# On the relations between the zeros of a polynomial and its Mahler measure

M. Ounaies, G. Rhin and J.-M. Sac-Épée111M. Ounaies, IRMA, France,
<EMAIL_ADDRESS>
G. Rhin, J.-M. Sac-Épée, IECL, France<EMAIL_ADDRESS>jean-
<EMAIL_ADDRESS>
###### Abstract

In this work, we are dealing with some properties relating the zeros of a
polynomial and its Mahler measure. We provide estimates on the number of real
zeros of a polynomial, lower bounds on the distance between the zeros of a
polynomial and non-zeros located on the unit circle and a lower bound on the
number of zeros of a polynomial in the disk $\\{|x-1|<1\\}$.

11footnotetext: 2000 Mathematics Subject Classification : 11R06, 11C08, 12D10

##  1 Introduction

For a polynomial $P\in\mathbb{C}[x]$ of the form

$P(x)=\displaystyle\sum_{j=0}^{d}a_{j}x^{j}=\displaystyle
a_{d}\prod_{k=1}^{d}(x-\mu_{k}),a_{d}\not=0,$

its Mahler measure [13] is given by

$M(P)=\exp\Big{(}\displaystyle\frac{1}{2\pi}\displaystyle\int_{0}^{2\pi}\log|P(e^{it})|dt\Big{)}.$

Using the Jensen’s formula [10], it can be expressed as

$M(P)=|a_{d}|\displaystyle\prod_{|\mu_{k}|\geq 1}|\mu_{k}|.$

For an algebraic number $\alpha$, its Mahler measure $M(\alpha)$ is that of
its minimal polynomial in $\mathbb{Z}[x]$. We refer the reader to [23] for a
survey of results on Mahler measure of algebraic numbers.

When $P\in\mathbb{Z}[x]$, clearly $M(P)\geq 1$ and by Kronecker’s theorem
[11], the only irreducible monic integer polynomials $P$ such that $M(P)=1$,
apart from the monomial $x$, are cyclotomic polynomials. In other words, the
only algebraic numbers $\alpha$ such that $M(\alpha)=1$ are $0$ and the roots
of unity.

In a paper from 1933 [12], Lehmer exhibited the polynomial
$P_{L}(x)=x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1$ whose Mahler measure
$M(P_{L})=1.176280\dots$ and raised the question whether there exists
$P\in\mathbb{Z}[x]$ such that $1<M(P)<M(P_{L})$.

This question is still open and the existence of a constant $\epsilon>0$ such
that all $P\in\mathbb{Z}[x]$ satisfy either $M(P)=1$ or $M(P)>1+\epsilon$ is
now known as Lehmer’s conjecture.

For irreducible noncyclotomic polynomials $P\in\mathbb{Z}[x]$ of degree $d$,
E. Dobrowolski [4] showed a lower bound of the form

$M(P)>1+C\left(\frac{\log\log d}{\log d}\right)^{3},$

and it is the best unconditional lower bound for $M(P)$ known thus far, up to
the constant $C$.

A polynomial $P$ of degree $d$ is said to be self-reciprocal if $\displaystyle
P(x)=x^{d}P\left(\frac{1}{x}\right)$. C. Smyth [22] proved an important
theorem in the positive direction of Lehmer’s conjecture:

######  Theorem 1.1.

[Smyth] If $P\in\mathbb{Z}[x]$ is not self-reciprocal and satisfies
$P(0)P(1)\not=0$, then $M(P)\geq\theta_{0}$, where $\theta_{0}=1.324717\dots$
is the real root of the polynomial $x^{3}-x-1$.

Let us also cite a theorem due to A. Schinzel [20]:

######  Theorem 1.2.

[Schinzel] If $P$ is a monic integer polynomial of degree $d$ whose all roots
are real and such that $P(0)P(1)P(-1)\not=0$, then

$M(P)\geq\left(\frac{1+\sqrt{5}}{2}\right)^{\frac{d}{2}}.$

We will denote by $E_{\theta}$ the set of monic irreducible polynomials
$P\in\mathbb{Z}[x]$ such that $1<M(P)<\theta\leq\theta_{0}$, where
$\theta_{0}$ is as in in Theorem 1.1. Many authors have focused on the
computational search for such polynomials. D. Boyd [2, 3] gave a complete list
of polynomials in $E_{1.3}$ up to degree 20. This list was extended by
Mossinghoff [15] up to degree 24, then by Flammang, Rhin and Sac-Épée [8] with
exhaustive results up to degree 40 and by Mossinghoff, Rhin and Wu [18] who
provided an exhaustive list up to degree 44.

M. Mossinghoff [17] maintains a website that collects all known polynomials in
$E_{1.3}$ of degree less than 180. New polynomials have been added to the list
available on this website, from the works of Rhin and Sac-Épée [19],
Mossinghoff, Pinner and Vaaler [16], and El Otmani, Maul, Rhin and Sac-Épée
[6].

In the present work, we are dealing with estimates about the zeros of a
polynomial with complex coefficients in relation with its Mahler measure.

Because of their importance for Lehmer’s conjecture, we are interested in
applying some of our results from Sections 4, 5 and 6 to obtain information on
the zero sets of polynomials in $E_{\theta}$.

The paper is organised as follows: Section 2 provides notations, preliminary
results and first properties of $E_{\theta}$.

Section 3 deals with lower bounds on $|\mu-\omega|$ where $P(\mu)=0$,
$|\omega|=1$, $P(\omega)\not=0$. Subsection 3.1 focusses on integer
polynomials while Subsection 3.2 deals with real or complex polynomials,
mainly self-reciprocal. The lower bounds involve either $M(P)$ or $L(P)$.

A. Dubickas [5] showed that for all $\varepsilon>0$, there exists
$D_{0}(\varepsilon)$ such that for all monic, irreducible, non-cyclotomic
polynomials $P\in\mathbb{Z}[x]$ of degree $d\geq D_{0}(\varepsilon)$ and for
all $\mu\in P^{-1}(0)$,

$|\mu-1|>\exp\left(-\left(\frac{\pi}{4}+\varepsilon\right)\sqrt{d\log d\log
M(P)}\right).$ (1.1)

Our Proposition 3.2 improves on (1.1) by substituting the constant
$\frac{\pi}{8}$ to $\frac{\pi}{4}$ in the special case where $P$ is self-
reciprocal.

For polynomials $P\in\mathbb{C}[x]$ of degree $d$, Theorem 3.3 provides the
following inequality for all $\mu\in P^{-1}(0)$:

$|\mu-1|\geq\frac{|P(1)|}{edL(P)}.$

In particular, if all the coefficients of $P$ are positive, then

$|\mu-1|\geq\frac{1}{ed}.$

In Section 4, we generalise Theorem 1.2 to polynomials in $\mathbb{C}[x]$ and
we strengthen the dependence on $P(0)$, $P(-1)$ and $P(1)$. Our Theorem 4.1
states that for monic polynomials $P\in\mathbb{C}[x]$ of degree $d$ having
$m\geq 1$ real zeros and such that $P(0)P(1)P(-1)\not=0$, the following holds:

$M(P)\geq\left(\frac{|P(1)P(-1)|^{\frac{1}{m}}+\left(4^{\frac{d}{m}}|P(0)|^{\frac{2}{m}}+|P(1)P(-1)|^{\frac{2}{m}}\right)^{\frac{1}{2}}}{2^{\frac{d}{m}}}\right)^{\frac{m}{2}}.$

In Section 5, we give upper bounds for the number of real zeros of
$\displaystyle P(x)=\sum_{j=1}^{d}a_{j}x^{j}\in\mathbb{C}[x]$ by terms
involving (besides its degree) its Mahler measure or its length $\displaystyle
L(P)=\sum_{j=1}^{d}|a_{j}|$.

A. Dubickas [5] showed that for polynomials $P\in\mathbb{Z}[x]$ of degree $d$
satisfying $P(0)P(-1)P(1)\not=0$, there is a constant $\alpha>0$ such that

$|P^{-1}(0)\cap\mathbb{R}|\leq\alpha\sqrt{d\log d\log M(P)}.$ (1.2)

Corollary 5.4 allows us to improve (1.2) by eliminating the term $\sqrt{\log
d}$ in the special case where $\log L(P)\leq\sqrt{d\log M(P)}$.

In Section 6, we focus on integer polynomials. The main result is the
following lower bound for the Mahler measure of a monic irreducible polynomial
$P\in\mathbb{Z}[x]$ of degree $d\geq D_{0}(\varepsilon)$ and satisfying $\log
M(P)=o(\frac{d}{\log d})$:

$\log M(P)\geq\left(C-\varepsilon\right)\frac{d}{K(P)^{2}\log d}\ \text{ with
}C=\frac{4}{\pi^{2}}\log^{2}\left(\frac{1+\sqrt{5}}{2}\right)$

and where we have denoted by
$K(P)=\min\left(\left|P^{-1}(0)\cap\\{|x-1|<1\\}\right|,\left|P^{-1}(0)\cap\\{|x+1|<1\\}\right|\right)$.

##  2 Notations and preliminary results

Consider a polynomial
$P(x)=\displaystyle\sum_{j=0}^{d}a_{j}x^{j}\in\mathbb{C}[x]$ of degree $d$. In
our estimates in relation with $P$, besides $M(P)$, we will use the following
quantities:

$\begin{split}&H(P)=\max_{0\leq j\leq d}|a_{j}|,\ \ \
L(P)=\sum_{j=0}^{d}|a_{j}|,\\\ &\|P\|=\sup_{|x|=1}|P(x)|,\ \ \
L_{2}(P)=\left(\sum_{j=0}^{d}|a_{j}|^{2}\right)^{\frac{1}{2}}.\end{split}$

The height $H(P)$ and the length $L(P)$ were introduced in [13] by K. Mahler
who compared them to $M(P)$, showing the inequalities

$\begin{split}&|a_{j}|\leq M(P){d\choose j}\ \ (j=0,\ldots,d),\\\ &M(P)\leq
L_{2}(P)\leq L(P)\leq 2^{d}M(P),\\\ &H(P)\leq 2^{d-1}M(P),\ \ \
M(P)\leq(d+1)^{\frac{1}{2}}H(P).\\\ \end{split}$ (2.1)

Using Cauchy-Schwarz inequality, we also have

$|P(1)|\leq\|P\|\leq L(P)\leq(d+1)^{\frac{1}{2}}L_{2}(P)$ (2.2)

and by Parseval formula,

$L_{2}(P)=\left(\frac{1}{2\pi}\int_{0}^{2\pi}|P(e^{i\theta})|^{2}d\theta\right)^{\frac{1}{2}}\leq\|P\|.$
(2.3)

Define the reciprocal of $P$ by

$P^{*}(x)=x^{d}P\left(\frac{1}{x}\right)=\sum_{j=0}^{d}a_{d-j}x^{j}.$

Clearly, $P$ is self-reciprocal iff $P=P^{*}$.

Let
$P(x)=\displaystyle\sum_{j=0}^{d}a_{j}x^{j}=\displaystyle\prod_{k=1}^{d}(x-\mu_{k})\in\mathbb{C}[x]$,
$a_{d}=1$, $a_{0}\not=0$. We will denote by

$Q(x)=a_{0}^{-1}P(x)P^{*}(x)=\prod_{k=1}^{d}(x-\mu_{k})\prod_{k=1}^{d}(x-\mu_{k}^{-1})=\sum_{j=0}^{2d}A_{j}x^{j}.$
(2.4)

Then $Q$ is monic and self-reciprocal and

$M(Q)=|a_{0}|^{-1}M(P)^{2},\ Q(1)=a_{0}^{-1}P(1)^{2},\
Q(-1)=(-1)^{d}a_{0}^{-1}P(-1)^{2}.$ (2.5)

Besides, the coefficients are given by

$A_{j}=a_{0}^{-1}\sum_{k=0}^{j}a_{k}a_{d-j+k}\ \ (j=0,\ldots,d)$

and they satisfy

$\sum_{j=0}^{d}|A_{j}|\leq|a_{0}^{-1}|\sum_{j=0}^{d}\sum_{k=0}^{j}|a_{k}||a_{d-j+k}|=|a_{0}^{-1}|\sum_{k=0}^{d}\left(|a_{k}|\sum_{j=k}^{d}|a_{j}|\right)\leq|a_{0}^{-1}|L(P)^{2}.$
(2.6)

For a set $A\subset\mathbb{C}$, we will denote by $|P^{-1}(0)\cap A|$ the
number of zeros of $P$ belonging to $A$ counted with multiplicities.

######  Proposition 2.1.

Let $r>1$ and $P$ be a monic polynomial in $\mathbb{C}[x]$. Then

$|P^{-1}(0)\cap\\{|x|>r\\}|<\frac{\log{M(P)}}{\log r}.$

###### Proof.

Put $N=|P^{-1}(0)\cap\\{|x|>r\\}|$. Then

$r^{N}<\prod_{\mu\in P^{-1}(0)\cap\\{|x|>r\\}}|\mu|\leq\prod_{\mu\in
P^{-1}(0)\cap\\{|x|\geq 1\\}}|\mu|=M(P).$

∎

For $1<\theta<\theta_{0}$, where $\theta_{0}$ is given by Theorem 1.1, let
$E_{\theta}$ be the set of integer monic irreducible polynomials $P$ such that
$1<M(P)\leq\theta$ and satisfying conditions c1 and c2:

  * c1.

$P$ is primitive, i.e., it cannot be written as $Q(x^{m})$ with $m\geq 2$,
$Q\in\mathbb{Z}[x]$,

  * c2.

$a_{j_{0}}>0$, where $j_{0}=\min\\{j\geq 1\text{ such that }a_{j}\not=0\\}$.
This condition is satisfied either by $P(x)$ or by $P(-x)$.

Conditions c1 and c2 are usually added by the authors looking for polynomials
with small Mahler measure via computational methods in order to avoid
redundancy, since

$M(P(x))=M(P(-x))\text{ and }M(P(x^{m}))=M(P(x)).$

######  Remark 2.2.

  * •

For any $P\in E_{\theta}$ and $\mu\in P^{-1}(0)$, $P$ is the minimal
polynomial of $\mu$ in $\mathbb{Z}[x]$.

  * •

By Theorem 1.1, all polynomials of $E_{\theta}$ are self-reciprocal.

  * •

Any self-reciprocal polynomial of odd degree is divisible by $(x+1)$. Thus all
polynomials in $E_{\theta}$ are of even degree.

  * •

By Kronecker’s theorem, the cyclotomic polynomials are excluded of
$E_{\theta}$ because their Mahler measure is equal to $1$.

The first properties satisfied by the zero set $\displaystyle
Z_{\theta}=\bigcup_{P\in E_{\theta}}P^{-1}(0)$ are listed in the proposition
below.

######  Proposition 2.3.

  1. 1.

All $P^{-1}(0)$, $P\in E_{\theta}$ are disjoint and they contain only simple
zeros.

  2. 2.

For all $P\in E_{\theta}$, $P^{-1}(0)$ is symmetric with regard to
$\\{|x|=1\\}$ and $\\{\Im x=0\\}$.

  3. 3.

$Z_{\theta}\subset\\{\frac{1}{\theta}\leq|x|\leq\theta\\}$.

  4. 4.

$Z_{\theta}\cap\\{\Im
x\not=0\\}\subset\\{\frac{1}{\sqrt{\theta}}\leq|x|\leq\sqrt{\theta}\\}$.

  5. 5.

For all $P\in E_{\theta}$ of degree $2n$, and for all $r>1$,

$|P^{-1}(0)\cap\\{\frac{1}{r}\leq|x|\leq r\\}|>2(n-\frac{\log\theta}{\log
r}).$

  6. 6.

$Z_{\theta}$ does not contain any root of unity.

  7. 7.

$Z_{\theta}$ does not intersect the imaginary axe.

###### Proof.

  1. 1.

This follows immediately after the uniqueness of the minimal polynomial of an
algebraic integer.

  2. 2.

This is because $P(x)=0\Leftrightarrow P(\bar{x})=0\Leftrightarrow P(1/x)=0$.

  3. 3.

Let $P\in E_{\theta}$. By symmetry of $P^{-1}(0)$ with regards to
$\\{|x|=1\\}$, it suffices to note that if $x\in P^{-1}(0)\cap\\{|x|>1\\}$
then $|x|\leq M(P)\leq\theta$.

  4. 4.

Let $P\in E_{\theta}$. By symmetry of $P^{-1}(0)$ with regards to
$\\{|x|=1\\}$ and $\\{\Im x=0\\}$, it suffices to note that if $x\in
P^{-1}(0)\cap\\{|x|>1\\}\cap\\{\Im x\not=0\\}$ then $|x|^{2}\leq
M(P)\leq\theta$.

  5. 5.

This is a consequence of Proposition 2.1 and 2.

  6. 6.

The minimal polynomials corresponding to the roots of unity are cyclotomic
polynomials, which are excluded from $E_{\theta}$.

  7. 7.

Assume that for some $y\in\mathbb{R}$ and $\displaystyle
P=\sum_{j=0}^{2n}a_{j}x^{j}\in E_{\theta}$ $(a_{2n}\not=0)$, $P(iy)=0$. Then
$\displaystyle i\Im(P(iy))=\sum_{k=0}^{n-1}a_{2k+1}(iy)^{2k+1}=0$. But at
least one of the coefficients $a_{2k+1}$ is not zero by condition c1. This
would contradict the fact that $P$ is the minimal polynomial of $iy$.

∎

For illustration purposes, the graph on the left in Figure 1 shows all known
roots belonging to $Z_{1.3}$, corresponding to polynomials of degree less than
180. In particular, one can visualize the properties of symmetry of this set.
The small Mahler measure polynomials from which these roots are derived are
listed on Michael Mossinghoff’s website [17].

Figure 1:  The known elements of $Z_{1.3}$ obtained from polynomials of degree
less than 180

##  3 Distance between zeros of a polynomial and non-zeros of modulus one

On Figure 1, we notice that the roots of the polynomials in $E_{1.3}$ seem to
stay far from the roots of unity. The results given in this section, when
specialised to polynomials in $E_{\theta}$, $\theta\leq\theta_{0}$, allow us
to estimate from below the distance between the zeros of $P\in E_{\theta}$ and
the roots of unity. Most of our results deal with polynomials which are self-
reciprocal of even degree like the ones in $E_{\theta}$.

###  3.1 Lower bounds involving the Mahler measure

For an algebraic number $\mu\not=1$ of degree $d$, Liouville inequality gives:

$|\mu-1|\geq 2^{1-d}M(\mu)^{-1}.$

We prove the following Liouville-type inequalities:

######  Proposition 3.1.

Assume that $P\in\mathbb{Z}[x]$ is monic, self-reciprocal of degree $2n$ and
does not vanish at the roots of unity. If $P(\mu)=0$ and $\omega^{m}=1$ or
$\omega^{m}=-1$ $(m\in\mathbb{N}^{*})$, then

$\text{ If }\mu\in\mathbb{R}\text{ or }|\mu|=1,\ \ |\mu-\omega|\geq
m^{-1}2^{1-n}M(P)^{-\frac{m}{2}},$ $\text{ If }\mu\notin\mathbb{R}\text{ and
}\ |\mu|\not=1,\ \ |\mu-\omega|\geq
m^{-1}2^{1-\frac{n}{2}}M(P)^{-\frac{m}{4}},$

###### Proof.

By substituting $P(-x)$ to $P(x)$, we may assume that $\omega^{m}=1$ and by
substituting $\bar{\omega}$ to $\omega$, we may assume that $\Im\mu\geq 0$. We
divide $P^{-1}(0)\cap\\{|x|\geq 1,\ \Im x\geq 0\\}$ into two sets:

$\begin{split}&\mu\in A\Leftrightarrow P(\mu)=0,\ |\mu|=1,\Im\mu>0\text{ or
}|\mu|>1,\Im\mu=0,\\\ &\mu\in B\Leftrightarrow P(\mu)=0,\
|\mu|>1,\Im\mu>0.\end{split}$

Then $P^{-1}(0)=\\{\mu,\mu^{-1},\ \mu\in
A\\}\cup\\{\mu,\mu^{-1},\bar{\mu},\bar{\mu}^{-1},\ \mu\in B\\}$ and

$\prod_{\mu\in P^{-1}(0)}|1-\mu^{m}|=M(P)^{m}\prod_{\mu\in
A}|1-\mu^{-m}|^{2}\prod_{\mu\in B}|1-\mu^{-m}|^{4}.\\\ $

$\displaystyle\prod_{\mu\in P^{-1}(0)}(1-\mu^{m})$ is a symmetric polynomial
in the zeros of $P$ thus it is an integer and as $P$ does not vanish at the
roots of unity, it is a non-zero integer. We deduce that

$\begin{split}1\leq M(P)^{\frac{m}{2}}\prod_{\mu\in
A}|1-\mu^{-m}|\prod_{\mu\in B}|1-\mu^{-m}|^{2}.\\\ \end{split}$

For all $\mu\in A\cup B$, we have $|1-\mu^{-m}|\leq 2$ thus

$\forall\mu\in A,\ \ 1\leq M(P)^{\frac{m}{2}}2^{n-1}|1-\mu^{-m}|,$
$\forall\mu\in B,\ \ 1\leq M(P)^{\frac{m}{2}}2^{n-2}|1-\mu^{-m}|^{2}.$

To obtain the desired estimate for all $\mu\in A\cup B$, we use the following

$|1-\mu^{-m}|\leq|\bar{\omega}-\mu^{-1}|\sum_{j=0}^{m-1}|\mu^{-j}|\leq
m|\bar{\omega}-\mu^{-1}|=m|\mu|^{-1}|\omega-\mu|\leq m|\omega-\mu|.$

If $|\mu|<1$, we apply the former to $\bar{\mu}^{-1}$. and we use
$|1-\bar{\mu}^{m}|\leq m|\omega-\mu|$. ∎

For integer polynomials of small Mahler measure, there are better lower bounds
due to Mignotte and Waldschmitt [14]. They showed that for all
$\varepsilon>0$, there exists $D_{0}(\varepsilon)$ such that for $d\geq
D_{0}(\varepsilon)$

$|\mu-1|>\exp\left(-(1+\varepsilon)\sqrt{d\log d\log M}\right).$

This was improved by Dubickas [5] as follows:

$|\mu-1|>\exp\left(-\left(\frac{\pi}{4}+\varepsilon\right)\sqrt{d\log d\log
M}\right).$ (3.1)

With a minor modification in Dubickas proof of Theorem 1 in [5], we obtain a
better result than (3.1) in the special case when $P$ is self-reciprocal. This
is displayed in the next proposition.

######  Proposition 3.2.

Assume that $P\in\mathbb{Z}[x]$ is monic, irreducible, self-reciprocal of
degree $d$ and satisfies $M(P)>1$. Let $P(\mu)=0$ and $\omega^{m}=1$ or
$\omega^{m}=-1$ $(m\in\mathbb{N}^{*})$. Then for all $\varepsilon>0$, there
exists $D_{0}(\varepsilon)$ such that for $d\geq D_{0}(\varepsilon)$

$|\omega-\mu|\geq\exp\left(-\left(\frac{\pi\sqrt{m}}{8}+\varepsilon\right)\sqrt{d\log
d\log M(P)}\right).$

If $\mu\notin\mathbb{R}$ and $|\mu|\not=1$, then

$|\omega-\mu|\geq\exp\left(-\left(\frac{\pi\sqrt{m}}{16}+\varepsilon\right)\sqrt{d\log
d\log M(P)}\right).$

###  3.2 Lower bounds involving the coefficients

######  Theorem 3.3.

Let $P\in\mathbb{C}[x]$ be of degree $d\in\mathbb{N}^{*}$. Let $\mu$ be a zero
of $P$ with multiplicity $m_{\mu}$ and $|\omega|=1$. Then,

$|\mu-\omega|\geq
d^{-1}\left(e^{-1}\frac{|P(\omega)|}{L(P)}\right)^{\frac{1}{m_{\mu}}}.$

In particular, if all the coefficients of $P$ are positive, then

$|\mu-1|\geq d^{-1}e^{-\frac{1}{m_{\mu}}}.$

If $|P(\omega)|=\|P\|$, then

$|\mu-\omega|\geq
d^{-1}\left(e^{-1}(d+1)^{-\frac{1}{2}}\right)^{\frac{1}{m_{\mu}}}.$

###### Proof.

We may assume $P(\omega)\not=0$. For all $\theta\in[0,2\pi]$,

$|P(\omega+d^{-1}e^{i\theta})|\leq(1+d^{-1})^{d}L(P)\leq eL(P).$

Let $\mu_{k}$ be the distinct zeros of $P$ with respective multiplicities
$m_{k}$. Applying Jensen’s formula to $\frac{P(x)}{P(\omega)}$ in the disk
$\\{|x-\omega|<d^{-1}\\}$, we find:

$\sum_{|\mu_{k}-\omega|\leq
d^{-1}}m_{k}\log\frac{d^{-1}}{|\mu_{k}-\omega|}=\frac{1}{2\pi}\int_{0}^{2\pi}\log\left|\frac{P(\omega+d^{-1}e^{i\theta})}{P(\omega)}\right|d\theta\leq\log\frac{eL(P)}{|P(\omega)|}.$

For each $k$, either $|\mu_{k}-\omega|\geq d^{-1}\geq
d^{-1}\frac{|P(\omega)|}{eL(P)}$ or, using the estimate above,

$\left(d|\mu_{k}-\omega|\right)^{m_{k}}\geq\frac{|P(\omega)|}{eL(P)}.$

In the case where all the coefficients of $P$ are positive, $P(1)=L(P)$.

In the case where $P(\omega)=\|P\|$, we use the inequalities (2.2) and (2.3):

$L(P)\leq(d+1)^{\frac{1}{2}}\|P\|=(d+1)^{\frac{1}{2}}|P(\omega)|.$

∎

The rest of this subsection deals with self-reciprocal polynomials. As any
self-reciprocal polynomial of odd degree is the product of $(x+1)$ and a self-
reciprocal polynomial of even degree, we will only consider polynomials of
degree $d=2n$.

######  Lemma 3.4.

Let $P(x)=\displaystyle\sum_{j=0}^{2n}a_{j}x^{j}$ be a self-reciprocal
polynomial in $\mathbb{C}[x]$ of degree $2n$. Let $|\omega|=1$,
$P(\omega)\not=0$ and $\rho\in(0,1)$. Then

$\begin{split}\sum_{\mu\in
P^{-1}(0),|\mu-\omega|\leq\rho}\log\frac{\rho}{|\mu-\omega|}\leq\log\left(1+\frac{\rho(1+(1-\rho)^{-n})}{|P(\omega)|}\sum_{j=0}^{n-1}(n-j)|a_{j}|\right).\end{split}$
(3.2)

If $\omega\in\\{-1,1\\}$,

$\begin{split}\sum_{\mu\in
P^{-1}(0),|\mu-\omega|\leq\rho}\log\frac{\rho}{|\mu-\omega|}\leq\log\left(1+\frac{\rho^{2}(1-\rho)^{-n}}{|P(\omega)|}\sum_{j=0}^{n-1}(n-j)^{2}|a_{j}|\right).\end{split}$
(3.3)

###### Proof.

Jensen’s formula applied to the function $\frac{x^{-n}P(x)}{P(\omega)}$ in the
disk $\\{|x-\omega|<\rho\\}$ gives

$\sum_{\mu\in
P^{-1}(0),|\mu-\omega|\leq\rho}\log\frac{\rho}{|\mu-\omega|}=\frac{1}{2\pi}\int_{0}^{2\pi}\log\left|\frac{(\omega+\rho
e^{i\theta})^{-n}P(\omega+\rho e^{i\theta})}{P(\omega)}\right|d\theta.$ (3.4)

Let $x$ be a non-zero complex number. A straightforward computation shows that

$\begin{split}x^{-n}P(x)&=\bar{\omega}^{n}P(\omega)-\sum_{j=0}^{n-1}a_{j}{\bar{\omega}}^{n-j}(x^{n-j}-\omega^{n-j})(x^{j-n}-\omega^{n-j}).\end{split}$

We use

$\begin{split}\left|(x^{n-j}-\omega^{n-j})(x^{j-n}-\omega^{n-j})\right|&=|x-\omega||x^{j-n}-\omega^{n-j}|\left|\sum_{k=0}^{n-j-1}x^{k}\omega^{-1-k}\right|\\\
&=|x-\omega|\left|\sum_{k=0}^{n-j-1}(x^{k+j-n}\omega^{-1-k}-x^{k}\omega^{n-j-1-k})\right|\\\
&\leq|x-\omega|(n-j)(1+\max(|x|^{-n},|x|^{n})\end{split}$

and we readily obtain

$\left|\frac{x^{-n}P(x)}{P(\omega)}\right|\leq
1+\frac{|x-\omega|(1+\max(|x|^{-n},|x|^{n})}{|P(\omega)|}\sum_{j=0}^{n-1}(n-j)|a_{j}|.$
(3.5)

(3.2) follows from (3.4), (3.5) and the fact that for all $\theta\in[0,2\pi]$,

$1-\rho\leq|\omega+\rho e^{i\theta}|\leq 1+\rho\leq(1-\rho)^{-1}.$ (3.6)

In the case where $\omega\in\\{1,-1\\}$, we write

$\begin{split}\left|(x^{n-j}-\omega^{n-j})(x^{j-n}-\omega^{n-j})\right|&=|x|^{-1}|x-\omega|^{2}\left|\sum_{k=0}^{n-j-1}x^{k}\omega^{-1-k}\right|\left|\sum_{k=0}^{n-j-1}x^{-k}\omega^{-1-k}\right|\\\
&\leq|x-\omega|^{2}(n-j)^{2}\max(|x|^{-n},|x|^{n})\end{split}$

which gives the inequality

$\left|\frac{x^{-n}P(x)}{P(\omega)}\right|\leq
1+\frac{|x-\omega|^{2}\max(|x|^{n},|x|^{-n})}{|P(\omega)|}\sum_{j=0}^{n-1}(n-j)^{2}|a_{j}|.$
(3.7)

(3.3) follows from (3.4), (3.6) and (3.7). ∎

######  Theorem 3.5.

Let $P(x)=\displaystyle\sum_{j=0}^{2n}a_{j}x^{j}$ be a self-reciprocal
polynomial in $\mathbb{R}[x]$ of degree $2n$. Assume $P(\mu)=0$, $|\omega|=1$,
$P(\omega)\not=0$ and $\delta>1$. Then

$\begin{split}\frac{1}{|\mu-\omega|}\leq\frac{n}{\log\delta}+1+\frac{1+\delta}{|P(\omega)|}\sum_{j=0}^{n-1}(n-j)|a_{j}|,\\\
\end{split}$ (3.8)

Besides, if $|\mu|\not=1$ then

$\frac{|\mu|}{|\mu-\omega|^{2}}\leq(\frac{n}{\log\delta}+1)^{2}+(\frac{n}{\log\delta}+1)\frac{1+\delta}{|P(\omega)|}\sum_{j=0}^{n-1}(n-j)|a_{j}|.$
(3.9)

For $\omega\in\\{-1,1\\}$,

$\frac{|\mu|}{|\mu-\omega|^{2}}\leq(\frac{n}{\log\delta}+1)^{2}+\frac{\delta}{|P(\omega)|}\sum_{j=0}^{n-1}(n-j)^{2}|a_{j}|,$
(3.10)

Besides, if $\mu\notin\mathbb{R}$ and $|\mu|\not=1$ then

$\frac{|\mu|^{2}}{|\mu-\omega|^{4}}\leq(\frac{n}{\log\delta}+1)^{4}+(\frac{n}{\log\delta}+1)^{2}\frac{\delta}{|P(\omega)|}\sum_{j=0}^{n-1}(n-j)^{2}|a_{j}|.$
(3.11)

###### Proof.

We apply Lemma 3.4 with $\rho=\frac{\log\delta}{n+\log\delta}$. Let $\mu\in
P^{-1}(0)$ and $|\omega|=1$, $P(\omega)\not=0$. We may assume that
$|\mu-\omega|\leq\rho$, otherwise the inequalities are trivial. Then, applying
(3.2),

$\begin{split}\frac{\rho}{|\mu-\omega|}&\leq\prod_{\underset{|\alpha-\omega|\leq\rho}{\alpha\in
P^{-1}(0)}}\frac{\rho}{|\alpha-\omega|}\leq
1+\frac{\rho(1+(1-\rho)^{-n})}{|P(\omega)|}\sum_{j=0}^{n-1}(n-j)|a_{j}|.\end{split}$

Now we use the inequality
$(1-\rho)^{-n}=\left(1+\frac{\log\delta}{n}\right)^{n}\leq\delta$ and we
obtain (3.8). In the rest of this proof, since $P(\bar{\mu}^{-1})=0$ and
$\frac{|\mu|}{|\mu-\omega|^{2}}$ is invariant by substituting $\bar{\mu}^{-1}$
to $\mu$, we may assume that $|\mu|\geq 1$.

If $|\mu|>1$, then $\bar{\mu}^{-1}\not=\mu$ and

$|\bar{\mu}^{-1}-\omega|=\frac{|\mu-\omega|}{|\mu|}\leq|\mu-\omega|\leq\rho.$

Thus

$\begin{split}\frac{\rho^{2}|\mu|}{|\mu-\omega|^{2}}&\leq\prod_{\underset{|\alpha-\omega|\leq\rho}{\alpha\in
P^{-1}(0)}}\frac{\rho}{|\alpha-\omega|}\leq
1+\frac{\rho(1+\delta)}{|P(\omega)|}\sum_{j=0}^{n-1}(n-j)|a_{j}|\end{split}$

and this proves (3.9).

Let us assume now that $\omega\in\\{-1,1\\}$. Since $P(\mu^{-1})=0$ and
$|\mu^{-1}-\omega|\leq|\mu-\omega|\leq\rho$, applying (3.3) we obtain

$\begin{split}\frac{\rho^{2}|\mu|}{|\mu-\omega|^{2}}\leq\prod_{\underset{|\alpha-\omega|\leq\rho}{\alpha\in
P^{-1}(0)}}\frac{\rho}{|\alpha-\omega|}\leq
1+\frac{\rho^{2}\delta}{|P(\omega)|}\sum_{j=0}^{n-1}(n-j)^{2}|a_{j}|,\end{split}$

which proves (3.10).

Finally, to prove (3.11), we note that if $|\mu|>1$ and $\mu\notin\mathbb{R}$,
then $\mu$, $\bar{\mu}$, $\mu^{-1}$ and $\bar{\mu}^{-1}$ are distinct zeros of
$P$ and all of them are located in the disk $\\{|x-\omega|\leq\rho\\}$. Thus

$\begin{split}\frac{\rho^{4}|\mu|^{2}}{|\mu-\omega|^{4}}&\leq\prod_{\underset{|\alpha-\omega|\leq\rho}{\alpha\in
P^{-1}(0)}}\frac{\rho}{|\alpha-\omega|}\leq
1+\frac{\rho^{2}\delta}{|P(\omega)|}\sum_{j=0}^{n-1}(n-j)^{2}|a_{j}|.\end{split}$

∎

In the remainder of this section, we will use the following notation: for a
polynomial $P$ of degree $2n$, we will write
$f(P)\gtrsim_{n\rightarrow\infty}g(n)$ if there exists an expression $h(n)$
only depending on $n$ such that $f(P)\geq h(n)$ and
$\frac{h(n)}{g(n)}\rightarrow_{n\rightarrow\infty}1$.

######  Corollary 3.6.

Let $P(x)=\displaystyle\sum_{j=0}^{2n}a_{j}x^{j}$ be a self-reciprocal
polynomial in $\mathbb{R}[x]$ of degree $2n$. Assume $H(P)\leq H$. Let
$P(\mu)=0$, $|\omega|=1$, $|P(\omega)|\geq 1$. Then

$|\mu-\omega|\gtrsim_{n\rightarrow\infty}H^{-1}n^{-2}.$

Besides, if $|\mu|\not=1$ then

$|\mu|^{-\frac{1}{2}}|\mu-\omega|\gtrsim_{n\rightarrow\infty}\left(2c^{-1}H^{-1}\right)^{\frac{1}{2}}n^{-\frac{3}{2}},$

where $c>1$ is such that $c+1=c\log c\ \ (c=3.594...)$.

For $\omega\in\\{-1,1\\}$:

$|\mu|^{-\frac{1}{2}}|\mu-\omega|\gtrsim_{n\rightarrow\infty}\left(3H^{-1}\right)^{\frac{1}{2}}n^{-\frac{3}{2}}.$

If $|\mu|\not=1$ and $\mu\notin\mathbb{R}$, then

$|\mu|^{-\frac{1}{2}}|\mu-\omega|\gtrsim_{n\rightarrow\infty}(2e^{-1})^{\frac{1}{2}}\left(3H^{-1}\right)^{\frac{1}{4}}n^{-\frac{5}{4}}.$

###### Proof.

We observe that

$\sum_{j=0}^{n-1}(n-j)|a_{j}|\leq\frac{n(n+1)}{2}H,$
$\sum_{j=0}^{n-1}(n-j)^{2}|a_{j}|\leq\frac{n(n+1)(2n+1)}{6}H.$

Then we apply Theorem 3.5 with different values of $\delta$ in order to get
the best upper bound: $\delta=1+\frac{1}{\sqrt{n}}$ in (3.8), $c$ in (3.9),
$\delta=1+\frac{1}{n^{\frac{1}{4}}}$ in (3.10) and finally $\delta=e^{2}$ in
(3.11). ∎

######  Corollary 3.7.

Let $P(x)=\displaystyle\sum_{j=0}^{2n}a_{j}x^{j}$ be a self-reciprocal
polynomial in $\mathbb{R}[x]$ of degree $2n$. Assume $a_{j}\geq 0$
$(j=0,\ldots,n)$. Let $P(\mu)=0$. Then

$|\mu|^{-\frac{1}{2}}|\mu-1|\gtrsim_{n\rightarrow\infty}\frac{A}{n}.$

Besides, if $|\mu|\not=1$ and $\mu\notin\mathbb{R}$, then

$|\mu|^{-\frac{1}{2}}|\mu-1|\gtrsim_{n\rightarrow\infty}\frac{B}{n},$

where $A$ and $B$ are given by

$A^{2}=4\left(a(2+\log a)\right)^{-1},a>1,a(\log a)^{3}=4\ \ (A=0.655...).$
$B^{4}=8\log b^{2}\left(b(\log b+2)\right)^{-1},b>1,b(\log b)^{2}(\log b-2)=8\
\ (B=0.984...).$

###### Proof.

$\sum_{j=0}^{n-1}(n-j)^{2}|a_{j}|\leq\frac{n^{2}}{2}L(P)$

and $L(P)=P(1)=\|P\|$. Then we apply Theorem 3.5 with $\delta=a$ in (3.10) and
with $\delta=b$ in (3.11). ∎

######  Corollary 3.8.

Let $P(x)=\displaystyle\sum_{j=0}^{2n}a_{j}x^{j}$ be a self-reciprocal
polynomial in $\mathbb{R}[x]$ of degree $2n$. Let $P(\mu)=0$, $|\omega|=1$,
$\|P\|=|P(\omega)|$. Then

$|\mu-\omega|\gtrsim_{n\rightarrow\infty}3^{\frac{1}{2}}2^{-\frac{1}{2}}n^{-\frac{3}{2}}.$

Besides, if $|\mu|\not=1$, then

$|\mu|^{-\frac{1}{2}}|\mu-\omega|\gtrsim_{n\rightarrow\infty}6^{\frac{1}{4}}c^{-\frac{1}{2}}n^{-\frac{5}{4}},$

where $c>1$ is such that $c+1=c\log c\ \ (c=3.594...)$. For
$\omega\in\\{-1,1\\}$:

$|\mu|^{-\frac{1}{2}}|\mu-\omega|\gtrsim_{n\rightarrow\infty}10^{\frac{1}{4}}n^{-\frac{5}{4}}.$

If $|\mu|\not=1$ and $\mu\notin\mathbb{R}$, then

$|\mu|^{-\frac{1}{2}}|\mu-\omega|\gtrsim_{n\rightarrow\infty}\left(2e^{-1}\right)^{\frac{1}{2}}10^{\frac{1}{8}}n^{-\frac{9}{8}}.$

###### Proof.

By Cauchy-Schwarz inequality,

$\sum_{j=0}^{n-1}(n-j)|a_{j}|\leq\frac{1}{\sqrt{1}2}\left(n(n+1)(2n+1)\right)^{\frac{1}{2}}L_{2}(P),$
$\sum_{j=0}^{n-1}(n-j)^{2}|a_{j}|\leq\frac{1}{\sqrt{60}}\left(n(n+1)(2n+1)(3n^{2}+3n-1)\right)^{\frac{1}{2}}L_{2}(P)$

and $L_{2}(P)\leq\|P\|=|P(\omega)|$. Then we apply Theorem 3.5 with
$\delta=1+\frac{1}{n^{\frac{1}{4}}}$ in (3.8), $c>1$ in (3.9),
$\delta=1+\frac{1}{n^{\frac{1}{8}}}$ in (3.10) and finally $\delta=e^{2}$ in
(3.11). ∎

##  4 A lower bound for the Mahler measure of complex polynomials

A result due to Schinzel [20] states that any monic polynomial
$P\in\mathbb{Z}[x]$ of degree $d$ whose all zeros are real (resp. positive)
and such that $P(-1)P(1)\not=0$ and $|P(0)|=1$ satisfies

$M(P)\geq\left(\frac{1+\sqrt{5}}{2}\right)^{\frac{d}{2}}\ (\text{resp.
}M(P)\geq\left(\frac{1+\sqrt{5}}{2}\right)^{d}).$

V. Flammang [7] generalised Schinzel’s theorem by proving that any monic
$P\in\mathbb{R}[x]$ of degree $d$ whose all zeros are real (resp. positive)
and such that $P(0)\not=0$, $|P(1)|\geq 1$, $|P(-1)|\geq 1$ satisfies

$M(P)\geq\left(\frac{1+\left(4|P(0)|^{\frac{2}{d}}+1\right)^{\frac{1}{2}}}{2}\right)^{\frac{d}{2}}\
(\text{resp.
}M(P)\geq\left(\frac{1+\left(4|P(0)|^{\frac{1}{d}}+1\right)^{\frac{1}{2}}}{2}\right)^{d}).$

J. Garza [9] showed that for monic polynomials $P\in\mathbb{Z}[x]$ satisfying
$P(0)P(-1)P(1)\not=0$ and having $m\geq 1$ real zeros, the following holds:

$M(P)\geq\left(\frac{1+\left(4^{\frac{d}{m}}+1\right)^{\frac{1}{2}}}{2^{\frac{d}{m}}}\right)^{\frac{m}{2}}.$

Our next theorem generalises the three above results to polynomials in
$\mathbb{C}[x]$. It also strengthens the dependence on $P(0)$, $P(1)$ and
$P(-1)$.

######  Theorem 4.1.

Let $P\in\mathbb{C}[x]$ be a monic polynomial of degree $d\geq 2$. Assume that
$P$ has $m\geq 1$ real zeros and that $P(0)P(1)P(-1)\not=0$, then

$M(P)\geq\left(\frac{|P(1)P(-1)|^{\frac{1}{m}}+\left(4^{\frac{d}{m}}|P(0)|^{\frac{2}{m}}+|P(1)P(-1)|^{\frac{2}{m}}\right)^{\frac{1}{2}}}{2^{\frac{d}{m}}}\right)^{\frac{m}{2}}$
(4.1)

with equality if and only if $P$ is of the form

$P=(z-i)^{d_{1}}(z+i)^{d_{2}}(z-a)^{d_{3}}(z+a)^{d_{4}}(z-a^{-1})^{d_{5}}(z+a^{-1})^{d_{6}},\
\ a>1.$ (4.2)

If $P$ has $n\geq 1$ positive zeros and $P(0)P(1)\not=0$, then

$M(P)\geq\left(\frac{|P(1)|^{\frac{1}{n}}+\left(4^{\frac{d}{n}}|P(0)|^{\frac{1}{n}}+|P(1)|^{\frac{2}{n}}\right)^{\frac{1}{2}}}{2^{\frac{d}{n}}}\right)^{n}$
(4.3)

with equality if and only if $P$ is of the form

$P=(z-a)^{d_{1}}(z-a^{-1})^{d_{2}},\ \ a>1.$

###### Proof.

Let $Q=P(0)^{-1}PP^{*}$ defined by (2.4). Let $\\{\beta_{k}\\}_{k=1}^{m}$ be
the real zeros of $Q$ outside $[-1,1]$ and $\\{\delta_{k}\\}_{k=1}^{f}$ its
non-real zeros such that either $|\delta_{k}|>1$ or $|\delta_{k}|=1$,
$\Im\delta_{k}>0$. Then $d=m+f$ and

$\begin{split}Q(1)&=\prod_{k=1}^{f}(1-\delta_{k})(1-\delta_{k}^{-1})\prod_{k=1}^{m}(1-\beta_{k})(1-\beta_{k}^{-1})\\\
Q(-1)&=\prod_{k=1}^{f}(1+\delta_{k})(1+\delta_{k}^{-1})\prod_{k=1}^{m}(1+\beta_{k})(1+\beta_{k}^{-1})\\\
|Q(1)Q(-1)|^{\frac{1}{2}}&=M(Q)\prod_{k=1}^{f}|1-\delta_{k}^{-2}|\prod_{k=1}^{m}(1-\beta_{k}^{-2})\\\
\end{split}$

Like in the proof of Schinzel’s theorem, we use the inequality [20][Lemma 3]

$\prod_{k=1}^{m}(1-\beta_{k}^{-2})\leq\left(1-\left(\prod_{k=1}^{m}\beta_{k}^{-2}\right)^{\frac{1}{m}}\right)^{m}\leq\left(1-M(Q)^{-\frac{2}{m}}\right)^{m},$

which follows from the concavity of the function $t\mapsto\log(1-e^{-t}),t>0$.
We obtain

$\begin{split}|Q(1)Q(-1)|^{\frac{1}{2}}\leq
M(Q)2^{d-m}\left(1-M(Q)^{-\frac{2}{m}}\right)^{m}.\end{split}$ (4.4)

We deduce that

$M(Q)^{\frac{2}{m}}-2^{1-\frac{d}{m}}|Q(1)Q(-1)|^{\frac{1}{2m}}M(Q)^{\frac{1}{m}}-1\geq
0$

and solving the inequality
$t^{2}-2^{1-\frac{d}{m}}|Q(1)Q(-1)|^{\frac{1}{2m}}t-1\geq 0$, we find

$2^{\frac{d}{m}}M(Q)^{\frac{1}{m}}\geq|Q(1)Q(-1)|^{\frac{1}{2m}}+\left(4^{\frac{d}{m}}+|Q(1)Q(-1)|^{\frac{1}{m}}\right)^{\frac{1}{2}}.$

We conclude the proof of (4.1) by using (2.5).

Equality in (4.1) is equivalent to equality in (4.4). It is attained iff

$\prod_{k=1}^{f}\frac{2}{|1-\delta_{k}^{-2}|}=\left(1-M(Q)^{-\frac{2}{m}}\right)^{-m}\prod_{k=1}^{m}(1-\beta_{k}^{-2})=1,$

which gives

$\delta_{1}^{2}=\ldots=\delta_{f}^{2}=-1,\ \ \
\beta_{1}^{2}=\ldots=\beta_{m}^{2}=a^{2}\text{ with }a>1.$

For the proof of (4.3), let $\\{\beta_{k}\\}_{k=1}^{n}$ be the positive zeros
of $Q$ and $\\{\beta_{k}\\}_{k=n+1}^{m}$ be the negative zeros of $Q$. Using
once again the concavity of the function $t\mapsto\log(1-e^{-t}),t>0$, we have

$\begin{split}|Q(1)|^{\frac{1}{2}}&=M(Q)^{\frac{1}{2}}\prod_{k=1}^{f}|1-\delta_{k}^{-1}|\prod_{k=1}^{n}(1-\beta_{k}^{-1})\prod_{k=n+1}^{m}(1-\beta_{k}^{-1})\\\
&\leq M(Q)^{\frac{1}{2}}2^{d-n}\prod_{k=1}^{n}(1-\beta_{k}^{-1})\\\ &\leq
M(Q)^{\frac{1}{2}}2^{d-n}\left(1-M(Q)^{-\frac{1}{n}}\right)^{n}.\end{split}$
(4.5)

We find

$M(Q)^{\frac{1}{n}}-2^{1-\frac{d}{n}}|Q(1)|^{\frac{1}{2n}}M(Q)^{\frac{1}{2n}}-1\geq
0.$

We proceed as in the end of the proof of (4.1) by substituting
$M(Q)^{\frac{1}{2n}}$ to $M(Q)^{\frac{1}{m}}$ and $|Q(1)|^{{\frac{1}{2n}}}$ to
$|Q(1)Q(-1)|^{\frac{1}{2m}}$.

Equality in (4.3) is equivalent to equalities in (4.5). It is attained iff

$\prod_{k=1}^{f}\frac{2}{|1-\delta_{k}^{-1}|}\prod_{k=n+1}^{m}\frac{2}{|1-\beta_{k}^{-1}|}=\left(1-M(Q)^{-\frac{1}{n}}\right)^{-n}\prod_{k=1}^{m}(1-\beta_{k}^{-1})=1,$

which gives

$f=0,\ \ \ n=m=d,\ \ \ \beta_{1}=\ldots=\beta_{n}=a\text{ with }a>1.$

∎

##  5 The number of real zeros

###  5.1 Bounds involving the Mahler measure

A. Dubickas [5] showed that for polynomials $P\in\mathbb{Z}[x]$ of degree $d$
such that $P(0)P(-1)P(1)\not=0$, there is a constant $D_{0}$ such that for all
$d\geq D_{0}$,

$|P^{-1}(0)\cap\mathbb{R}|\leq 1.085\sqrt{d\log d\log M(P)}.$ (5.1)

A bound of this size does not apply to polynomials in $\mathbb{R}[x]$. For
example, the polynomials of the form $P=\left(x-d^{\frac{1}{d}}\right)^{d}$
satisfy $M(P)=d$,

$\sqrt{d\log d\log M(P)}=\sqrt{d}\log d\text{ \ \ while }\ \
|P^{-1}(0)\cap\mathbb{R}|=d=\frac{d\log M(P)}{\log d}\ .$

Our next theorem shows that the bound $\displaystyle\frac{d\log(2M(P))}{\log
d}$ holds, up to a small multiplicative constant, for polynomials in
$P\in\mathbb{C}[x]$ such that $|P(1)P(-1)|\geq 1$ and $|P(0)|\geq 1$.

######  Theorem 5.1.

Let $P\in\mathbb{C}[x]$ be a monic polynomial of degree $d\geq 2$. Assume
$P(0)P(1)P(-1)\not=0$. Then

$\begin{split}|P^{-1}(0)\cap\mathbb{R}|&\leq\max\left(\frac{d\log
2+\log|P(0)|-\log\left|P(1)P(-1)\right|}{-\log\left(\sinh\left(\frac{\log
d}{d}\right)\right)},\frac{d\log\left(M(P)^{2}|P(0)|^{-1}\right)}{\log
d}\right).\\\ \end{split}$ (5.2)

We note that $\displaystyle-\log\left(\sinh\left(\frac{\log
d}{d}\right)\right)\sim\log d$.

######  Remark 5.2.

The equality in (5.2) is attained by polynomials of the form (4.2) with
$a=d^{\frac{1}{d}}$. If we denote by $m=|P^{-1}(0)\cap\mathbb{R}|$, they
satisfy

$M(P)^{2}|P(0)|^{-1}=d^{\frac{m}{d}},\ \
|P(0)|^{-1}|P(1)P(-1)|=2^{d}\left(\sinh\left(\frac{\log
d}{d}\right)\right)^{m}.$

Thus

$\frac{d\log
2+\log|P(0)|-\log\left|P(1)P(-1)\right|}{-\log\left(\sinh\left(\frac{\log
d}{d}\right)\right)}=\frac{d\log\left(M(P)^{2}|P(0)|^{-1}\right)}{\log d}=m.$

###### Proof of Theorem 5.1.

Let us repeat the beginning of the proof of Theorem 4.1 with the same
notations. We assume that $\displaystyle m\geq\frac{d\log M(Q)}{\log d}$ and
we use inequality (4.4):

$\begin{split}|Q(1)Q(-1)|^{\frac{1}{2}}&\leq
M(Q)2^{d-m}\left(1-M(Q)^{-\frac{2}{m}}\right)^{m}\\\
&=2^{d}\left(\sinh\left(\frac{\log M(Q)}{m}\right)\right)^{m}\\\ &\leq
2^{d}\left(\sinh\left(\frac{\log d}{d}\right)\right)^{m}\\\ \end{split}$

We deduce that

$m\leq\frac{d\log
2-\frac{1}{2}\log\left|Q(1)Q(-1)\right|}{-\log\left(\sinh\left(\frac{\log
d}{d}\right)\right)}.$

The result follows by using (2.5).

∎

###  5.2 Bounds involving the Mahler measure and the length

A theorem announced by E. Schmidt in 1932 states that for polynomials of the
form $P(x)=\displaystyle\sum_{j=0}^{d}a_{j}x^{j}$, $a_{0}a_{d}\not=0$,
$a_{j}\in\mathbb{C}$, there is an absolute constant $C>0$ such that

$|P^{-1}(0)\cap\mathbb{R}|^{2}\leq
C\,d\log\left(\frac{L(P)}{\sqrt{|a_{0}a_{d}|}}\right).$ (5.3)

I. Schur [21] gave an elementary proof of (5.3) and showed that the best
possible constant $C$ is $4$.

P. Borwein, T. Erdélyi and G. Kós [1] showed that whenever
$|a_{0}|=|a_{d}|=1$, $|a_{j}|\leq 1$, there is an absolute constant $C>0$ such
that

$|P^{-1}(0)\cap\mathbb{R}|\leq C\sqrt{d}.$

Our next result gives an upper bound depending both on the Mahler measure and
on the length of the polynomial.

######  Theorem 5.3.

Let $P\in\mathbb{C}[x]$ be a monic polynomial of degree $d$. Assume
$P(0)P(1)P(-1)\not=0$. Then

$\begin{split}|P^{-1}(0)\cap\mathbb{R}|&\leq
2\sqrt{c}\left(d\log\frac{M(P)^{2}}{|P(0)|}\right)^{\frac{1}{2}}+(c+1)\log\frac{M(P)^{2}}{|P(0)|}\\\
&+(\log c)^{-1}\log(1+d^{2})+(\log
c)^{-1}\log\left(\frac{L(P)^{2}}{2|P(1)|^{2}}+\frac{L(P)^{2}}{2|P(-1)|^{2}}\right),\\\
\end{split}$ (5.4)

where $c>1$ is such that $c\log c=1+c\ \ (c=3.594...)$.

###### Proof.

Once again we use $Q=P(0)^{-1}PP^{*}$ and we may assume that $M(Q)>1$
otherwise there are no real zeros at all. We apply Lemma 3.4 to $Q$ with
$\rho=\frac{\gamma}{\gamma+1}$, $\gamma=\sqrt{\frac{(c+1)\log c\log M(Q)}{d}}$
and first with $\omega=1$:

$\begin{split}2\log c\,|Q^{-1}(0)\cap\\{1<x\leq
1+\frac{\rho}{c}\\}|&\leq\sum_{\underset{1<\mu\leq
1+\frac{\rho}{c}}{Q(\mu)=0}}\left(\log\frac{\rho}{|\mu-1|}+\log\frac{\rho}{|\mu^{-1}-1|}\right)\\\
&\leq\sum_{\underset{|\mu-1|\leq\rho}{Q(\mu)=0}}\log\frac{\rho}{|\mu-1|}\\\
&\leq\log\left(1+\frac{\rho^{2}(1-\rho)^{-d}d^{2}}{|Q(1)|}\sum_{k=0}^{d-1}|A_{k}|\right).\end{split}$
(5.5)

Repeating the argument with $\omega=-1$,

$2\log c\,|Q^{-1}(0)\cap\\{-1-\frac{\rho}{c}\leq
x<-1\\}|\leq\log\left(1+\frac{\rho^{2}(1-\rho)^{-d}d^{2}}{|Q(-1)|}\sum_{k=0}^{d-1}|A_{k}|\right).$
(5.6)

Adding (5.5) and (5.6) and using the inequality
$(1+u)(1+v)\leq\left(1+\frac{u+v}{2}\right)^{2}$,

$\begin{split}\log c\,|Q^{-1}(0)\cap\\{1<|x|\leq
1+\frac{\rho}{c}\\}|&\leq\log\left(1+d^{2}\rho^{2}(1-\rho)^{-d}\left(\frac{1}{2|Q(1)|}+\frac{1}{2|Q(-1)|}\right)\sum_{k=0}^{d-1}|A_{k}|\right)\\\
&\leq\log\left(\left(\frac{1}{2|Q(1)|}+\frac{1}{2|Q(-1)|}\right)\sum_{k=0}^{d}|A_{k}|\right)+\log(1+d^{2}\rho^{2})-d\log(1-\rho)\\\
\end{split}$

Now we use the inequality $-\log(1-\rho)\leq\frac{\rho}{1-\rho}=\gamma$, (2.5)
and (2.6) to obtain

$\log c|Q^{-1}(0)\cap\\{1<|x|\leq
1+\frac{\rho}{c}\\}|\leq\log\left(\frac{L(P)^{2}}{2|P(1)|^{2}}+\frac{L(P)^{2}}{2|P(-1)|^{2}}\right)+\log(1+d^{2})+\gamma
d.$

On the other hand, by Proposition 2.1 and the inequality
$\log(1+\frac{\rho}{c})\geq\frac{\rho}{c+1}$,

$|Q^{-1}(0)\cap\\{|x|>1+\frac{\rho}{c}\\}|\leq\frac{\log
M(Q)}{\log(1+\frac{\rho}{c})}\leq\log M(Q)(c+1)\rho^{-1}=\log
M(Q)(c+1)(1+\gamma^{-1})$ $\leq(c+1)\log M(Q)+\sqrt{c\log M(Q)d}.$

Finally,

$\begin{split}|P^{-1}(0)\cap\mathbb{R}|&=|Q^{-1}(0)\cap\\{|x|>1\\}|\\\
&\leq(\log
c)^{-1}\log\left(\frac{L(P)^{2}}{2|P(1)|^{2}}+\frac{L(P)^{2}}{2|P(-1)|^{2}}\right)+\\\
&+(\log c)^{-1}\log(1+d^{2})+(c+1)\log M(Q)+2\sqrt{c\log M(Q)d}.\\\
\end{split}$

We conclude the proof of (5.4) by applying (2.5). ∎

######  Corollary 5.4.

Let $P\in\mathbb{Z}[x]$ be of degree $d$. Assume $P(0)P(1)P(-1)\not=0$. Then

$\begin{split}|P^{-1}(0)\cap\mathbb{R}|&\leq 2\sqrt{2c\,d\log M(P)}+2(c+1)\log
M(P)\\\ &+(\log c)^{-1}\log(1+d^{2})+2(\log c)^{-1}\log L(P),\end{split}$
(5.7)

where $c>1$ is such that $c\log c=1+c\ \ (c=3.594...)$.

If moreover, $\log L(P)\leq c_{1}\sqrt{d\log M(P)}$ for some constant
$c_{1}>0$, then for all $\varepsilon>0$, there exists $D_{0}(\varepsilon)$
such that for $d\geq D_{0}(\varepsilon)$,

$|P^{-1}(0)\cap\mathbb{R}|\leq(c_{2}+\varepsilon)\sqrt{d\log M(P)},$ (5.8)

where $c_{2}=2\left(\sqrt{2c}+c+1+(\log c)^{-1}c_{1}\right)$.

######  Remark 5.5.

Inequality (5.8) improves the bound (5.1) given by Dubickas in the particular
case where $L(P)\leq c_{1}\sqrt{d\log M(P)}$.

###### Proof.

Let $P=a_{d}x^{d}+\ldots+a_{0}$ with $a_{0},\ldots,a_{d}\in\mathbb{Z}$,
$a_{d}\not=0$ and $P(0)P(1)P(-1)\not=0$. We apply Theorem 5.3 to the
polynomial $\tilde{P}=a_{d}^{-1}P$. Inequality (5.7) follows after observing
that

$\begin{split}\frac{M(\tilde{P})^{2}}{|\tilde{P}(0)|}&=\frac{M(P)^{2}}{a_{d}|\tilde{P}(0)|}\leq
M(P)^{2}\\\
\frac{L(\tilde{P})^{2}}{2|\tilde{P}(1)|^{2}}+\frac{L(\tilde{P})^{2}}{2|\tilde{P}(-1)|^{2}}&=\frac{L(P)^{2}}{2|P(1)|^{2}}+\frac{L(P)^{2}}{2|P(-1)|^{2}}\leq
L(P)^{2}.\end{split}$

For the proof of (5.8), we may assume that $M(P)>1$, otherwise by Kronecker’s
theorem $P$ has no real zeros. Then, we know from [4] that for some constant
$\sigma>0$,

$\log M(P)\geq\sigma\left(\frac{\log\log d}{\log d}\right)^{3}.$ (5.9)

Thus for all $\varepsilon>0$, there exists $D_{0}(\varepsilon)$ such that for
$d\geq D_{0}(\varepsilon)$,

$(\log c)^{-1}\log(1+d^{2})\leq\varepsilon\sqrt{d\log M(P)}.$

On the other hand, we may also assume that $\log M(p)\leq d$ because otherwise
$\sqrt{d\log M(P)}\geq d$ and (5.8) is again trivially satisfied.

Since $\log M(p)\leq\sqrt{d\log M(p)}$, (5.8) follows immediately from (5.7)
and the assumption that $\log L(P)\leq c_{1}\sqrt{d\log M(P)}$. ∎

##  6 The number of zeros in $\\{|x-1|<1\\}$.

In [5], A. Dubickas used a Vandermonde determinant which, in its simplified
form is defined by

$d(x)=\det(x^{(j-1)l})_{1\leq j,l\leq N}=\prod_{1\leq v<u\leq N}(x^{u}-x^{v})\
\ \ (N\in\mathbb{N}^{*}).$

Following the lines of his work, we put

$R(x)=|d(x)|\prod_{1\leq v<u\leq N}|x|^{-v}=\prod_{j=1}^{N-1}|x^{j}-1|^{N-j}.$

In [5], the author applies Hadamard’s inequality to obtain:

$R(x)\leq\max(1,|x|)^{\frac{(N-1)N(N+1)}{6}}N^{\frac{N}{2}}.$ (6.1)

######  Lemma 6.1.

Assume $P\in\mathbb{Z}[x]$ is monic, has degree $d$ and does not vanish at the
roots of unity. Then for all integer $N\geq 2$,

$|P(1)|\leq N^{\frac{d}{N-1}}M(P)^{\frac{N+1}{3}}.$

######  Remark 6.2.

  * •

By Kronecker’s theorem, either $P=x^{d}$ or $M(P)>1$.

  * •

By (2.1) and (2.3), we already had $|P(1)|\leq 2^{d}M(P)$. We retrieve this
inequality if we put $N=2$ in Lemma 6.1.

###### Proof.

Let $\mu_{1},\ldots,\mu_{d}$ be the zeros of $P$. Then

$\begin{split}R(\mu_{k})&=|\mu_{k}-1|^{\frac{(N-1)N}{2}}\prod_{j=1}^{N-1}\left|\sum_{l=0}^{j-1}\mu_{k}^{l}\right|^{N-j}\\\
\end{split}$

and

$\prod_{k=1}^{d}R(\mu_{k})=|P(1)|^{\frac{(N-1)N}{2}}\prod_{k=1}^{d}\prod_{j=1}^{N-1}\left|\sum_{l=0}^{j-1}\mu_{k}^{l}\right|^{N-j}.$

Observing that

$\prod_{k=1}^{d}\prod_{j=1}^{N-1}\left(\sum_{l=0}^{j-1}\mu_{k}^{l}\right)^{N-j}\in\mathbb{Z}^{*},$

and using the inequality (6.1), we deduce that

$\begin{split}|P(1)|^{\frac{(N-1)N}{2}}&\leq\prod_{k=1}^{d}R(\mu_{k})\leq
N^{\frac{dN}{2}}M(P)^{\frac{(N-1)N(N+1)}{6}}.\end{split}$

The result follows easily. ∎

We will need the following result due to Zhang [25] and for which Zagier gave
an elementary proof in [24]:

######  Theorem 6.3.

[Zhang, Zagier]Let $\omega$ be a primitive $6$th root of unity. Let
$P\in\mathbb{Z}[x]$ be a polynomial of degree $d$ such that
$P(0)P(1)P(\omega)\not=0$. Denote $P_{*}(x)=P(1-x)$. Then

$M(P)M(P_{*})\geq\left(\frac{1+\sqrt{5}}{2}\right)^{\frac{d}{2}}.$

######  Theorem 6.4.

Let $P\in\mathbb{Z}[x]$ be monic, irreducible, of degree $d$ and satisfying
$0<\log M(P)\leq\phi(d)$ where $\phi(d)=o(\frac{d}{\log d})$. Then for all
$\varepsilon>0$, there exists $D_{0}(\varepsilon)$ such that for $d\geq
D_{0}(\varepsilon)$,

$|P^{-1}(0)\cap\\{|x-1|<1\\}|\geq\left(\frac{2}{\pi}\log\frac{1+\sqrt{5}}{2}-\varepsilon\right)\sqrt{\frac{d}{\log
d\log M(P)}}.$

This leads to the lower bound

$\log
M(P)\geq\left(\frac{2}{\pi}\log\frac{1+\sqrt{5}}{2}-\varepsilon\right)^{2}\frac{d}{K(P)^{2}\log
d},$ (6.2)

where
$K(P)=\min\left(\left|P^{-1}(0)\cap\\{|x-1|<1\\}\right|,\left|P^{-1}(0)\cap\\{|x+1|<1\\}\right|\right)$.

###### Proof.

We apply Lemma 6.1 with $\displaystyle N=\left[\sqrt{\frac{d\log d}{\log
M(P)}}\right]+2.$ We obtain

$\begin{split}\frac{N+1}{3}\log M(P)&\leq\log M(P)+\frac{1}{3}\sqrt{d\log
d\log M(P)},\\\ d\frac{\log N}{N-1}&\leq\sqrt{\frac{d\log M(P)}{\log
d}}\log\left(2\sqrt{\frac{d\log d}{\log M(P)}}\right)\\\ &=\sqrt{d\log d\log
M(P)}\left(\frac{1}{2}+\frac{\log(4\log d)}{2\log
d}\right)-\sqrt{\frac{d}{\log d}}\sqrt{\log M(P)}\log\sqrt{\log M(P)}\\\
&\leq\sqrt{d\log d\log M(P)}\left(\frac{1}{2}+\frac{\log(4\log d)}{2\log
d}\right)+e^{-1}\sqrt{\frac{d}{\log d}}.\\\ \end{split}$ $|P(1)|\leq
M(P)\exp\left(\sqrt{d\log d\log M(P)}\left(\frac{5}{6}+\frac{\log(4\log
d)}{2\log d}\right)+\sqrt{\frac{d}{\log d}}\right).$

Denote by $\mu_{1},\ldots,\mu_{d}$ the zeros of $P$. By Theorem 6.3, we have

$\prod_{|1-\mu_{k}|\geq
1}|1-\mu_{k}|\geq\left(\frac{1+\sqrt{5}}{2}\right)^{\frac{d}{2}}M(P)^{-1}.$

We immediately deduce that

$\begin{split}\prod_{|1-\mu_{k}|<1}|1-\mu_{k}|&\leq|P(1)|\left(\frac{1+\sqrt{5}}{2}\right)^{-\frac{d}{2}}M(P)\\\
&\leq M(P)^{2}\exp\left({\sqrt{d\log d\log
M(P)}}\left(\frac{5}{6}+\frac{\log(4\log d)}{2\log
d}\right)+\sqrt{\frac{d}{\log
d}}-\frac{d}{2}\log\left(\frac{1+\sqrt{5}}{2}\right)\right)\\\ \end{split}$
(6.3)

Denote by $J=|P^{-1}(0)\cap\\{|x-1|<1\\}|$ and let $\varepsilon>0$. Using the
lower bound given by A. Dubickas (see [5] or (3.1)), there exists
$D_{0}(\varepsilon)$ such that for all $d\geq D_{0}(\varepsilon)$

$\prod_{|1-\mu_{k}|\leq
1}|1-\mu_{k}|\geq\exp\left(-J(\frac{\pi}{4}+\varepsilon^{\prime})\sqrt{d\log
d\log M(P)}\right)$ (6.4)

where $\varepsilon^{\prime}>0$ is chosen small enough with regards to
$\varepsilon$. Using (6.3), (6.4) and $\log M(P)\leq\phi(d)=o(\frac{d}{\log
d})$, we find for $d\geq D_{0}(\varepsilon)$

$J\geq\left(\frac{2}{\pi}\log\frac{1+\sqrt{5}}{2}-\varepsilon\right)\sqrt{\frac{d}{\log
d\log M(P)}}.$

As the same apply to $P(-x)$, we have the same estimate for
$|P^{-1}(0)\cap\\{|x+1|<1\\}|$ and we obtain (6.2). ∎

##  7 Acknowledgments

We sincerely thank the anonymous reviewer, whose comments and suggestions
helped us to improve and clarify our manuscript.

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# Detection Of Primary User Emulation Attack (PUEA) In Cognitive Radio
Networks Using One-Class Classification

Bishal Chhetry  North Eastern Regional Institute of Science and Technology,
Nirjuli, Itanagar, Arunachal Pradesh 791109, India<EMAIL_ADDRESS>Ningrinla Marchang<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Opportunistic usage of spectrum owned by licensed (or primary) users is the
cornerstone on which the Cognitive Radio technology is built. Unlicensed (or
secondary) users that thus use the spectrum rely opportunistically on spectrum
sensing to determine the presence of primary user signal. In such a context,
an attacker may mimic the behavior of a primary user (PU) to deceive the
secondary users (SUs) into believing that a PU signal is present whereas it is
not. Such an attack is known as the Primary User Emulation Attack (PUEA). A
malicious user may launch a PUEA with the intention of grabbing the vacant
bands for its own transmission. Another reason may be to simply disrupt the
functioning of the Cognitive Radio Network (CRN). This work investigates the
use of one-class classification for detecting PUEA in an infrastructure-based
CRN. We opine that sensing data collected at the fusion center mainly for
Collaborative Spectrum Sensing (CSS) can be exploited to characterize a PU
signal. The PU signal features thus learned can aid in distinguishing a PU
signal from a PU signal emulation. In particular, we investigate the use of
one-class classification techniques, viz., Isolation Forest, Support Vector
Machines (SVM), Minimum Covariance Determinant(MCD) and Local Outlier
Factor(LOF) for detection of PUEA attacks. Simulation results support the
validity of using one-class classification for detection of PUEA.

###### keywords:

Primary User Emulation Attacks (PUEA), one-class classification, outlier
detection, cognitive radio network security.

##  1 Introduction

The last few decades have witnessed the mushrooming of wireless technologies
and services which has led to crowding the ISM bands, which will eventually
lead to ‘ _radio traffic jam_ ’. Ironically, studies have shown that whereas
some bands are overcrowded, some others, and a lot of the licensed bands in
particular, lie fallow most of the time and in most areas. This has led to the
emergence of cognitive radio technology. The basic idea of this technology is
to make opportunistic use of these fallow bands without causing harmful
interference to licensed users, which own these bands. These licensed users
are also known as primary users (PUs). Thus in CRNs, unlicensed or secondary
users (SUs) look out for unused licensed spectrum bands with the help of
spectrum sensing. One important underlying principle upon which opportunistic
usage in CRN works is that SUs must not cause harmful interference to PUs.
This implies two things. First, only when an SU detects that a band is unused,
then only, it can use the band for its transmission. Second, when an SU is
transmitting in an unused band and the PU starts its transmission, the SU must
vacant the band immediately.

Unfortunately, such a requirement opens the door for attacks such as the as
the Primary user emulation (PUE) attack. This attack happens when an attacker
(which could be an SU acting maliciously) emulates a PU signal. Consequently,
if other honest SUs happen to be using the associated channel, they will
evacuate from the channel. On the other hand, if they happen to be waiting for
this channel to be free, they will not be able to use it as they would
mistakenly sense that PU signal is present. An inside attacker, viz., a
malicious SU, may launch a PUEA for grabbing the channel selfishly for itself.
On the other hand, an outside attacker may do so for causing a DoS (Denial of
Service) attack. Hence, the PUEA greatly limits the spectrum access
opportunity of SUs.

Whereas there are several PUEA mitigation techniques in the literature, unlike
them, this work proposes the use of one-class classification for the same. The
basic motivation behind this work is: ‘The sensing data collected at the
fusion center (FC) forms a rich audit data-set for understanding the features
of a PU Transmission. One-class classification is a perfect fit, as it can use
this imbalanced data-set for training a model. The trained model can be used
for detecting any PU-like transmission which do not fit into this class of
_‘PU transmissions.’_

The contributions of this paper can be summarized as follows:

  * 1.

One-class classification is used for detecting PUEA. Four classification
algorithms are investigated, viz., Isolation Forest, Support Vector Machine,
MCD and LOF.

  * 2.

Audit data of the sensing reports collected at the FC are processed to
characterize the behaviour of a PU transmission, thereby generating the
training data-set.

  * 3.

Simulation results support the validity of using one-class classification for
PUEA detection.

The rest of this paper is structured as follows. Section 2 reviews related
existing works, which is followed by the system model in section 3. Then, we
explain the development of the proposed scheme in section 4. In section 5, we
present the results and performance analysis which is followed by the
conclusions in section 6.

##  2 Related Work

Research on detecting PUE attacks is an ongoing activity. A lot of work in
this area has been reported in the literature [1] \- [17]. In [1], an
analytical model and a practical mechanism to detect PUEA in CRNs (a.k.a.
dynamic spectrum access networks) are presented. In a similar work [2], an
analytical approach using Fenton’s approximation and Markov inequality to
obtain a lower bound on the probability of a successful PUEA is presented, in
which a fading wireless environment is considered. In [3], we find a survey on
security threats and related protection techniques for Software Defined Radio
(SDR) and Cognitive Radio (CR) technologies. Estimation techniques (e.g.,
maximum likelihood estimation) and the learning methods (e.g., mean-field
approach) can also be used for mitigating PUEA, wherein individual spectrum
decision is obtained by characterizing the received power at good secondary
user through a flexible log-normal sum approximation method [4].

An example of a technique based on localization is presented in [5], in which
a transmitter verification scheme, called LocDef (localization-based defense)
is given which verifies whether a given signal is that of an incumbent
transmitter by estimating its location and observing its signal
characteristics. The scheme presented in [6] uses the activity pattern of a
signal through spectrum sensing, such as the ON and OFF periods of the signal.
Then, the observed signal activity pattern is reconstructed through a
reconstruction model. The reconstruction error is used to distinguish a signal
activity pattern of a PU from a signal activity pattern of a PUEA attacker. A
comparatively involved scheme [7] uses cryptographic signatures and wireless
link signatures (derived from physical radio channel characteristics) to
enable primary user detection in the presence of PUEA attackers. In [8], a
framework for security and resilience is applied to spectrum sensing
functionality to ensure that it is robust even in the event of failures and
PUEA.

In [9], the authors develop a distributed spectrum decision protocol in which
SUs make individual spectrum decisions and then exchange individual sensing
results with their one-hop neighbors to increase resilience to PUEA. In [10],
a mechanism for detection and prevention of PUEA is presented, which employs
Time-Distance with signal Strength Evaluation (TDSE) and Extreme Machine
Learning (EML) algorithm. Whereas our work also uses machine learning, it
relies only on the audit data of sensing reports received at the FC. However,
in [10], the authors use time duration of traversal, RSS from the oncoming
transmitter and the ideal time of the PU. A novel algorithm for detecting non-
intelligent primary user emulation attack is presented in [11]. Neyman-Pearson
composite hypothesis test (NPCHT) and Wald’s sequential probability ratio test
(WSPRT) are employed in [12] to detect PUEA. In [13], the authors describe the
special characteristics of cognitive radio and CRN and present an analysis the
potential security threats that arise due to these characteristics. A method
to detect the PUE attack of mobile PUs by exploiting the correlations between
RF signals and acoustic information with the aim of verifying the existence of
wireless microphones is found in [14]. In [15], a spectrum decision protocol
resilient to PUEA in dynamic spectrum access networks is proposed. Device-
specific features have been employed to propose a passive, non-parametric
classification method to determine the number of PUs that are transmitting in
the spectrum. In [17], an algorithm for the detection of PUEA and jamming
attacks in CRN is proposed, which is founded on the sparse coding of the
compressed received signal over a channel-dependent dictionary.

Our work is motivated by the above tools and techniques. However, none of them
have used one-class classification which we perceive is a suitable technique
for detecting PUEA. Moreover, just by analyzing the sensing reports received
at the FC, a PU transmission can be characterised and the derived features are
used to drive the classification model. To the best of our knowledge, our work
is the first one which employs sensed data at the FC to train one-class
classification models for detecting PUEA. Representative algorithms such as
Isolation Forest (IF), SVM, MCD and LOF are employed in our work.

##  3 System Model

We assume a CRN consisting of $n$ SUs, wherein all the SUs collaborate to
sense the presence of a PU. We assume that the PU and the SUs are stationary,
and the PU is away from the SU. No two SUs have the same location. The system
model is illustrated in Fig. 1.

For simplicity, we consider a single-channel system. Each SU takes the help of
an energy detector for sensing the channel locally [4] and sends its sensing
report to the FC. Here, we assume a continuous reporting CRN and so, each SU
sends the energy level (in dB) of the received signal to the FC. The FC uses
some fusion rule to make the final decision about whether the PU is present or
not. However, how the decision is made is not the focus of our study. Rather,
it focuses on exploiting these sensing reports (raw energy levels) received at
the FC for mitigating PUEA.

Time for each SU is divided into fixed-size slots and these slots are
synchronized. Each slot is again subdivided into two sub-slots: i) sensing,
and ii) data transmission, wherein the first sub-slot is for sensing and the
second one for data transmission. In each sensing sub-slot, an SU senses the
channel and sends the sensing report to the FC. Thus, in this sub-slot, the FC
receives the reports (energy values), $y_{1},y_{2},..,y_{n}$ from the $n$ SUs.

Figure 1: PUEA in an Infrastructure-based CRN

The relationship between the transmit signal power and the received signal
power is modelled with the help of the path loss model and the log-normal
shadowing of a communication channel as in [4]. Assuming the path loss
exponent, $\alpha$, the received signal energy $P_{r}$ is proportional to
r$-$$\alpha$, where $r$ is the distance between the transmitter and the
receiver. Additionally, $P_{r}$ is also proportional to a shadowing random
variable G. Here, G=$\ e^{a\beta}$, where $a=\frac{ln10}{10}$ and $\beta$
follows a normal distribution, $N(0,\sigma^{2}$). Thus, when a transmitter
employs energy $P_{t}$ to transmit signals, then the received signal energy,
$P_{r}$ at the receiver end is given by [4]:

$\displaystyle
P_{r}=P_{t}r\textsuperscript{-$\alpha$}e\textsuperscript{a$\beta$}$ (1)

At the FC, these received energy values are processed to create the data-set,
which is described later. The built data-set is used for one-class
classification with the goal of mitigating PUEA. As seen in Fig. 1, a PUEA can
be launched by an outsider with the aim of disrupting the functioning of the
CRN. Alternatively, it can also be launched by an insider, i.e., one among the
SUs.

##  4 PUEA Detection as a One-Class Classification Problem

This section describes how we employ one-class classification for detecting
PUEA attack in a CRN. First we describe how the detection problem can be
transformed into a one-class classification problem. Second, we present how
data-sets are formed and can be used for analysis. Third, we explain in brief
the classification algorithms, viz., Isolation Forest, SVM, MCD and LOF that
have been used in our work.

###  4.1 One-Class Classification

In classification, it may happen that the training set is such that all or
most of the examples in the set are from one class (say, the positive class),
and none or very little from the negative class. Such a classification is
called one-class classification. After the model is trained using such a data-
set, a new example which is fed to the trained model would be classified into
either the positive class of the negative class. When the training set
consists of only examples from one class, an input example which does not get
classified into this class is also called an outlier. Hence, one-class
classification techniques are also known as outlier detection techniques.

###  4.2 Motivation

Any transmission which imitates a PU transmission falls under the gambit of
PUEA. Since the only requirement of this attack is that normal SUs should
unwittingly sense it as a PU signal, there could be several ways this attack
is carried out. For instance, an attacker could position itself as close as
possible to a PU station and launch an attack. Alternatively, it could emulate
the PU signal (e.g., PU’s transmission power). Thus, it is a challenging task
to profile a PUEA behaviour. On the other hand, profiling of a PU behaviour is
easier comparatively, since a PU does not change its form over long periods of
time. For instance, most of the PU stations are stationary stations (e.g., a
TV tower) which do not generally change its transmission characteristics over
long periods of time. In conclusion, the idea is to build a training data-set
that consists of only PU signal examples. Consequently, one-class
classification is a perfect fit for detecting PUEA.

###  4.3 Building the Data-set

At each slot, when the FC receives the sensing reports (energy levels) from
the $n$ SUs, it collects them and calculates the mean, variance, median, upper
quartile and lower quartile of the $n$ values and creates a training example.
Thus, each training example consists of five input features: mean, variance,
median, upper quartile, lower quartile. We include output feature (i.e.,
label) as ‘1’ (positive class) when PU is present. On the other hand, if the
example is from a PUEA, it would be labelled ‘-1’ (negative class). So, if we
collect data from $T$ slots to create the data-set, there would be $T$
examples. in which PU is present, there will be $T$ examples.

###  4.4 Classification Algorithms

The data gathered at the FC for a good PU is processed so as to train a
classifier. Subsequently, we can employ the trained classifier to predict
whether the signal is from a PU or an attacker. We use four classifiers, viz.,
Isolation Forest, Support Vector Machine (SVM), Minimum Covariance Determinant
(MCD) and Local Outlier Factor (LOF).

####  4.4.1 Isolation Forest

Isolation Forest [18] is a tree-based anomaly detection algorithm. Anomalies
are isolated with the help of tree structures. The trees are created in such a
way that isolated examples tend to have a relatively short depth in the trees
as compared to normal examples which are less isolated, and thereby exhibit a
greater depth in the trees.

####  4.4.2 Support Vector Machine (SVM)

Initially, the SVM algorithm was developed for binary classification. However,
it can be employed for one-class classification [19]. For SVM to behave as a
one-class classification technique, the algorithm works by capturing the
density of the majority class. Then, examples on the extremes of the density
function are classified as outliers. The SVM thus modified is known as One-
Class SVM.

####  4.4.3 Minimum Covariance Determinant (MCD)

MCD [20] algorithm works by creating a a hypersphere (ellipsoid) in such a way
that it covers the normal data (examples). Hence, any example that is not
covered by this shape is detected as an outlier. MCD is an efficient
implementation of this technique for multivariate data.

(a) Inside SU Region (b) Outside SU Region

Figure 2: Locations of PU and Attacker

####  4.4.4 Local Outlier Factor(LOF)

Another approach [21] to detect outliers is to identify those examples that
are far from the other examples, which form larger groups in the feature
space. This approach works by determining the LOF degree of each example with
the help of nearest neighbors concept. This technique is known to be more
reliable for less number of features as compared to when the number of
features is high.

(a) $10\%$ PUEA data (b) $20\%$ PUEA data

Figure 3: Accuracy: [Inside SU region]

(a) $10\%$ PUEA data (b) $20\%$ PUEA data

Figure 4: Precision: [Inside SU region]

(a) $10\%$ PUEA data (b) $20\%$ PUEA data

Figure 5: Recall: [Inside SU region]

(a) $10\%$ PUEA data (b) $20\%$ PUEA data

Figure 6: F1-score: [Inside SU region]

##  5 Results and Discussion

This section presents the simulation results. We consider a CRN consisting of
40 SUs and 10,000 sensing slots. We assume that the SUs occupy some region of
$100\times 100$ sq. units. Then, we randomly generate the locations of the SUs
in this region. Next, we choose a location each for the PU and the attacker.
Depending on how their locations are chosen, we consider two scenarios: (i)
when the PU and the attacker are within the same region as the SUs as depicted
in Fig. 2(a), and (ii) when the PU and the attacker are outside the SUs’
region (refer Fig. 2(b)). In both the figures, $D$ denotes the distance
between the PU and the attacker.

The model described in Section 3 is used to create the data-set. We assume the
same paramater values used in [4]. Thus, we let $P_{t}=10$ and $\sigma^{2}=8$
when PU is transmitting. Whereas, when the attacker is transmitting, the
channel parameter is $\sigma^{2}$ = 4, 8, or 12.

Using the above model, the energy levels received from the PU at the SUs are
generated for each slot. Then, they are processed to generate the examples,
each example consisting of input features, viz., the mean, variance, median,
upper quartile and lower quartile of for each slot. Here, we assume that a
common control channel is used to send the energy levels from the SUs to the
FC and that there is no channel error. Hence, the data-set is created at the
FC. Since 10,000 slots are considered, the size of the training data-set is
10,000.

Similarly, for testing purpose, the energy levels that are received at the SUs
from the attacker are artificially generated using the same model. Due to lack
of availability of real-world data-set w.r.t. PUEA, we build an artificial
data-set as explained above.

We say that a true positive has occurred if a malicious PUE attack is detected
as malicious, whereas a false positive has occurred if a geniune PU is
incorrectly detected as an attacker. Similarly, we say that a true negative
has occurred if a good PU is detected as a genuine PU, whereas a false
negative has occured if an attacker is identified as a genuine PU.

Let TP, TN, FP, FN and N denote the count of true positives, true negatives,
false positives, false negatives and the total number of examples. Note that
$N=TP+TN+FP+FN$. The performance parameters considered are: i) $Accuracy$: It
gives a measure of how many cases are correctly identified. It is defined by
term, $(TP+TN)/N$. It is most useful when all the classes are equally
important. ii) $Precision$: It gives a measure of how many out of all the
predicted positive cases are correctly identified as positive cases. It is
defined by term, $TP/(TP+FP)$. Thus, this value will be high when FP is less.
iii) $Recall$: It gives a measure of how many out of all the actual positive
cases are correctly identified as positive cases. It is defined by term,
$TP/(TP+FN)$. Thus, this value will be high when FN is less. iv) $F1-Score$:
It is the harmonic mean of Precision and Recall. Thus, this value will be high
when the count of incorrectly classified cases is less. The values of these
metrics shown in the following result graphs are in terms of percentage.

The simulation results are divided into two parts based on the location of the
PU and the attacker w.r.t. the region where the SUs lie. We consider two
scenarios as shown in Fig. 2. This helps us to determine whether such factors
have any impact on the detection system.

###  5.1 PU Inside SU Region

In the scenario when the PU and the attackers are within the region where the
SUs lie (refer Fig. 2(a)), we compare the results for varying distances
between the PU and the attacker. Moreover, we vary the percentage of PUEA
examples in the test set and determine the results. It may be noted here that
the training set consists of only examples associated with the PU signal
(i.e., 10,000 examples) as explained in section 4.3. However, with regard to
the test set, we initially form it using the training set (consisting of
10,000 examples) and PUEA examples whose count is $10\%$ of the size of the
training set. Thus, the test set consists of 11,000 examples. Next, we
increase the percentage of the PUEA examples in the test set from $10\%$ to
$20\%$. The accuracy shown by different algorithms for varying values of ‘D’
are shown in Fig. 3. The label ‘D=5’ indicates the plot when the distance
between the PU and the attacker is 5 units. Similar labels in the rest of the
graphs have similar meanings. In Fig. 3(a), we find that LOF and MCD
outperform the other algorithms with an accuracy of $90\%$. We find that the
accuracy drops a little when the percentage of PUEA examples is $20\%$ (refer
Fig. 3(b)) as compared to when it is $10\%$ (refer Fig. 3(a)). This is due to
the increase in the number of unseen examples which are not in the training
set. We also find that there is no perceivable difference between the results
when the values of $D$ are varied.

k | IF | SVM | MCD | LOF
---|---|---|---|---
| acc.(%) | acc.(%) | acc.(%) | acc.(%)
2 | 84 | 84 | 90 | 90
5 | 83 | 84 | 90 | 90
10 | 84 | 84 | 90 | 90
20 | 84 | 85 | 90 | 90
Table 1: k-fold cross validation: 10% PUEA data [Inside SU region] k | IF | SVM | MCD | LOF
---|---|---|---|---
| acc.(%) | acc.(%) | acc.(%) | acc.(%)
2 | 77 | 79 | 83 | 82
5 | 77 | 79 | 82 | 82
10 | 77 | 79 | 82 | 82
20 | 76 | 78 | 82 | 82
Table 2: k-fold cross validation: 20% PUEA data [Inside SU region]

When the data-sets are imbalanced as in our study, accuracy may not be a good
parameter to measure performance. One important metric is precision. Fig. 4
illustrates the corresponding precision values for the same scenarios.
Precision is recorded to be above $90\%$ when the percentage of PUEA data in
the test set is $10\%$ (refer Fig. 4(a)) whereas it is above $80\%$ when the
corresponding percentage is $20\%$ (refer Fig. 4(b)) due to the reason stated
earlier that there are more unseen examples in this case as compared to the
earlier case, viz., $10\%$ case.

Recall is illustrated in Fig. 5. For both the scenarios (graphs), recall is
above $90\%$ for all the algorithms. Moreover, the F1-score is also well above
$90\%$ for all the algorithms for both scenarios (refer Fig. 6).

(a) $10\%$ PUEA data (b) $20\%$ PUEA data

Figure 7: Accuracy: [Outside SU region]

(a) $10\%$ PUEA data (b) $20\%$ PUEA data

Figure 8: Precision: [Outside SU region]

(a) $10\%$ PUEA data (b) $20\%$ PUEA data

Figure 9: Recall: [Outside SU region]

(a) $10\%$ PUEA data (b) $20\%$ PUEA data

Figure 10: F1-score: [Outside SU region]

Additionally, we employ $k$-fold cross validation on the two test data-sets.
We partition the data-set into $k$ fixed-sized divisions of equal size. Then,
we choose one partition as the test data and the rest $k-1$ partitions as the
training data. We, then train the model using the training data and test using
the test data. This process is repeated for $k$ iterations by choosing a
distinct partition as the test data in each iteration. The performance results
shown are determined by averaging over the $k$ iterations. Tables 1 and 2 give
the accuracy for different values of $k$. Here, the value of $k$ does not seem
to have any significant effect on the accuracy.

###  5.2 PU Outside SU Region

The results presented in the previous subsection are for the case when the PU
and the attacker are in the same region as the SUs (as illustrated in Fig.
2(a)). In this subsection, we present the results when they are outside the
region (as illustrated in Fig. 2(b)). This results are obtained so as to check
whether there is any effect of varying distance and orientation between the PU
and the SUs.

The corresponding graphs that depict accuracy are given in Fig. 7. We find
that even if the PU and attacker are farther away from the SUs, the
performance of the ML-based detection system is good. This can be seen also
w.r.t. other performance metrics such as _Precision_ , _Recall_ and _F1-score_
as illustrated in Figs. 8 to 10.

k | IF | SVM | MCD | LOF
---|---|---|---|---
| acc.(%) | acc.(%) | acc.(%) | acc.(%)
2 | 82 | 84 | 90 | 90
5 | 82 | 84 | 89 | 90
10 | 82 | 84 | 89 | 90
20 | 82 | 84 | 90 | 90
Table 3: k-fold cross validation: 10% PUEA data [Outside SU region] k | IF | SVM | MCD | LOF
---|---|---|---|---
| acc.(%) | acc.(%) | acc.(%) | acc.(%)
2 | 75 | 78 | 82 | 82
5 | 75 | 78 | 82 | 82
10 | 75 | 78 | 82 | 82
20 | 74 | 78 | 82 | 82
Table 4: k-fold cross validation: 20% PUEA data [Outside SU region]

For this scenario, we also apply $k$-fold cross validation. The corresponding
accuracy results are given in Tables 3 and 4 for different values of $k$. Here
again, the value of $k$ does not seem to have any significant effect on the
accuracy.

##  6 Conclusions

This research studies the use of supervised machine learning, viz., one-class
classification as a tool for detecting PUEA in infrastructure-based CRN.
Classifiers such as Isolation Forest (IF), SVM, MCD and LOF are examined. We
show through simulation how audit data of the sensing reports collected at the
FC can be processed to characterize the behaviour of a PU transmission,
thereby generating the training data-set. To the best of our knowledge, there
has been no similar study that employs the rich sensed data collected at the
FC for characterizing a PU transmission. Subsequently, we employ one-class
classification algorithms and train them using the data-set thus created to
build models that detect PUEA. Of all the techniques, SVM performs the worst
with inconsistent performance. Isolation Forest (IF) performs better than SVM
but not as good as MCD and LOF. MCD and LOF are the most robust, showing
consistently very good performance.

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