Now integrating both sides of (61) with respect to $k$ yields that, for
$\\{|k|>K_{\mathfrak{a},\mathfrak{1}}\\},$

$\tilde{\mathrm{m}}_{\mathfrak{a}}(P_{k})=\tilde{\mathrm{m}}_{\mathfrak{1}}(P_{k})+\tilde{g}(\mathfrak{a}),$

where $\tilde{g}:\mathbb{R}_{>0}^{n}\rightarrow\mathbb{C}.$ Taking real parts
on both sides, we get

(62)
$\mathrm{m}_{\mathfrak{a}}(P_{k})=\mathrm{m}_{\mathfrak{1}}(P_{k})+g(\mathfrak{a}),$

for $\\{|k|>K_{\mathfrak{a},\mathfrak{1}}\\},$ where
$g=\mathop{\mathrm{Re}}\left(\tilde{g}\right).$

Let $U_{\mathfrak{a}}$ be the unbounded open connected component of
$\mathbb{C}\setminus\mathcal{K}_{\mathfrak{a}}$ which contains the region
$\left\\{|k|>K_{\mathfrak{a},\mathfrak{1}}\right\\}.$ As both sides of (62)
are harmonic on $O_{\mathfrak{a}}:=U_{\mathfrak{a}}\cap U_{\mathfrak{1}},$ the
equality can be extended to $O_{\mathfrak{a}}.$ In other words, for $k\in
O_{\mathfrak{a}},$ we have
$\mathrm{m}_{\mathfrak{a}}(P_{k})=\mathrm{m}_{\mathfrak{1}}(P_{k})+g(\mathfrak{a}).$

It only remains to express $g$ explicitly in terms of $\mathfrak{a}.$ Consider
the functions

$a_{j}\frac{\partial\tilde{\mathrm{m}}_{\mathfrak{a}}(P_{k})}{\partial
a_{j}}=\frac{1}{(2\pi
i)^{n}}\int_{|x_{1}|=a_{1},\dots,\widehat{|x_{j}|=a_{j}},\dots,|x_{n}|=a_{n}}\left(\int_{|x_{j}|=a_{j}}\frac{\partial_{x_{j}}P_{k}}{P_{k}}dx_{j}\right)\frac{dx_{1}}{x_{1}}\cdots\widehat{\frac{dx_{j}}{x_{j}}}\cdots\frac{dx_{n}}{x_{n}}$

for all $j=1,\dots,n.$ Here $\ \widehat{\textbullet}\ $ denotes that the term
is omitted from the expression. Now, again following the steps for
$2$-variable case we conclude that
$a_{j}\frac{\partial\tilde{\mathrm{m}}_{\mathfrak{a}}(P_{k})}{\partial a_{j}}$
is constant depending only on $a_{j}.$ More precisely, we find that

(63) $a_{j}\frac{\partial\tilde{\mathrm{m}}_{\mathfrak{a}}(P_{k})}{\partial
a_{j}}=\nu_{\mathfrak{a},k}^{j},$

where $\nu_{\mathfrak{a},k}^{j}$ is the difference between the number of
zeroes (counting multiplicities) of
$P_{k}(a_{1},\dots,a_{j-1},x_{j},a_{j+1},\dots,x_{n})$ inside the circle
$\mathbb{T}_{a_{j}}^{1},$ denoted by $Z_{\mathfrak{a},k}^{j},$ and the order
of the pole of $P_{k}(a_{1},\dots,a_{j-1},x_{j},a_{j+1},\dots,x_{n})$ at
$x_{j}=0,$ denoted by $P_{\mathfrak{a},k}^{j}.$ In other words,

$\nu_{\mathfrak{a},k}^{j}=Z_{\mathfrak{a},k}^{j}-P_{\mathfrak{a},k}^{j}.$

We also note that $\nu_{\mathfrak{a},k}^{j}$ is independent of $k$ when $k\in
O_{\mathfrak{a}},$ and only depends on $\mathfrak{a}$ and the polynomial
$P=k-P_{k}.$ Integrating (63) with respect to $a_{j}$ for $j=1,\dots,n,$ and
then substituting those values in (62) lead to the following theorem.

######  Theorem 7.1.

Let $\mathfrak{a}=(a_{1},\dots,a_{n})\in(\mathbb{R}_{>0})^{n}.$ Let
$P_{k}(x_{1},\dots,x_{n})=k-P(x_{1},\dots,x_{n})\in\mathbb{C}[x_{1}^{\pm},\dots,x_{n}^{\pm}],$
such that $P$ has no constant term. Denote $U_{\mathfrak{a}}$ the unbounded
open connected component of $\mathbb{C}\setminus\mathcal{K}_{\mathfrak{a}}$
containing some neighbourhood of $k=\infty.$ Then, for $k\in
U_{\mathfrak{a}}\cap U_{\mathfrak{1}},$

$\mathrm{m}_{\mathfrak{a}}(P_{k})=\mathrm{m}(P_{k})+\sum_{j=1}^{n}\nu_{\mathfrak{a},k}^{j}\log
a_{j},$

where $\nu_{\mathfrak{a},k}^{j}$ is defined as above, and
$\mathrm{m}_{\mathfrak{1}}(P_{k})=\mathrm{m}(P_{k}).$ Moreover, for $k\in
U_{\mathfrak{a}}\cap U_{\mathfrak{1}}$ and $j=1,\dots,n,$
$\nu_{\mathfrak{a},k}^{j}$ only depends on $\mathfrak{a}.$

Next, after multiplying $P_{k}$ with a suitable power of $x_{j},$ we can
factorise $P_{k}$ in linear factors with coefficients in
$\overline{\mathbb{C}(x_{1},\dots,\widehat{x_{j}},\dots,x_{n})}$ as

$P_{k}(x_{1},\dots,x_{n})=x_{j}^{-v_{j}}P_{F,k}^{j}(x_{1},\dots,\widehat{x_{j}},\dots,x_{n})\prod_{l=1}^{d_{n}}\left(x_{j}-X_{l,k,j}\left(x_{1},\dots,\widehat{x_{j}},\dots,x_{n}\right)\right),$

where $d_{j}$ is the degree of $P_{k}$ as a polynomial in $x_{j},$ $X_{l,k,j}$
are algebraic functions of $(x_{1},\dots,\widehat{x_{j}},\dots,x_{n})$ for
$l=1,\dots,d_{n},$ $P_{F,k}^{j}$ is the leading coefficient with respect to
the variable $x_{j},$ and $v_{j}$ is the largest power of $x_{j}^{-1}$ in
$P_{k}.$ Let $P_{f,k}^{j}(x_{1},\dots,\widehat{x_{j}},\dots,x_{n})$ denote the
"constant" coefficient with respect to the variable $x_{j}.$ Then

$P_{F,k}^{j}(x_{1},\dots,\widehat{x_{j}},\dots,x_{n})\prod_{j=1}^{d_{n}}X_{l,k,j}(x_{1},\dots,\widehat{x_{j}},\dots,x_{n})=P_{f,k}^{j}(x_{1},\dots,\widehat{x_{j}},\dots,x_{n}).$

For
$\left(u_{1},\dots,\widehat{u_{j}},\dots,u_{n}\right)\in\mathbb{T}_{a_{1},\dots,\widehat{a_{j}},\dots,a_{n}}^{n-1},$
let
$\varrho_{\mathfrak{a},k}^{j}\left(u_{1},\dots,\widehat{u_{j}},\dots,u_{n}\right)$
be the number of zeroes (counting multiplicities) of
$P_{k}(u_{1},\dots,u_{j-1},x_{j},u_{j+1},\dots,u_{n})$ inside the circle
$\mathbb{T}_{a_{j}}^{1}.$ Then, from the above discussion, we have
$P_{\mathfrak{a},k}^{j}=v_{j},$ and

$\varrho_{\mathfrak{a},k}^{j}\left(a_{1},\dots,\widehat{a_{j}},\dots,a_{n}\right)=Z_{\mathfrak{a},k}^{j}=\nu_{\mathfrak{a},k}^{j}+P_{\mathfrak{a},k}^{j}=\nu_{\mathfrak{a},k}^{j}+v_{j}.$

An analogous argument as in the proof of Proposition 3.2 yields the following
proposition.

######  Proposition 7.2.

Let $k\notin\mathcal{K}_{\mathfrak{a}}.$ Then
$\varrho_{\mathfrak{a},k}^{j}(x_{1},\dots,\widehat{x_{j}},\dots,x_{n})$ is
constant for all
$(x_{1},\dots,\widehat{x_{j}},\dots,x_{n})\in\mathbb{T}_{a_{1},\dots,\widehat{a_{j}},\dots,a_{n}}^{n-1}.$

We omit the proof of the proposition here since it is an immediate extension
of Proposition 3.2, which follows from an induction argument on $n\geq 2.$

Then Proposition 7.2, along with a similar argument as in the proof of Theorem
1.3, establishes the following extension of Theorem 1.3 for several variable
case.

######  Theorem 7.3.

Let $\mathfrak{a}=(a_{1},\dots,a_{n})\in\mathbb{R}_{>0}^{n}.$ Let
$k_{0}\in\mathbb{C}\setminus\mathcal{K}_{\mathfrak{a}}$ such that $k_{0}$
belongs to one of the bounded open connected components of
$\mathbb{C}\setminus\mathcal{K}_{\mathfrak{a}}.$ We denote by
$V_{\mathfrak{a},k_{0}}$ the bounded open connected component containing
$k_{0}.$

  1. (I)

For $j=1,\dots,n,$ if all the roots of
$P_{k_{0}}(a_{1},\dots,a_{j-1},x_{j},a_{j+1},\dots,x_{n})$ lie entirely inside
the circle $\mathbb{T}_{a_{j}}^{1},$ then, for all $k\in
V_{\mathfrak{a},k_{0}},$

$\mathrm{m}_{\mathfrak{a}}(P_{k})=\nu_{\mathfrak{a},k}^{j}\log
a_{j}+\mathrm{m}_{a_{1},\dots,\widehat{a_{j}},\dots,a_{n}}\left(P_{F,k}^{j}\right).$

  2. (II)

For $j=1,\dots,n,$ if all the roots of
$P_{k_{0}}(a_{1},\dots,a_{j-1},x_{j},a_{j+1},\dots,x_{n})$ lie entirely
outside the circle $\mathbb{T}_{a_{j}}^{1},$ then, for all $k\in
V_{\mathfrak{a},k_{0}},$

$\mathrm{m}_{\mathfrak{a}}(P_{k})=\nu_{\mathfrak{a},k}^{j}\log
a_{j}+\mathrm{m}_{a_{1},\dots,\widehat{a_{j}},\dots,a_{n}}\left(P_{f,k}^{j}\right).$

##  8\. Conclusion

There are several directions for further exploration. The most immediate
question one can ask is how to evaluate $\mathrm{m}_{a,b}(Q_{r})$ when
$r\in\mathcal{R}_{a,b}.$ A primary observation in this case is that the
integration path is not necessarily closed (and in most cases it is not). This
turns out to be a challenging problem since the integration path in this case
cannot be easily identified as a cycle in the homology group. We face a
similar obstacle while evaluating the Mahler measure of $Q_{r}(x,y)$ on the
bounded connected components when the number of roots of $Q_{r}(a,y)$
(counting multiplicity) (or $Q_{r}(x,b)$) inside $\mathbb{T}_{b}^{1}$ (or
$\mathbb{T}_{a}^{1}$) is strictly less than the degree of the polynomials. In
this situation it is frequently required to integrate the algebraic functions
coming from the factorisation of $Q_{r}(x,y)$ (when considered as a polynomial
in either $x$ or $y$), on paths which are not closed. These similar challenges
also extend to the $n$-variable cases when $n\geq 3.$

A different direction would be to consider the family of rational polynomials

$P_{k}(x_{1},\dots,x_{n})=k-\frac{P(x_{1},\dots,x_{n})}{Q(x_{1},\dots,x_{n})}\in\mathbb{C}(x_{1},\dots,x_{n}),\quad\quad\mbox{for}\
k\in\mathbb{C}.$

Our method of proof for Theorems 1.2 and 1.3 extends to this type of rational
polynomials when $Q(x_{1},\dots,x_{n})$ is a monomial, which essentially
recovers Theorems 7.1 and 7.3.

The expression of $\nu_{\mathfrak{a},k}^{j}$ in (63) appears in the work of
Forsberg, Passare, and Tsikh [23], where it is denoted as the order of an
element in the complement of the Amoeba associated to the respected
polynomial. Our theorems also re-establish certain properties of the Ronkin
function associated to amoebas mentioned in [23]. Therefore, it would be also
natural to explore the generalized Mahler measure in terms of the Ronkin
function associated to amoebas in more depth.

## Appendix: Explicit derivations of the region $\mathcal{R}_{a,b}$

In this Appendix, we include the proofs of Lemmas 5.4 and 5.5 for completion.

###### Proof of Lemma 5.4.

We first recall that, for $r$ given by (32),

(64)
$\max_{r\in\mathcal{R}_{a,b}}|\mathop{\mathrm{Im}}(r)|=\left|a-a^{-1}\right|+\left|b-b^{-1}\right|.$

Using property (B) from Section $5$ regarding the symmetric nature of the
compact set $\mathcal{R}_{a,b},$ we can focus on proving the statement when
$\alpha,\beta\in[0,\pi),$ and the other cases will follow from similar
arguments. This particular choice implies $\sin\alpha,\sin\beta>0.$ Also note
that $\mathcal{R}_{a,b}$ is invariant under the change of variables $a\mapsto
a^{-1}$ and $b\mapsto b^{-1}.$ Therefore, we can further restrict ourselves to
$a,b>1.$ The above assumptions allow us to restate (64) as

$\max_{r\in\mathcal{R}_{a,b}}|\mathop{\mathrm{Im}}(r)|=a-a^{-1}+b-b^{-1}=2\sinh
x+2\sinh y,$

where $a=e^{x},$ $b=e^{y},$ $a-a^{-1}=2\sinh x$ and $b-b^{-1}=2\sinh y.$

Let $\theta\in[0,\pi)$ such that

$\displaystyle\left[\left(a-a^{-1}\right)\sin\alpha+\left(b-b^{-1}\right)\sin\beta\right]=(a-a^{-1}+b-b^{-1})\sin\theta,$
$\displaystyle\Longleftrightarrow$ $\displaystyle\sinh x\sin\alpha+\sinh
y\sin\beta=(\sinh x+\sinh y)\sin\theta.$

Squaring both sides and then collecting the coefficients of
$\sinh^{2}x,\sinh^{2}y$ and $\sinh x\sinh y,$ we find that

$(\sin^{2}\alpha-\sin^{2}\theta)\sinh^{2}x+(\sinh^{2}\beta-\sinh^{2}\theta)\sinh^{2}y-2\sinh
x\sinh y(\sin\alpha\sin\beta-\sin^{2}\theta)=0.$

With the help of trigonometric identities and relations among hyperbolic
functions, the above equality becomes

(65) $\displaystyle\left(\cosh x+\cosh y\right)^{2}\cos^{2}\theta-\left(\cosh
x\cos\alpha+\cosh y\cos\beta\right)^{2}$ $\displaystyle=$
$\displaystyle\sin^{2}\alpha+\sin^{2}\beta-2\sin^{2}\theta+2\cosh(x-y)\left(\sin\alpha\sin\beta-\sin^{2}\theta\right)$
$\displaystyle+4\cosh x\cosh y\sin^{2}\left(\frac{\alpha+\beta}{2}\right).$

Notice that, in order to prove the lemma, we need to find certain conditions
involving $\sinh x$ and $\sinh y$ such that the left hand side (LHS) of (65)
is at least $0.$ Indeed, if LHS is $\geq 0,$ then

$r=\left(\mathop{\mathrm{Re}}(r),\mathop{\mathrm{Im}}(r)\right)=\left(2\cosh
x\cos\alpha+2\cosh y\cos\beta,2\sinh x\sin\alpha+2\sinh y\sin\beta\right)$

satisfies (37), which is essentially what Lemma 5.4 conveys.

Since $\alpha,\beta\in[0,\pi),$ the concavity of the sine function in
$[0,\pi)$ implies that

$4\sin^{2}\left(\frac{\alpha+\beta}{2}\right)\geq\left(\sin\alpha+\sin\beta\right)^{2}.$

Then, the definition of $\sin\theta$ (in terms of $\sin\alpha,\sin\beta,\sinh
x$ and $\sinh y$), the above discussion, and a further simplification of the
right hand side (RHS) of (65) imply that

$\displaystyle\mbox{RHS}\ \geq$ $\displaystyle\sin^{2}\alpha\left[1+\cosh
x\cosh
y-\frac{\sinh^{2}x}{\sinh^{2}\left(\frac{x+y}{2}\right)}\right]+\sin^{2}\beta\left[1+\cosh
x\cosh y-\frac{\sinh^{2}y}{\sinh^{2}\left(\frac{x+y}{2}\right)}\right]$
$\displaystyle+2\sin\alpha\sin\beta\left[\cosh\left(\frac{x-y}{2}\right)+\cosh
x\cosh y-\frac{\sinh x\sinh y}{\sinh^{2}\left(\frac{x+y}{2}\right)}\right].$

Therefore, when the coefficients of $\sin^{2}\alpha,\sin^{2}\beta$ and
$\sin\alpha\sin\beta$ are all $\geq 0,$ we have that both RHS and LHS are at
least $0.$ Notice that

(66) $\left[1+\cosh x\cosh
y-\frac{\max\\{\sinh^{2}x,\sinh^{2}y\\}}{\sinh^{2}\left(\frac{x+y}{2}\right)}\right]\geq
0$

implies that the coefficient of $\sin\alpha\sin\beta$ is also $\geq 0.$
Therefore, (66) is a sufficient condition to prove the statement, as it is
essentially a restatement of the condition in the statement of the lemma when
$x\neq y,$ and this concludes the proof. ∎

Since the bounded component of $\mathbb{C}\setminus\mathcal{R}_{a,b}$ only
exists when $a\neq b,$ we can assume that $a>b>0$ to prove Lemma 5.5. The
symmetric nature of $\mathcal{R}_{a,b}$ again narrows our proof down to the
case $a>b>1.$ Since

$a+a^{-1}=2\cosh x,\ a-a^{-1}=2\sinh x,\ b+b^{-1}=2\cosh y,\ \mbox{and}\
b-b^{-1}=2\sinh y,$

we only need to consider the case $x>y>0.$

###### Proof of Lemma 5.5.

Recall that every $r\in\mathcal{R}_{a,b}$ is of the form

$r=\left(a+a^{-1}\right)\cos\alpha-\left(b+b^{-1}\right)\cos\beta+i\left[\left(a-a^{-1}\right)\sin\alpha-\left(b-b^{-1}\right)\sin\beta\right],$

where $\alpha,\beta\in[-\pi,\pi).$ The expression of $r\in\mathcal{R}_{a,b}$
above implies that

$\min_{r\in\mathcal{R}_{a,b}}|r|=\left(a+a^{-1}\right)-\left(b+b^{-1}\right).$

To prove the statement, we need to find conditions on $a$ and $b$ such that
the inner boundary of $\mathcal{R}_{a,b}$ is an ellipse with centre at the
origin, and the length of major and minor axes are
$2\left(a-a^{-1}\right)-2\left(b-b^{-1}\right)$ and
$2\left(a+a^{-1}\right)-2\left(b+b^{-1}\right),$ respectively. We also note
that the major axis is the line joining
$i\left[\left(a-a^{-1}\right)-\left(b-b^{-1}\right)\right]$ and
$-i\left[\left(a-a^{-1}\right)-\left(b-b^{-1}\right)\right],$ and the minor
axis is the line joining $\left(a+a^{-1}\right)-\left(b+b^{-1}\right)$ and
$\left(b+b^{-1}\right)-\left(a+a^{-1}\right).$ Therefore, we need to
investigate $\mathop{\mathrm{Im}}(r)$ for $r\in\mathcal{R}_{a,b},$ and find
conditions on $a$ and $b$ such that

(67) $\min_{r\in\mathcal{R}_{a,b}\cap
i\mathbb{R}}\left|\mathop{\mathrm{Im}}(r)\right|=\left(a-a^{-1}\right)-\left(b-b^{-1}\right),$

and, for $\phi\in[0,\pi),$

$\displaystyle\left(a-a^{-1}\right)\sin\alpha-\left(b-b^{-1}\right)\sin\beta=\left[\left(a-a^{-1}\right)-\left(b-b^{-1}\right)\right]\sin\phi$
(68) $\displaystyle\Rightarrow$
$\displaystyle\left|\left(a+a^{-1}\right)\cos\alpha-\left(b+b^{-1}\right)\cos\beta\right|\geq\left|\left[\left(a+a^{-1}\right)-\left(b+b^{-1}\right)\right]\cos\phi\right|.$

Note that, because of the symmetry of the region $\mathcal{R}_{a,b},$ it
suffices to consider the case when $\alpha\in[0,\pi)$ and $\beta\in[0,\pi).$

Since, $r\in\mathcal{R}_{a,b}$ implies that
$\bar{r},-\bar{r},-r\in\mathcal{R}_{a,b},$ we can restrict ourselves to the
upper half plane $\mathbb{H}=\\{z\in\mathbb{C}:\mathop{\mathrm{Im}}(z)>0\\}.$
Recall that $r\in\mathcal{R}_{a,b}$ can also be expressed as the points on the
ellipses

$E_{x,z_{\tau}}:\left|r-z_{\tau}-2\right|+\left|r-z_{\tau}+2\right|=2\left(a+a^{-1}\right)=4\cosh
x,$

where