## Preliminary Version

## 1\. Introduction and Notation

Research on the distribution of the zeros of algebraic polynomials has a long
and rich history. In fact all the papers [1–48] in our list of references are
just some of the papers devoted to this topic. The study of the number of real
zeros trigonometric polynomials and the number of unimodular zeros (that is,
zeros lying on the unit circle of the complex plane) of algebraic polynomials
with various constraints on their coefficients are the subject of quite a few
of these. We do not try to survey these in our introduction.

Let $S\subset{\mathbb{C}}$. Let ${\Cal{P}}_{n}(S)$ be the set of all algebraic
polynomials of degree at most $n$ with each of their coefficients in $S$. A
polynomial $P$ of the form

$P(z)=\sum_{j=0}^{n}{a_{j}z^{j}}\,,\qquad a_{j}\in{\mathbb{C}}\,,$ $None$

is called conjugate-reciprocal if

$\overline{a}_{j}=a_{n-j}\,,\qquad j=0,1,\ldots,n\,.$ $None$

A polynomial $P$ of the form (1.1) is called plain-reciprocal or self-
reciprocal if

$a_{j}=a_{n-j},\qquad j=0,1,\ldots,n\,.$ $None$

If a conjugate reciprocal polynomial $P$ has only real coefficients, then it
is obviously plain-reciprocal. We note also that if

$P(z)=\sum_{j=0}^{2n}{a_{j}z^{j}}\,,\qquad a_{j}\in{\mathbb{C}}\,,$

is conjugate-reciprocal, then there are $\theta_{j}\in{\mathbb{R}}$,
$j=1,2,\ldots\,n$, such that

$T(t):=P(e^{it})e^{-int}=a_{n}+\sum_{j=1}^{n}{2|a_{j+n}|\cos(jt+\theta_{j})}\,.$

If the polynomial $P$ above is plain-reciprocal, then

$T(t):=P(e^{it})e^{-int}=a_{n}+\sum_{j=1}^{n}{2a_{j+n}\cos(jt)}\,.$

In this paper, whenever we write “$P\in{\Cal{P}}_{n}(S)$ is conjugate-
reciprocal” we mean that $P$ is of the form (1.1) with each $a_{j}\in S$
satisfying (1.2). Similarly, whenever we write “$P\in{\Cal{P}}_{n}(S)$ is
self-reciprocal” we mean that $P$ is of the form (1.1) with each $a_{j}\in S$
satisfying (1.3). This is going to be our understanding even if the degree of
$P\in{\Cal{P}}_{n}(S)$ is less than $n$. It is easy to see that
$P\in{\Cal{P}}_{n}(S)$ is self-reciprocal and $n$ is odd, then $P(-1)=0$.
Associated with an algebraic polynomial $P$ of the form (1.1) we introduce the
numbers

$\text{\rm NC}(P):=|\\{j\in\\{0,1,\ldots,n\\}:a_{j}\neq 0\\}|\,.$

Here, and in what follows $|A|$ denotes the number of elements of a finite set
$A$. Let $\text{\rm NZ}(P)$ denote the number of real zeros (by counting
multiplicities) of an algebraic polynomial $P$ on the unit circle. Associated
with a trigonometric polynomial

$T(t)=\sum_{j=0}^{n}{a_{j}\cos(jt)}$

we introduce the numbers

$\text{\rm NC}(T):=|\\{j\in\\{0,1,\ldots,n\\}:a_{j}\neq 0\\}|\,.$

Let $\text{\rm NZ}(T)$ denote the number of real zeros (by counting
multiplicities) of a trigonometric polynomial $T$ in a period (of length
$2\pi$). Let $\text{\rm NZ}^{*}(T)$ denote the number of sign changes of a
trigonometric polynomial $T$ in a period (of length $2\pi$). The quotation
below is from [7]. “Let $0\leq n_{1}<n_{2}<\cdots<n_{N}$ be integers. A cosine
polynomial of the form $T(\theta)=\sum_{j=1}^{N}{\cos(n_{j}\theta)}$ must have
at least one real zero in a period. This is obvious if $n_{1}\neq 0$, since
then the integral of the sum on a period is $0$. The above statement is less
obvious if $n_{1}=0$, but for sufficiently large $N$ it follows from
Littlewood’s Conjecture simply. Here we mean the Littlewood’s Conjecture
proved by S. Konyagin [27] and independently by McGehee, Pigno, and Smith [35]
in 1981. See also [14, pages 285-288] for a book proof. It is not difficult to
prove the statement in general even in the case $n_{1}=0$ without using
Littlewood’s Conjecture. One possible way is to use the identity

$\sum_{j=1}^{n_{N}}{T\left(\frac{(2j-1)\pi}{n_{N}}\right)}=0\,.$

See [28], for example. Another way is to use Theorem 2 of [36]. So there is
certainly no shortage of possible approaches to prove the starting observation
of this paper even in the case $n_{1}=0$.

It seems likely that the number of zeros of the above sums in a period must
tend to $\infty$ with $N$. In a private communication B. Conrey asked how fast
the number of real zeros of the above sums in a period tends to $\infty$ as a
function $N$. In [12] the authors observed that for an odd prime $p$ the
Fekete polynomial

$f_{p}(z)=\sum_{k=0}^{p-1}{\binom{k}{p}z^{k}}$

(the coefficients are Legendre symbols) has $\sim\kappa_{0}p$ zeros on the
unit circle, where $0.500813>\kappa_{0}>0.500668$. Conrey’s question in
general does not appear to be easy.

Littlewood in his 1968 monograph ‘Some Problems in Real and Complex Analysis´
[33, problem 22] poses the following research problem, which appears to still
be open: ‘If the $n_{m}$ are integral and all different, what is the lower
bound on the number of real zeros of $\sum_{m=1}^{N}\cos(n_{m}\theta)$?
Possibly $N-1$, or not much less.´ Here real zeros are counted in a period. In
fact no progress appears to have been made on this in the last half century.
In a recent paper [7] we showed that this is false. There exist cosine
polynomials $\sum_{m=1}^{N}\cos(n_{m}\theta)$ with the $n_{m}$ integral and
all different so that the number of its real zeros in a period is
$O(N^{9/10}(\log N)^{1/5})$ (here the frequencies $n_{m}=n_{m}(N)$ may vary
with $N$). However, there are reasons to believe that a cosine polynomial
$\sum_{m=1}^{N}\cos(n_{m}\theta)$ always has many zeros in a period.”

Let

${\Cal{L}}_{n}:=\left\\{P:\,P(z)=\sum_{j=0}^{n}{a_{j}z^{j}}\,,\enskip
a_{j}\in\\{-1,1\\}\right\\}\,.$

Elements of ${\Cal{L}}_{n}$ are often called Littlewood polynomials of degree
$n$. Let

${\Cal{K}}_{n}:=\left\\{P:\,P(z)=\sum_{j=0}^{n}{a_{j}z^{j}}\,,\enskip
a_{j}\in{\mathbb{C}},\enskip|a_{0}|=|a_{n}|=1,\enskip|a_{j}|\leq
1\right\\}\,.$

Observe that ${\Cal{L}}_{n}\subset{\Cal{K}}_{n}$. In [10] we proved that any
polynomial $P\in{\Cal{K}}_{n}$ has at least $8n^{1/2}\log n$ zeros in any open
disk centered at a point on the unit circle with radius $33n^{-1/2}\log n$.
Thus polynomials in ${\Cal{K}}_{n}$ have quite a few zeros near the unit
circle. One may naturally ask how many unimodular roots a polynomial in
${\Cal{K}}_{n}$ can have. Mercer [36] proved that if a Littlewood polynomial
$P\in{\Cal{L}}_{n}$ of the form (1.1) is skew reciprocal, that is,
$a_{j}=(-1)^{j}a_{n-j}$ for each $j=0,1,\ldots,n$, then it has no zeros on the
unit circle. However, by using different elementary methods it was observed in
both [17] and [37] that if a Littlewood polynomial $P$ of the form (1.1) is
self-reciprocal, that is, $a_{j}=a_{n-j}$ for each $j=0,1,\ldots,n$, $n\geq
1$, then it has at least one zero on the unit circle. Mukunda [37] improved
this result by showing that every self-reciprocal Littlewood polynomial of odd
degree has at least $3$ zeros on the unit circle. Drungilas [15] proved that
every self-reciprocal Littlewood polynomial of odd degree $n\geq 7$ has at
least $5$ zeros on the unit circle and every self-reciprocal Littlewood
polynomial of even degree $n\geq 14$ has at least $4$ zeros on the unit
circle. In [4] two types of Littlewood polynomials are considered: Littlewood
polynomials with one sign change in the sequence of coefficients and
Littlewood polynomials with one negative coefficient, and the numbers of the
zeros such Littlewood polynomials have on the unit circle and inside the unit
disk, respectively, are investigated. Note that the Littlewood polynomials
studied in [4] are very special. In [7] we proved that the average number of
zeros of self-reciprocal Littlewood polynomials of degree $n$ is at least
$n/4$. However, it is much harder to give decent lower bounds for the
quantities

$\text{\rm NZ}_{n}:=\min_{P}{\text{\rm NZ}(P)}\,,$

where $\text{\rm NZ}(P)$ denotes the number of zeros of a polynomial $P$ lying
on the unit circle and the minimum is taken for all self-reciprocal Littlewood
polynomials $P\in{\Cal{L}}_{n}$. It has been conjectured for a long time that
$\lim_{n\rightarrow\infty}\text{\rm NZ}_{n}=\infty$. In [21] we showed that
$\lim_{n\rightarrow\infty}\text{\rm NZ}(P_{n})=\infty$ whenever
$P_{n}\in{\Cal{L}}_{n}$ is self-reciprocal and
$\lim_{n\rightarrow\infty}{|P_{n}(1)|}=\infty$. This follows as a consequence
of a more general result, see Corollary 2.3 in [21], stated as Corollary 1.5
here, in which the coefficients of the self-reciprocal polynomials $P_{n}$ of
degree at most $n$ belong to a fixed finite set of real numbers. In [7] we
proved the following result.

###### Theorem 1.1

If the set $\\{a_{j}:j\in{\mathbb{N}}\\}\subset{\mathbb{R}}$ is finite, the
set $\\{j\in{\mathbb{N}}:a_{j}\neq 0\\}$ is infinite, the sequence $(a_{j})$
is not eventually periodic, and

$T_{n}(t)=\sum_{j=0}^{n}{a_{j}\cos(jt)}\,,$

then $\lim_{n\rightarrow\infty}{\text{\rm NZ}(T_{n})}=\infty\,.$

In [6] Theorem 1.1 is stated without the assumption that the sequence
$(a_{j})$ is not eventually periodic. However, as the following example shows,
Lemma 3.4 in [7], dealing with the case of eventually periodic sequences
$(a_{j})$, is incorrect. Let

$\begin{split}T_{n}(t):=&\cos
t+\cos((4n+1)t)+\sum_{k=0}^{n-1}{(\cos((4k+1)t)-\cos((4k+3)t))}\cr=&\frac{1+\cos((4n+2)t)}{2\cos
t}+\cos t\,.\cr\end{split}$

It is easy to see that $T_{n}(t)\neq 0$ on
$[-\pi,\pi]\setminus\\{-\pi/2,\pi/2\\}$ and the zeros of $T_{n}$ at $-\pi/2$
and $\pi/2$ are simple. Hence $T_{n}$ has only two (simple) zeros in a period.
So the conclusion of Theorem 1.1 above is false for the sequence $(a_{j})$
with $a_{0}:=0$, $a_{1}:=2$, $a_{3}:=-1$, $a_{2k}:=0$, $a_{4k+1}:=1$,
$a_{4k+3}:=-1$ for every $k=1,2,\ldots$. Nevertheless, Theorem 1.1 can be
saved even in the case of eventually periodic sequences $(a_{j})$ if we assume
that $a_{j}\neq 0$ for all sufficiently large $j$. See Lemma 3.11 in [22]
where Theorem 1 in [7] is corrected as

###### Theorem 1.2

If the set $\\{a_{j}:j\in{\mathbb{N}}\\}\subset{\mathbb{R}}$ is finite,
$a_{j}\neq 0$ for all sufficiently large $j$, and

$T_{n}(t)=\sum_{j=0}^{n}{a_{j}\cos(jt)}\,,$

then $\lim_{n\rightarrow\infty}{\text{\rm NZ}(T_{n})}=\infty\,.$

It was expected that the conclusion of the above theorem remains true even if
the coefficients of $T_{n}$ do not come from the same sequence, that is,

$T_{n}(t)=\sum_{j=0}^{n}{a_{j,n}\cos(jt)}\,,$

where the set

$S:=\\{a_{j,n}:j\in\\{0,1,\ldots,n\\},\,n\in{\mathbb{N}}\\}\subset{\mathbb{R}}$

is finite and

$\lim_{n\rightarrow\infty}{|\\{j\in\\{0,1,\ldots,n\\},\,a_{j,n}\neq
0\\}|}=\infty\,.$

Associated with an algebraic polynomial

$P(z)=\sum_{j=0}^{n}{p_{j}z^{j}}\,,\qquad p_{j}\in{\mathbb{C}}\,,$

let

$\text{\rm NC}_{k}(P):=\left|\left\\{u:0\leq u\leq n-k+1,\enskip
a_{u}+a_{u+1}+\cdots+a_{u+k-1}\neq 0\right\\}\right|\,.$

In [15] we proved the following results.

###### Theorem 1.3

If $S\subset{\mathbb{R}}$ is a finite set, $P_{2n}\in{\Cal{P}}_{2n}(S)$ are
self-reciprocal polynomials,

$T_{n}(t):=P_{2n}(e^{it})e^{-int}\,,$

and

$\lim_{n\rightarrow\infty}{\text{\rm NC}_{k}(P_{2n})}=\infty$

for every $k\in{\mathbb{N}}$, then

$\lim_{n\rightarrow\infty}{\text{\rm
NZ}(P_{2n})}=\lim_{n\rightarrow\infty}{\text{\rm NZ}(T_{n})}=\infty\,.$

Some of the most important consequences of the above theorem obtained in [15]
are stated below.

###### Corollary 1.4

If $S\subset{\mathbb{R}}$ is a finite set, $P_{n}\in{\Cal{P}}_{n}(S)$ are
self-reciprocal polynomials, and

$\lim_{n\rightarrow\infty}{|P_{n}(1)|}=\infty\,,$

then

$\lim_{n\rightarrow\infty}{\text{\rm NZ}(P_{n})}=\infty\,.$

###### Corollary 1.5

Suppose the finite set $S\subset{\mathbb{R}}$ has the property that

$s_{1}+s_{2}+\cdots+s_{k}=0\,,\enskip s_{1},s_{2},\ldots,s_{k}\in
S\,,\enskip\text{\rm implies}\enskip s_{1}=s_{2}=\cdots=s_{k}=0\,,$

that is, any sum of nonzero elements of $S$ is different from $0$. If
$P_{n}\in{\Cal{P}}_{n}(S)$ are self-reciprocal polynomials and

$\lim_{n\rightarrow\infty}{\text{\rm NC}(P_{n})}=\infty\,,$

then

$\lim_{n\rightarrow\infty}{\text{\rm NZ}(P_{n})}=\infty\,.$

J. Sahasrabudhe [43] examined the case when $S\subset{\mathbb{Z}}$ is finite.
Exploiting the assumption that the coefficients are integer he proved that for
any finite set $S\subset{\mathbb{Z}}$ a self-reciprocal polynomial
$P\in{\Cal{P}}_{2n}(S)$ has at least

$c\,\left(\log\log\log|P(1)|\right)^{1/2-\varepsilon}-1$

zeros on the unit circle of ${\mathbb{C}}$ with a constant $c>0$ depending
only on $M=M(S):=\max\\{|z|:z\in S\\}$ and $\varepsilon>0$.

Let $\phi(n)$ denote the Euler’s totient function defined as the number of
integers $1\leq k\leq n$ that are relative prime to $n$. In an earlier version
of his paper Sahasrabudhe [43] used the trivial estimate $\phi(n)\neq\sqrt{n}$
for $n\geq 3$ and he proved his result with the exponent $1/4-\varepsilon$
rather than $1/2-\varepsilon$. Using the nontrivial estimate $\phi(n)\geq
n/8(\log\log n)$ [40] for all $n>3$ allowed him to prove his result with
$1/2-\varepsilon$.

In the papers [6], [21], [42] the already mentioned Littlewood Conjecture,
proved by Konyagin [27] and independently by McGehee, Pigno, and B. Smith
[35], plays a key role, and we rely on it heavily in the proof of the main
results of this paper as well. This states the following.

###### Theorem 1.6

There is an absolute constant $c>0$ such that

$\int_{0}^{2\pi}{\Big{|}\sum_{j=1}^{m}{a_{j}e^{i\lambda_{j}t}}\Big{|}\,dt}\geq
c\gamma\log m\,$

whenever $\lambda_{1},\lambda_{2},\ldots,\lambda_{n}$ are distinct integers
and $a_{1},a_{2},\ldots,a_{m}$ are complex numbers of modulus at least
$\gamma>0$. Here $c=1/30$ is a suitable choice.

This is an obvious consequence of the following result a book proof of which
has been worked out by Lorentz and DeVore in [13, pages 285-288].

###### Theorem 1.7

If $\lambda_{1}<\lambda_{2}<\cdots<\lambda_{m}$ are integers and
$a_{1},a_{2},\ldots,a_{m}$ are complex numbers, then

$\int_{0}^{2\pi}{\Big{|}\sum_{j=1}^{m}{a_{j}e^{i\lambda_{j}t}}\Big{|}\,dt}\geq\frac{1}{30}\sum_{j=1}^{m}{\frac{|a_{j}|}{j}}\,.$

Associated with a finite set $\\{A\subset{\mathbb{C}}$ we will use the
notation

$M(A):=\max{|z|}:z\in A\\}$

throughout the paper.

## 2\. New Results

###### Theorem 2.1

If $S\subset{\mathbb{Z}}$ is a finite set, $P\in{\Cal{P}}_{2n}(S)$ is a self-
reciprocal polynomial,

$T(t):=P(e^{it})e^{-int}\,,$

and $M=M(S):=\max\\{|z|:z\in S\\}$, then

$\text{\rm NZ}^{*}(T_{n})\geq c(\log\log\log|P(1)|)^{1-\varepsilon}-1$

with a constant $c>0$ depending only on $\varepsilon>0$ and $M=M(S)$.

###### Corollary 2.2

If $S\subset{\mathbb{Z}}$ is a finite set, $P\in{\Cal{P}}_{n}(S)$ is a self-
reciprocal polynomial, and $M=M(S):=\max\\{|z|:z\in S\\}$, then

$\text{\rm NZ}(P)\geq c(\log\log\log|P(1)|)^{1-\varepsilon}-1$

with a constant $c>0$ depending only on $\varepsilon>0$ and $M=M(S)$.

This improves the exponent $1/2-\varepsilon$ to $1-\varepsilon$ in a recent
breakthrough result [41] by Julian Sahasrabudhe. We note that in both
Sahasrabudhe’s paper and this paper the assumption that the finite set $S$
contains only integers is deeply exploited. Our next result is an obvious
consequence of Corollary 2.2.

###### Corollary 2.3

If the set $S\subset{\mathbb{Z}}$ is finite

$T(t)=\sum_{j=0}^{n}{a_{j}\cos(jt)}\,,\qquad a_{j}\in S\,,$

then

$\text{\rm NZ}^{*}(T)\geq c(\log\log\log|T(0)|)^{1-\varepsilon}-1$

with a constant $c>0$ depending only on $\varepsilon>0$ and $M=M(S)$.

## 3\. Lemmas

Let ${\Cal{P}}_{n}^{c}$ denote the set of all algebraic polynomials of degree
at most $n$ with complex coefficients. Our first four lemmas are quite similar
to some of the lemmas used in [21], but some modifications in the formulation
of these lemmas and their proofs are needed. It will be convenient for us to
define $\log 0:=-\infty$.

###### Lemma 3.1

If $S\subset{\mathbb{C}}$ is a finite set,

$S_{k}:=\\{s_{1}+s_{2}+\cdots+s_{k}:\,s_{j}\in S\cup\\{0\\}\\}\,,$
$\gamma:=\min_{z\in S_{k}\setminus\\{0\\}}{|z|}\,,\qquad
M:=M(S):=\max\\{|z|:z\in S\\}\,,$

$P\in{\Cal{P}}_{2n}(S)$, $H(z):=z^{k}-1$, and

$\text{\rm NC}(PH)\leq\mu\,,$ $None$

then

$\int_{-\delta}^{\delta}{|P(e^{it})|\,dt}>\frac{\gamma}{30}\,\log(\text{\rm
NC}_{k}(P))-\frac{\pi^{2}\mu M}{\delta}$

for every $\delta\in(0,\pi)$.

###### Demonstration Proof

We define

$G(z):=\sum_{j=0}^{k-1}{z^{j}}$

so that $H(z)=G(z)(z-1)$. As $P\in{\Cal{P}}_{2n}(S)$ and the set $S$ is
finite, the set $S^{*}$ is also finite. By Theorem 1.3 there is an absolute
constant $c>0$ such that

$\int_{0}^{2\pi}{|(PG)(e^{it})|\,dt}\geq c\gamma\log(\text{\rm NC}(PG))\geq
c_{1}\gamma\log(\text{\rm NC}_{k}(P))\,.$ $None$

We define

$M_{k}:=M(S_{k})=\max\\{|z|:z\in S_{k}\\}\leq k\max\\{|z|:z\in S\\}\leq kM\,.$

Observe that

$|(PG)(e^{it})|=\frac{1}{|e^{it}-1|}\,|(PH)(e^{it})|\leq\frac{\mu
M}{|e^{it}-1|}=\frac{\mu M_{k}}{|2\sin(t/2)|}\leq\frac{\pi\mu
M_{k}}{|2t|}\,,\quad t\in(-\pi,\pi)\,.$

It follows that

$\int_{[-\pi,\pi]\setminus[-\delta,\delta]}{|(PG)(e^{it})|\,dt}\leq
2\pi\frac{\pi\mu M_{k}}{2\delta}=\frac{\pi^{2}\mu kM}{\delta}\,.$ $None$

Now (3.2) and (3.3) give

$\begin{split}\int_{-\delta}^{\delta}{|P(e^{it})|\,dt}&\geq\frac{1}{k}\int_{-\delta}^{\delta}{|(PG)(e^{it})|\,dt}\cr&=\frac{1}{k}\left(\int_{0}^{2\pi}{|(PG(e^{it})|\,dt}-\int_{[-\pi,\pi]\setminus[-\delta,\delta]}{|(PG)(e^{it})|\,dt}\right)\cr&\geq\frac{1}{k}\,c\gamma\log(\text{\rm
NC}_{k}(P))-\frac{\pi^{2}\mu
kM}{k\delta}\cr&\geq\frac{1}{k}\,c\gamma\log(\text{\rm
NC}_{k}(P))-\frac{\pi^{2}\mu M}{\delta}\end{split}$

∎

###### Lemma 3.2

If $S\subset{\mathbb{R}}$ is a finite set, $P\in{\Cal{P}}_{2n}(S)$ is self-
reciprocal, $H(z):=z^{k}-1$,

$\text{\rm NC}(P_{2n}H)\leq\mu\,,$ $None$ $T(t):=P(e^{it})e^{-int}\,,\qquad
R(x):=\int_{0}^{x}{T(t)\,dt}\,,$

$M:=M(S)=\max\\{|z|:z\in S\\}$, and $0<\delta\leq(2k)^{-1}$, then

$\max_{x\in[-\delta,\delta]}{|R_{n}(x)|}<42k(\mu+1)M\,.$

###### Demonstration Proof

Let

$T(t)=a_{0}+\sum_{j=1}^{n}{2a_{j}\cos(jt)}\,,\qquad a_{j}\in S\,.$

Observe that (3.1) implies that

$|\\{j:k\leq j\leq n,\,a_{j-k}\neq a_{j}\\}|\leq\mu\,.$ $None$

We have

$R(x)=a_{0}x+\sum_{j=1}^{n}{\frac{2a_{j}}{j}\,\sin(jx)}\,.$

Now (3.4) implies that

$R_{n}(x)=a_{0}x+\sum_{m=1}^{u}{F_{m,k}(x)}\,,$

where

$F_{m,k}(x):=\sum_{j=0}^{n_{m}}{\frac{2A_{m,k}\sin((j_{m}+jk)x)}{j_{m}+jk}}$

with some $A_{m,k}\in S$, $m=1,2,\ldots,u$, $j_{m}\in{\mathbb{N}}$, and
$n_{m}\in{\mathbb{N}}$, where $u\leq k\mu$ (we do not know much about $j_{m}$
and $n_{m}$). Since the set $S\subset[-M,M]$ is finite, it is sufficient to
prove that

$\max_{x\in[-\delta,\delta]}{|F_{m,k}(x)|}\leq 21\,,\qquad m=1,2,\ldots,u\,,$

that is, it is sufficient to prove that if

$F(x):=\sum_{j=0}^{\nu}{\frac{\sin((j_{0}+jk)x)}{j_{0}+jk}}\,,$

then

$\max_{x\in[-\delta,\delta]}{|F(x)|}=\max_{x\in[0,\delta]}{|F(x)|}\leq 21\,,$
$None$

Note that the equality in (3.5) holds as $F$ is odd. To prove the inequality
in (3.5) let $x\in(0,\delta]$, where $0<\delta\leq(2k)^{-1}$. We break the sum
as

$F=R+S\,,$ $None$

where

$R(x):=\sum_{j=0\atop j_{0}+jk\leq
x^{-1}}^{\nu}{\frac{\sin((j_{0}+jk)x)}{j_{0}+jk}}$

and

$S(x):=\sum_{j=0\atop
x^{-1}<j_{0}+jk}^{\nu}{\frac{\sin((j_{0}+jk)x)}{j_{0}+jk}}\,.$

Here

$\begin{split}|R(x)|\leq&\sum_{j=0\atop j_{0}+jk\leq
x^{-1}}^{\nu}{\left|\frac{\sin((j_{0}+jk)x)}{j_{0}+jk}\right|}\leq(x^{-1}+1)||x|\cr\leq&1+|x|\leq
1+\delta=1+(2k)^{-1}\leq\frac{3}{2}\,,\cr\end{split}$ $None$

where each term in the sum in the middle is estimated by

$\left|\frac{\sin((j_{0}+jk)x)}{j_{0}+jk}\right|\leq\left|\frac{(j_{0}+jk)x)}{j_{0}+jk}\right|=|x|\,,$

and the number of terms in the sum in the middle is clearly at most
$x^{-1}+1$. Further, using Abel rearrangement, we have

$S(x)=-\frac{B_{v}(x)}{j_{0}+vk}+\frac{B_{u}(x)}{j_{0}+uk}+\sum_{j=0\atop
x^{-1}<j_{0}+jk}^{\nu}{B_{j}(x)}\left(\frac{1}{j_{0}+jk}-\frac{1}{j_{0}+(j+1)k}\right)$

with

$B_{j}(x):=B_{j,k}(x):=\sum_{h=0}^{j}{\sin((j_{0}+hk)x)}$

and with some $u,v\in{\mathbb{N}}_{0}$ for which $x^{-1}<j_{0}+(u+1)k$ and
$x^{-1}<j_{0}+(v+1)k$. Hence,

$|S(x)|\leq\left|\frac{B_{v}(x)}{j_{0}+vk}\right|+\left|\frac{B_{u}(x)}{j_{0}+uk}\right|+\sum_{j=0\atop
x^{-1}<j_{0}+jk}^{\nu}{|B_{j}(x)}|\left(\frac{1}{j_{0}+jk}-\frac{1}{j_{0}+(j+1)k}\right)\,.$
$None$

Observe that $x\in(0,\delta]$, $0<\delta\leq(2k)^{-1}$, $x^{-1}<j_{0}+(w+1)k$,
and $w\in{\mathbb{N}}_{0}$ imply

$x^{-1}<j_{0}+(w+1)k<2(j_{0}+wk)\qquad\text{\rm if}\enskip w\geq 1\,,$

and

$2k\leq\delta^{-1}\leq x^{-1}<j_{0}+k\qquad\text{\rm if}\enskip w=0\,,$

and hence

$\frac{1}{j_{0}+wk}\leq 2x\,,\qquad w\in{\mathbb{N}}_{0}\,.$ $None$

Observe also that $x\in(0,\delta]$ and $0<\delta\leq(2k)^{-1}$ imply that
$0<x<\pi k^{-1}$. Hence, with $z=e^{ix}$ we have

$\begin{split}|B_{j}(x)|&=\left|\frac{1}{2}\text{\rm
Im}\left(\sum_{h=0}^{j}{z^{j_{0}+hk}}\right)\right|\leq\left|\frac{1}{2}\sum_{h=0}^{j}{z^{j_{0}+hk}}\right|=\left|\frac{1}{2}\sum_{h=0}^{j}{z^{hk}}\right|\cr&=\left|\frac{1}{2}\,\frac{1-z^{(j+1)k}}{1-z^{k}}\right|\leq\frac{1}{2}\,|1-z^{(j+1)k}|\,\frac{1}{|1-z^{k}|}\leq\frac{1}{|1-z^{k}|}\cr&\leq\frac{1}{\sin(kx/2)}\leq\frac{\pi}{kx}\,.\cr\end{split}$
$None$

Combining (3.8), (3.9), and (3.10), we conclude

$|S(x)|\leq\frac{\pi}{kx}\,2x+\frac{\pi}{kx}\,2x+\frac{\pi}{kx}\,2x\leq\frac{6\pi}{k}\,.$
$None$

Now (3.6), (3.7), and (3.11) give the inequality in (3.5) as $3/2+6\pi/k\leq
21$. ∎

###### Lemma 3.3

If $R$ is a continuously differentiable real-valued function on the interval
$[-\delta,\delta]$, $\delta>0$,

$L:=\int_{-\delta}^{\delta}{|R^{\prime}(x)|\,dx}\qquad and\qquad
N:=\max_{x\in[-\delta,\delta]}{|R(x)|}\,,$

then there is an $\eta\in[-N,N]$ such that $R-\eta$ has at least $L(2N)^{-1}$
distinct zeros in $[-\delta,\delta]$.

###### Lemma 3.4

If $S\subset{\mathbb{R}}$ is a finite set,

$S_{k}:=\\{s_{1}+s_{2}+\cdots+s_{k}:\,s_{j}\in S\cup\\{0\\}\\}\,,$
$\gamma:=\min_{z\in S_{k}\setminus\\{0\\}}{|z|}\,,\qquad
M:=M(S):=\max\\{|z|:z\in S\\}\,,$

$P\in{\Cal{P}}_{2n}(S)$ is self-reciprocal, $T(t):=P(e^{it})e^{-int}$,
$H(z):=z^{k}-1$, and

$\text{\rm NC}(PH)\leq\mu\,,$ $None$

then

$\text{\rm NZ}^{*}(T)\geq\left(\frac{\gamma}{30}\,\log(\text{\rm
NC}_{k}(P))-2k\pi^{2}\mu M\right)(84k(\mu+1)M)^{-1}\,.$

###### Demonstration Proof

Let $0<\delta:=(2k)^{-1}$. Let $R$ be defined by

$R(x):=\int_{0}^{x}{T(t)\,dt}\,.$

Observe that $|T(x)|=|P(e^{ix})|$ for all $x\in{\mathbb{R}}$, and hence Lemma
3.1 yields that

$\begin{split}\int_{-\delta}^{\delta}{|R^{\prime}(x)|\,dx}&\,=\int_{-\delta}^{\delta}{|T(x)|\,dx}=\,\int_{-\delta}^{\delta}{|P(e^{ix})|\,dx}\cr&\,>\frac{\gamma}{30}\,\log(\text{\rm
NC}_{k}(P))-\frac{\pi^{2}\mu
M}{\delta}\cr&\,>\frac{\gamma}{30}\,\log(\text{\rm NC}_{k}(P))-2k\pi^{2}\mu
M\,,\cr\end{split}$

while by Lemma 3.2 we have

$\max_{x\in[-\delta,\delta]}{|R(x)|}<42k(\mu+1)M\,.$

Therefore, by Lemma 3.3 there is an $\eta\in{\mathbb{R}}$ such that $R-\eta$
has at least

$\left(\frac{\gamma}{30}\,\log(\text{\rm NC}_{k}(P))-2k\pi^{2}\mu
M\right)(84k(\mu+1)M)^{-1}\,.$

distinct zeros in $[-\delta,\delta]$. However,
$T_{n}(x)=(R_{n}-c_{n})^{\prime}(x)$ for all $x\in{\mathbb{R}}$, and hence

$\text{\rm NZ}^{*}(T)\geq\left(\frac{\gamma}{30}\,\log(\text{\rm
NC}_{k}(P))-2k\pi^{2}\mu M\right)(84k(\mu+1)M)^{-1}$

follows by Rolle’s Theorem. ∎

The following lemma, in which the assumption $S\subset{\mathbb{Z}}$ is
crucial, is simple to prove. It is stated as Lemma 8 in [42]. Its
straightforward proof given in [42] is reduced to the fact that a determinant
of integer entries is an integer, and hence if it is not $0$, then its modulus
is at least $1$.

###### Lemma 3.5

For $d\in{\mathbb{N}}$ let $A$ be an $d\times d$ invertible matrix with
entries from $S\subset{\mathbb{Z}}$. If $A{\mathbb{x}}=\mathbb{b}$ with

$\mathbb{x}=(x(1),x(2),\ldots,x(d))\in{\mathbb{C}}^{d}\qquad\text{\rm
and}\qquad\mathbb{b}=(b(1),b(2),\ldots,b(d))\in{\mathbb{C}}^{d}\,,$

then

$\max\\{|x(1)|,|x(2)|,\ldots,|x(d)|\\}\leq
M^{d-1}d^{d/2}\max\\{|b(1)|,|b(2)|,\ldots,|b(d)|\\}\,,$

where $M:=M(S):=\max\\{|z|:z\in S\\}$.

For integers $1\leq D\leq N$ we call

$(x(1+r),x(2+r),\ldots,x(D+r))\in{\mathbb{C}}^{N}\,,\qquad
r=0,1,\ldots,N-D\,,$

the $D$-tuples of

$(x(1),x(2),\ldots,x(N))\in{\mathbb{C}}^{N}\,.$

The following lemma is Lemma 9 in [42].

###### Lemma 3.6

For $u,v,D,t\in{\mathbb{N}}$, let $S\subset{\mathbb{Z}}$ be a finite set such
that $v-u>|S|^{D}+3D$, and let

$(x(u+1),x(u+2),\ldots,x(v))\in S^{v-u}\,.$

Let $V$ denote the linear space spanned by the $D$-tuples

$(x(r+1),x(r+2),\ldots,x(r+D))\in S^{D}\,,\qquad r=u,u+1,\ldots,v-D\,,$

over ${\mathbb{R}}$. If $\text{\rm dim}(V)=t<D$, then there are

$(x_{j}(u+D),x_{j}(u+D+1),\ldots,x_{j}(v-D))\in{\mathbb{C}}^{v-u-2D+1}\,,\qquad
j=1,2,\ldots,t\,,$

such that

$x(r)=x_{1}(r)+x_{2}(r)+\cdots+x_{t}(r)\,,\qquad r\in[u+D,v-D]\,,$

where

$(x_{j}(u+D),x_{j}(u+D+1),\ldots,x_{j}(v-D))\in{\mathbb{C}}^{v-u-2D+1}$

are periodic with period $\alpha_{j}\leq 16t\log\log(t+3)$ for each
$j=1,2,\ldots,t$.

Let $f$ be a continuous, even, real-valued function on $K:={\mathbb{R}}$ (mod
$2\pi$) which changes sign on $(0,\pi)$ exactly at $t_{1}<t_{2}<\cdots<t_{d}$.
We define the companion polynomial $Q$ of $f$ by

$e^{-idt}P(e^{i\theta}):=(-1)^{p}\prod_{j=1}^{d}{(\cos t-\cos t_{j})}\,,$

where $p\in\\{0,1\\}$ is chosen so that $f(t)e^{-idt}P(e^{it})\geq 0$ for all
$t\in(-\pi,\pi)$. Observe that

$Q(z):=(-1)^{p}\prod_{j=1}^{d}{(z-e^{it_{j}})(z-e^{-it_{j}})}$

is a monic self-reciprocal algebraic polynomial of degree $2d$ with real
coefficients and with constant term $1$.

Associated with $m\in{\mathbb{N}}$ let $d_{m}:=\text{\rm LCM}(1,2,\ldots,m)$.
It is shown in [40] that $d_{m}<3^{m}$ holds for all $m\in{\mathbb{N}}$, and
this upper bound will be useful for us later in this paper. We remark though
that $\lim_{m\rightarrow\infty}{d_{m}/e^{m}}=1$ holds and it is equivalent to
the Prime Number Theorem, see [47].

###### Lemma 3.7

Suppose $S\subset{\mathbb{Z}}$ is a finite set, $P\in{\Cal{P}}_{2n}(S)$ is
self-reciprocal, and $T(t):=P(e^{it})e^{-int}$ has exactly $2d$ sign changes
in $(-\pi,\pi)$. Let $Q$ be the companion polynomial of $T$ and let

$F(z):=P(z)(z^{d_{m}}-1)^{2}Q(z)=\sum_{j=0}^{2n+2d_{m}+2d}{a_{j}z^{j}}\,,$
$None$

where $m:=\lfloor 32d\log\log(2d+3)\rfloor$, and let

$\\{j\in[0,2n+2d_{m}+2d]:a_{j}\neq 0\\}=\\{j_{1}<j_{2}<\cdots<j_{q}\\}\,.$

If for the integers $1\leq u\leq v\leq q$

$|a_{j_{k}}|<(4M)^{-2d}(2d+1)^{-d-1/2}\,,\qquad k\in[u,v]\,,$ $None$

where $M:=M(S):=\max\\{|z|:z\in S\\}$, then

$v-u<(|S|+2)^{4m+2}+6d+3\,,$

where $|S|$ denotes the number of elements in the set $S$.

###### Demonstration Proof

Let

$S^{*}:=\\{s_{1}-s_{2}:\,s_{1},s_{2}\in S\cup\\{0\\}\\}\subset{\mathbb{Z}}$

and

$S^{**}:=\\{s_{1}-s_{2}:\,s_{1},s_{2}\in S^{*}\\}\subset{\mathbb{Z}}\,.$

Obviously

$|S^{*}|\leq(|S|+1)^{2},\qquad|S^{**}|\leq|S^{*}|^{2}\leq(|S|+1)^{4}\,,$
$None$

and

$M(S^{**})\leq 2M(S^{*})\leq 4M(S)\,.$ $None$

Put $D:=2d+1$. Suppose to the contrary that

$v-u\geq(|S|+2)^{4m+2}+3D\,.$ $None$

Let

$G(z):=P(z)(z^{d_{m}}-1):=\sum_{j=0}^{2n+d_{m}}{\alpha_{j}z^{j}}\,,\qquad\alpha_{j}\in
S^{*}\,,$

and

$H(z):=G(z)(z)(z^{d_{m}}-1)=P(z)(z^{d_{m}}-1)^{2}=\sum_{j=0}^{2n+2d_{m}}{\beta_{j}z^{j}}\,,$
$None$

where

$\beta_{j}:=\alpha_{j-d_{m}}-\alpha_{j}\in S^{**},\qquad
j=0,1,\ldots,2n+2d_{m}\,,$ $None$

with

$\alpha_{j-d_{m}}:=0,\qquad j=0,1,\ldots,d_{m}-1\,.$

Observe that $H\in{\Cal{P}}_{2n+d_{m}}(S^{**})$. Let $V$ denote the linear
space spanned by the $D$-tuples

$(\beta_{j},\beta_{j+1},\ldots,\beta_{j+D-1})\,,\qquad
j\in[j_{u},j_{v}-D+1]\,,$

over ${\mathbb{R}}$. Lemma 3.6 together with (3.13) and (3.15) implies that
$\text{\rm dim}(V)=t<D$. It follows from (3.13) and (3.15) that

$j_{v}-j_{u}\geq v-u\geq(|S|+2)^{4m+2}+3D>|S^{**}|^{m}+3D\,.$

As $t<D=2d+1$, we have $\lfloor 16t\log\log(t+3)\rfloor\leq\lfloor
32d\log\log(2d+3)\rfloor=m$. Applying Lemma 3.7 we obtain that

$(\beta_{j_{u}+D},\beta_{j_{u}+D+1},\ldots,\beta_{j_{v}-D+1})$

is periodic with period $d_{m}$, that is,

$\beta_{r+d_{m}}=\beta_{r}\,,\qquad r\in[j_{u}+D,j_{v}-d_{m}-D]\,.$ $None$

We claim that

$\beta_{r}=0,\qquad r\in[j_{u}+D,j_{v}-D-|S^{*}|d_{m}]\,.$ $None$

Indeed, if $\beta_{r}\neq 0$ for some $r\in[j_{u}+D,j_{v}-D-|S^{*}|d_{m}]\,.$,
then (3.18) and (3.19) give

$\begin{split}\alpha_{r+hd_{m}}&=\alpha_{r}+\sum_{j=1}^{h}{(\alpha_{r+jd_{m}}-\alpha_{r+(j-1)d_{m}})}=\alpha_{r}-\sum_{j=1}^{h}{\beta_{r+jd_{m}}}\cr&=\alpha_{r}-h\beta_{r}\in
S^{*},\qquad h=0,1,\ldots,|S^{*}|\,,\cr\end{split}$

exhibiting $|S^{*}|+1$ distinct elements of $S^{*}$, which is impossible. It
follows from (3.12), (3.17), and (3.20) that

$a_{j}=0,\qquad j\in[j_{u}+2D,j_{v}-D-|S^{*}|d_{m}]\,,$

hence, recalling $d_{m}<3^{m}$ and (3.14), we obtain

$\begin{split}v-u&\leq|S^{*}|d_{m}+3D<|S^{*}|3^{m}+3D<(|S|+1)^{2}(|S|+2)^{4m}+3D\cr&\leq(|S|+2)^{4m+2}+3D\,,\cr\end{split}$

which contradicts (3.16). In conclusion

$v-u<(|S|+2)^{4m+2}+3D=(|S|+2)^{4m+2}+6d+3\,.$

∎

###### Lemma 3.8

Under the assumptions of Lemma 3.7 we have

$\log q\leq 60(4M)^{2d+1}(2d+1)^{d+3/2}(|S|+2)^{4m+2}+6d+3)\,.$

###### Demonstration Proof

Let $L:=(|S|+2)^{4m+2}+6d+3$, and $r:=\lfloor q/L\rfloor$. Observe that

$P(e^{it})e^{-int}(e^{id_{m}t}-1)^{2}e^{-id_{m}t}Q(e^{it}))e^{-idt}$

is real and nonnegative for all $t\in{\mathbb{R}}$. Combining this with
Theorem 1.7 and Lemma 3.7 we obtain

$\begin{split}&\,\int_{0}^{2\pi}{P(e^{it})e^{-int}(e^{id_{m}t}-1)^{2}e^{-id_{m}t}Q(e^{it}))e^{-idt}\,dt}\cr=&\int_{0}^{2\pi}{|P_{2n}(e^{it})(e^{id_{m}t}-1)^{2}Q(e^{it})|\,dt}\cr\geq&\frac{1}{30}\,\sum_{k=1}^{q}{\frac{|a_{j_{k}}|}{k}}\geq\frac{1}{30}\,\sum_{j=1}^{r}\sum_{k=(j-1)L+1}^{jL}{\frac{|a_{j_{k}|}}{k}}\cr\geq&\frac{1}{30}(4M)^{-2d}(2d+1)^{-d-1/2}\sum_{j=1}^{r}{\frac{1}{jL}}\cr\geq&\frac{1}{30}(4M)^{-2d}(2d+1)^{-d-1/2}L^{-1}\log
r\cr\end{split}$ $None$

On the other hand, using orthogonality, we have

$\int_{0}^{2\pi}{P(e^{it})e^{-int}(e^{id_{m}t}-1)^{2}e^{-id_{m}t}Q(e^{it})e^{-idt}\,dt}\leq
4M(2d+1)\,.$ $None$

Combining (3.21) and (3.22) we conclude

$\frac{1}{30}(4M)^{-2d}(2d+1)^{-d-1/2}L^{-1}\log r\leq 4M(2d+1)\,,$

and hence

$\begin{split}\log q=\log r+\log L&\leq 60(4M)^{2d+1}(2d+1)^{d+3/2}L\cr&\leq
60(4M)^{2d+1}(2d+1)^{d+3/2}((|S|+2)^{4m+2}+6d+3)\,.\cr\end{split}$

∎

Our final lemma follows easily from Lemma 3.6.

###### Lemma 3.9

If $S\subset{\mathbb{Z}}$ is a finite set, $P\in{\Cal{P}}_{2n}(S)$,
$0\not\equiv R\in{\Cal{P}}_{u}^{c}$,

$\text{\rm NC}(PR)\leq\nu\,,$

$v:=\lfloor 16u\log\log(u+3)\rfloor$, $k:=d_{v}$, and $H(z)=z^{k}-1$, then

$\text{\rm NC}(PH)\leq\mu:=(\nu+1)(k+|S|^{u+1}+3(u+1)+2)\,.$

## 4\. Proof of the Theorems

###### Demonstration Proof of Theorem 2.1

Suppose $S\subset{\mathbb{Z}}$ is a finite set, $P\in{\Cal{P}}_{2n}(S)$ is
self-reciprocal, and $T(t):=P(e^{it})e^{-int}$ has exactly $2d$ sign changes
in $(-\pi,\pi)$. Let $Q$ be the companion polynomial of $T$. Let

$F(z):=P(z)(z^{d_{m}}-1)^{2}Q(z)=\sum_{j=0}^{2n+2d_{m}+2d}{a_{j}z^{j}}\,,$

where $m:=\lfloor 32d\log\log(2d+3)\rfloor$. Let

$\\{j\in[0,2n+2d_{m}+2d]:a_{j}\neq 0\\}=:\\{j_{1}<j_{2}<\cdots<j_{q}\\}\,.$

Lemma 3.8 together with $|S|+2\leq 2M(S)+3=2M+3\leq 5M$ and $4m+2d+3\leq 5m$
implies

$\log q\leq 60(5M)^{5m}(2d+1)^{d+3/2}\,,$ $None$

and hence applying Lemma 3.9 with $u:=2d_{m}+2d$, we have

$\text{\rm NC}(PH)\leq\mu:=(q+1)(d_{v}+|S|^{u+1}+3(u+1)+2)\,.$ $None$

with $v:=\lfloor 16u\log\log(u+3)\rfloor$, $k:=d_{v}$, and $H(z)=z^{k}-1$.

Observe that if $S\subset{\mathbb{Z}}$ and

$S_{k}:=\\{s_{1}+s_{2}+\cdots+s_{k}:\,s_{j}\in S\cup\\{0\\}\\}\,,$

then

$1\leq\gamma:=\min\\{|z|:z\in S_{k}\setminus\\{0\\}\\}\,.$

Lemma 3.4 gives

$84k(\mu+1)kM(2d)+2k\pi^{2}\mu M\geq\log(\text{\rm NC}_{k}(P))\,.$

Combining this with $m:=\lfloor 32d\log\log(2d+3)\rfloor$, $u:=2d_{m}+2d$,
$v:=\lfloor 16u\log\log(u+3)\rfloor$, $k=d_{v}<3^{v}$, $d_{m}<3^{m}$, (4.1),
(4.2), $|S|\leq 2M(S)+1=2M+1$, and the inequality $a+b\leq ab$ valid for all
$a\geq 1$ and $b\geq 1$, we obtain

$\begin{split}&\log(84k(\mu+1)M2d)+2k\pi^{2}\mu
M)\leq\log(84k(\mu+1)M(2d+1)\cr\leq&\,6+\log k+\log\nu+\log
M+\log(2d+1)\cr\leq&\,6+\log k+\log(q+1)+\log
d_{v}+\log(|S|^{u+1}+\log(3u+5))+\log M+\log(2d+1)\cr\leq&\,6+(\log 3)v+\log
2+60(5M)^{5m}(2d+1)^{d+3/2}+(\log 3)v+(u+1)\log(2M+1)\cr+&\log(3u+5)+\log
M+\log(2d+1)\cr\leq&\,7+2(\log
3)16u\log\log(u+3)+60(5M)^{5m}(2d+1)^{d+3/2}+(2\cdot
3^{m}+2d+1)\log(2M+1)\cr+&(\log 2+(\log 3)m)\log(2d+1)\log(2M+1)+\log
M+\log(2d+1)\cr\end{split}$

and hence

$\begin{split}&\log(84k(\mu+1)M2d)+2k\pi^{2}\mu M)\cr\leq&7+2(\log 3)16(2\cdot
3^{m}+2d)\log\log(2\cdot 3^{m}+2d)+60(5M)^{5m}(2d+1)^{d+3/2}+\cr+&(2\cdot
3^{m}+2d+1)\log(2M+1)+(\log 2+(\log 3)m)\log(2d+1)\log(2M+1)\cr+&\log
M+\log(2d+1)\,.\cr\end{split}$

and it follows that there is an absolute constant $c_{1}>0$ such that

$c_{1}((d+1)\log(d+2)+(d+1)\log\log(2d+3)(1+\log M))\geq\log\log\log(\text{\rm
NC}_{k}(P))\,.$ $None$

It is easy to see that

$\text{\rm NC}_{k}(P)\geq\frac{k|P(1)|-2k^{2}M}{kM}=\frac{|P(1)|}{M}-k\,.$

Therefore if $|P(1)|\geq 2kM$, then $\text{\rm
NC}_{k}(P)\geq\frac{1}{2}|P(1)|$, and the theorem follows from (4.3) after a
straightforward calculus. If $|P(1)|<2kM$, then it follows from (4.3) that

$\log\log\log(|P(1)|)<\log\log\log(2kM)\leq c_{2}\log(d+2)\leq c_{2}(2d+1)$

with an absolute constant $c_{2}>0$, and the theorem follows. ∎

###### Demonstration Proof of Corollary 2.2

Let $S\subset{\mathbb{Z}}$ be a finite set. If $P\in{\Cal{P}}_{2\nu}(S)$ is
self-reciprocal, then the corollary follows from Theorem 2.1. If
$P\in{\Cal{P}}_{2\nu+1}(S)$ is self-reciprocal, then
$\widetilde{P}\in{\Cal{P}}_{2\nu+2}^{c}(S^{*})$ defined by

$\widetilde{P}(z):=(z+1)P(z)\in{\Cal{P}}_{2\nu+2}^{c}(S^{*})$

is also self-reciprocal, where the fact that $S\subset{\mathbb{Z}}$ is finite
implies that the set

$S^{*}:=\\{s_{1}+s_{2}:\,s_{1},s_{2}\in S\cup\\{0\\}\\}\subset{\mathbb{Z}}\,,$

is also finite. Observe also that

$M(S^{*})=\max\\{|z|:z\in S^{*}\\}=2\max\\{|z|:z\in S\\}=2M(S)$

and

$\widetilde{P}(1)=2P(1)\,.$

Hence applying Theorem 2.1 to $\widetilde{P}\in{\Cal{P}}_{2\nu+2}(S^{*})$, we
obtain the statement of the corollary for $P\in{\Cal{P}}_{2\nu+1}(S)$ from
Theorem 2.1 again. ∎

###### Demonstration Proof of Corollary 2.3

The corollary follows from Theorem 2.1 and the fact that for every
trigonometric polynomial $T$ of the form

$T(t):=a_{0}+\sum_{j=1}^{n}{a_{j}\cos(jt)}\,,\qquad a_{j}\in{\mathbb{Z}}\,,$

there is a self-reciprocal algebraic polynomial $P$ of the form

$P(z)=2a_{0}z^{n}+\,\sum_{j=1}^{n}{a_{j}(z^{n+j}+z^{n-j})}\,,\qquad
a_{j}\in{\mathbb{Z}}\,,$

such that

$2T(t):=P(e^{it})e^{-int}\,.$

∎

## References

  * 1 V.V. Andrievskii and H-P. Blatt, Discrepancy of Signed Measures and Polynomial Approximation, Springer, 2002.
  * 2 A. Bloch and G. Pólya,, On the roots of certain algebraic equations, Proc. London Math. Soc. 33 (1932), 102–114.
  * 3 P. Borwein, Computational Excursions in Analysis and Number Theory, Springer, 2002.
  * 4 P. Borwein, S. Choi, R. Ferguson, and J. Jankauskas, On Littlewood polynomials with prescribed number of zeros inside the unit disk, Canad. J. of Math. 67 (2015), 507–526.
  * 5 P. Borwein and T. Erdélyi, On the zeros of polynomials with restricted coefficients, Illinois J. Math. 41 (1997), no. 4, 667–675.
  * 6 P. Borwein and T. Erdélyi, Lower bounds for the number of zeros of cosine polynomials in the period: a problem of Littlewood, Acta Arith. 128 (2007), no. 4, 377–384.
  * 7 P. Borwein, T. Erdélyi, R. Ferguson, and R. Lockhart, On the zeros of cosine polynomials: solution to a problem of Littlewood, Ann. Math. Ann. (2) 167 (2008), no. 3, 1109–1117.
  * 8 P. Borwein, T. Erdélyi, and G. Kós, Littlewood-type problems on $[0,1]$, Proc. London Math. Soc. 79 (1999), 22–46.
  * 9 P. Borwein, T. Erdélyi, and G. Kós, The multiplicity of the zero at $1$ of polynomials with constrained coefficients, Acta Arith. 159 (2013), no. 4, 387–395.
  * 10 P. Borwein, T. Erdélyi, and F. Littmann, Zeros of polynomials with finitely many different coefficients, Trans. Amer. Math. Soc. 360 (2008), 5145–5154.
  * 11 D. Boyd, On a problem of Byrne’s concerning polynomials with restricted coefficients, Math. Comput. 66 (1997), 1697–1703.
  * 12 B. Conrey, A. Granville, B. Poonen, and K. Soundararajan, Zeros of Fekete polynomials, Ann. Inst. Fourier (Grenoble) 50 (2000), 865–889.
  * 13 R.A. DeVore and G.G. Lorentz, Constructive Approximation, Springer-Verlag, 1993.
  * 14 Y. Do, H. Nguyen, and V. Vu, Real roots of random polynomials: expectation and repulsion (to appear).
  * 15 P. Drungilas, Unimodular roots of reciprocal Littlewood polynomials, J. Korean Math. Soc. 45 (2008), no. 3, 835–840.
  * 16 A. Edelman and E. Kostlan, How many zeros of a random polynomial are real?, Bull. Amer. Math. Soc. (N.S.) 32 (1995), 1–37. , Erratum:, Bull. Amer. Math. Soc. (N.S.) 33 (1996), 325.
  * 17 T. Erdélyi, On the zeros of polynomials with Littlewood-type coefficient constraints, Michigan Math. J. 49 (2001), 97–111.
  * 18 T. Erdélyi, An improvement of the Erdős-Turán theorem on the distribution of zeros of polynomials, C. R. Acad. Sci. Paris, Ser. I 346 (2008), no. 5, 267–270.
  * 19 T. Erdélyi, Extensions of the Bloch-Pólya theorem on the number of real zeros of polynomial, J. Théor. Nombres Bordeaux 20 (2008), no. 2, 281–-287.
  * 20 T. Erdélyi, Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at $1$, Acta Arith. 172 (2016), no. 3, 271–284.
  * 21 T. Erdélyi, The number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set, Acta Arith. 176 (2016), no. 2, 177–200.
  * 22 P. Erdős and A. C. Offord, On the number of real roots of a random algebraic equation, Proc. London Math. Soc. 6 (1956), 139–160.
  * 23 P. Erdős and P. Turán, On the distribution of roots of polynomials, Ann. Math. 57 (1950), 105–119.
  * 24 D. Hanson, On the product of the primes, Canad.Math. Bull. 15 (1972), 33–37.
  * 25 M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49 (1943), 314–320.
  * 26 M. Kac, On the average number of real roots of a random algebraic equation. II, Proc. London Math. Soc. 50 (1949), 390–408.
  * 27 S.V. Konyagin, On a problem of Littlewood, Mathematics of the USSR, Izvestia 18 (1981), 205–225.
  * 28 S.V. Konyagin and V.F. Lev, Character sums in complex half planes, J. Theor. Nombres Bordeaux 16 (2004), no. 3, 587–606.
  * 29 J.E. Littlewood, On the mean values of certain trigonometrical polynomials, J. London Math. Soc. 36 (1961), 307–334.
  * 30 J.E. Littlewood, On the real roots of real trigonometrical polynomials (II), J. London Math. Soc. 39 (1964), 511–552.
  * 31 J.E. Littlewood, On polynomials $\sum\pm z^{m}$ and $\sum e^{\alpha_{m}i}z^{m}$, $z=e^{\theta i}$, J. London Math. Soc. 41 (1966), 367–376.
  * 32 J.E. Littlewood, Some Problems in Real and Complex Analysis, Heath Mathematical Monographs, 1968.
  * 33 J.E. Littlewood and A.C. Offord, On the number of real roots of a random algebraic equation. II, Proc. Cambridge Philos. Soc. 35 (1939), 133–148.
  * 34 J.E. Littlewood and A.C. Offord, On the number of real roots of a random algebraic equation. III, Rec. Math. [Mat. Sbornik] N.S. 54 (1943), 277–286.
  * 35 O.C. McGehee, L. Pigno, and B. Smith, Hardy’s inequality and the $L_{1}$ norm of exponential sums, Ann. Math. 113 (1981), 613–618.
  * 36 I.D. Mercer, Unimodular roots of special Littlewood polynomials, Canad. Math. Bull. 49 (2006), no. 3, 438–447.
  * 37 K. Mukunda,, Littlewood Pisot numbers, J. Number Theory 117 (2006), no. 1, 106–121.
  * 38 H. Nguyen, O. Nguyen, and V. Vu, On the number of real roots of random polynomials (to appear).
  * 39 I.E. Pritsker and A.A. Sola, Expected discrepancy for zeros of random algebraic polynomials, Proc. Amer. Math. Soc. 142 (2014), 4251-4263.
  * 40 J.B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94.
  * 41 E. Schmidt, Über algebraische Gleichungen vom Pólya-Bloch-Typos, Sitz. Preuss. Akad. Wiss., Phys.-Math. Kl. (1932), 321.
  * 42 I. Schur, Untersuchungen über algebraische Gleichungen, Sitz. Preuss. Akad. Wiss., Phys.-Math. Kl. (1933), 403–428.
  * 43 J. Sahasrabudhe, Counting zeros of cosine polynomials: on a problem of Littlewood (to appear arXiv:1610.07680 [math.CA]).
  * 44 B. Solomyak, On the random series $\sum\pm\lambda^{n}$ (an Erdős problem), Ann. Math. 142 (1995), 611–625.
  * 45 G. Szegő, Bemerkungen zu einem Satz von E. Schmidt uber algebraische Gleichungen, Sitz. Preuss. Akad. Wiss., Phys.-Math. Kl. (1934), 86–98.
  * 46 T. Tao and V. Vu, Local universality of zeros of random polynomials, IMRN (2015) (to appear).
  * 47 G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, 1995.
  * 48 V. Totik and P. Varjú, Polynomials with prescribed zeros and small norm, Acta Sci. Math. (Szeged) 73 (2007), 593–612.


# Interaction of cosmological domain walls with large classical objects, like
planets and satellites, and the flyby anomaly

De-Chang Dai1,2111communicating author: De-Chang Dai,
email<EMAIL_ADDRESS>Djordje Minic3, Dejan Stojkovic4 1 Center for
Gravity and Cosmology, School of Physics Science and Technology, Yangzhou
University, 180 Siwangting Road, Yangzhou City, Jiangsu Province, P.R. China
225002  2 CERCA/Department of Physics/ISO, Case Western Reserve University,
Cleveland OH 44106-7079  3 Department of Physics, Virginia Tech, Blacksburg,
VA 24061, U.S.A.  4 HEPCOS, Department of Physics, SUNY at Buffalo, Buffalo,
NY 14260-1500, U.S.A.

###### Abstract

Cosmological domain walls can be formed as a result of symmetry breaking at
any epoch during the evolution of our universe. We study their interaction
with a classical macroscopic object, like Earth or a satellite in Earth’s
orbit. We set up an action that includes the interaction term between the
massive classical object and the scalar field that the domain wall is made of.
We use numerical calculations to solve the coupled equations of motion which
describe the crossing between the domain wall and the classical object.
Depending on the strength of the interaction, relative velocity and size, the
object can be either stopped by the wall, or it can pass through it inducing
deformations in the wall that cost energy. At the same time, the coupling to
the scalar filed might change the object’s mass during the crossover. The fact
that satellites in Earth’s orbit (or planets in Sun’s orbit) can change their
mass and/or lose energy interacting with walls can be used as a new domain
wall detection probe. For example, a typical velocity precision of a satellite
is about $0.5$ mm/s, which directly puts an upper limit on its mass change to
$\Delta M/M\lessapprox 5\times 10^{-17}$. Alternatively, a known satellite
flyby anomaly can easily be explained as an interaction with a closed domain
wall. We also show that the presence of matter modifies the scalar filed
potential and can locally create a bubble of the true vacuum, and thus trigger
the decay of the false vacuum. For a critical bubble which is able to expand,
such an interaction with the domain wall must be strong enough.

##  I introduction

Domain walls are sheet-like topological defects, which may be created in the
early Universe when a discrete symmetry is spontaneously broken. A domain wall
separates the space into two distinguishable vacuum states. The fundamental
constants (like particle masses, fine structure constant etc) can change after
crossing the domain wall Kibble:1976sj ; Vilenkin:1982ks ; Vilenkin:1984ib .
The energy density of the the domain wall network might dominate the total
energy density of the Universe if the domain walls are generated at early
times Zeldovich et al. (1975). This problem could be solved if the walls are
unstable or using some other elaborate mechanism Vilenkin:1981zs ;
Gelmini:1988sf ; Larsson:1996sp ; Dvali:1995cc ; Stojkovic:2005zh ;
Stojkovic:2004hz . On the other hand, if the walls are formed in the late time
epochs of the universe, they will be light and thus not easily detectable.
Such domain walls can even play the role of dark matter candidates or generate
other interesting structures Dai:2020rnc .

Since domain wall can change the fundamental constants, there are many
experiments invented to test their existence. These methods include mass
difference causing acceleration McNally:2019lcg , variance of the fine
structure constant Roberts:2019sfo , electric dipole moment measurements
Stadnik:2014cea , measuring magnetometers Pospelov:2012mt ; Pustelny et al.
(2013); JacksonKimball:2017qgk ; Afach:2021pfd , satellite synchronization
Derevianko:2013oaa ; Roberts:2017hla ; Kalaydzhyan:2017jtv  and gravitational
wave detectors Hall:2016usm ; Jaeckel:2020mqa ; Grote:2019uvn .

The interaction between the domain wall and regular matter can be classified
using a collection of gauge-invariant operators parameterizing the Standard
Model interactions with the fields that make up the composition of the wall
(see e.g. Essig:2013lka ). However, in these studies matter was considered to
be a fundamental point-like particle which will not distort the domain wall
during the crossing. In contrast, this paper focuses on classical objects of
finite sizes. Interaction of black holes with domain walls was studied in
Stojkovic:2004wy ; Frolov:2003mc ; Frolov:2004wy ; Frolov:2004bq ;
Frolov:1998td ; Christensen:1998hg , however no study was performed for finite
but regular classical objects. Earth or a satellite in Earth’s orbit are in
general such classical objects which, in principle, can distort a domain wall.
We find that when a macroscopic object encounters the domain wall, it may be
reflected or it may pass through depending on the strength of the interaction,
relative velocity and size. During the encounter, the object loses its energy
to deformations of the wall. In addition, the object changes its mass by
passing from one side of the wall to another. We track motions of satellites
in Earth’s orbit with a great precision, so minor changes in satellites’
parameters are observable in principle. For example, a typical velocity
precision of a satellite is about $0.5$ mm/s, which directly puts an upper
limit on its mass change to $\Delta M/M\lessapprox 5\times 10^{-17}$.
Therefore, this effect can offer a new method for detection of domain walls by
observing whether a satellite’s velocity gets suddenly changed. The famous
flyby anomaly – anomalous increase in speed observed during a planetary flyby
by a satellite – is very similar to this effect.

In the second part of the paper we show how matter can trigger the decay of
the false vacuum. The presence of matter modifies the scalar filed potential
and can locally create a bubble of the true vacuum. For a critical bubble
which is able to expand, interaction with the domain wall must be strong
enough.

##  II Interaction of a scalar field with classical objects

Consider a solid massive spherically symmetric object, with a uniform mass
distribution and radius $R$ in its own rest frame. The object is moving with
velocity $v_{O}$ in $z$-direction. The location of the center of the object
along the $z$-axis is labeled by $z_{O}$. The normalized mass distribution
function of the object in axially symmetric (around $z$-axis) coordinates is

$f(z,r,z_{O},v_{O})=\Theta(R-\sqrt{r^{2}+\frac{(z-z_{O})^{2}}{1-v_{O}^{2}}}),$
(1)

where $\Theta(x)$ is the Heaviside function. The radial coordinate is
$r^{2}=x^{2}+y^{2}$, while $1/\sqrt{1-v_{O}^{2}}$ is the standard Lorentz
contraction factor. The relativistic action for this object is

$S_{m}=-m\int\sqrt{1-v_{O}^{2}}dt,$ (2)

where $m$ is the object’s mass in the absence of any couplings.

Consider now a domain wall made of a scalar field $\phi$. The scalar field’s
action is

$S_{\phi}=\int\frac{(\partial_{t}\phi)^{2}}{2}-\frac{(\nabla\phi)^{2}}{2}-V(\phi)d^{4}x,$
(3)

The coupling between the massive object and the scalar field can be described
by

$S_{\rm int}=-A\int f(\vec{r})I(\phi)d^{4}x,$ (4)

where $A$ is the coupling constant, while $f(\vec{r})$ is given in Eq. (1).
$I(\phi)$ is a coupling function, which is model dependent. The equations of
motion that follow from the total action $S_{\rm tot}=S_{m}+S_{\phi}+S_{\rm
int}$ are

$\displaystyle\partial_{t}^{2}\phi-\partial_{r}^{2}\phi-\frac{1}{r}\partial_{r}\phi-\partial_{z}^{2}\phi+V^{\prime}+Af(\vec{r})I^{\prime}=0$
(5)
$\displaystyle\frac{d}{dt}\Big{(}\frac{v_{O}}{\sqrt{1-v_{O}^{2}}}M\Big{)}=A\int_{0}^{R}(I(\phi_{d})-I(\phi_{u})2\pi
rdr$ (6) $\displaystyle
M=m+A\int_{0}^{R}(I(\phi_{u})+I(\phi_{d}))\sqrt{R^{2}-r^{2}}2\pi rdr,$ (7)

where $\phi_{u}=\phi(r,z_{u})$, $\phi_{d}=\phi(r,z_{d})$, with
$z_{d}=z_{O}-\sqrt{R^{2}-r^{2}}\sqrt{1-v_{O}^{2}}$ and
$z_{u}=z_{O}+\sqrt{R^{2}-r^{2}}\sqrt{1-v_{O}^{2}}$. The parameters $z_{d}$ and
$z_{u}$ are the limits within which the object moves in $z$-direction. $M$ is
the object’s effective mass which is modified by the presence of $\phi$. The
effective potential of the scalar field can be now written as

$V_{eff}=V+AfI.$ (8)

###  II.1 Concrete model

We choose to work with a well known scalar field potential that provides the
existence of domain walls

$\displaystyle V=B(\phi^{2}-\Lambda^{2})^{2},$ (9)

where $B$ and $\Lambda$ are constants. In particular, $\Lambda$ is the vacuum
expectation value of the scalar field. One dimensional kink and anti-kink
solutions are described by the following profile

$\phi=\Lambda\tanh(\pm\Lambda\sqrt{2B}\frac{z-vt-z_{i}}{\sqrt{1-v^{2}}}),$
(10)

where $z_{i}$ is the initial position of the kink, while $v$ is it’s velocity.
Without any loss of generality we boost the object to a frame where the domain
wall is at rest initially, i.e. $v=0$, and its center is located at $z_{i}=0$.
These are the initial conditions of the domain wall before it interacts with
the massive object. The kink solution is shown in the plot $t=0$ in Fig. 2.
The energy per unit area of the domain wall is

$\rho=\frac{4}{3}\sqrt{2B}\Lambda^{3}.$ (11)

We take the coupling function to be

$I=(\phi+\Lambda)^{2}.$ (12)

With this choice, the object will not acquire any extra mass at
$\phi=-\Lambda$ vacuum, so $M=m$ before the object encounters the wall. Fig. 1
shows that the presence of the object inevitably modifies the potential. The
true and false vacuum states in empty space are different from the same states
inside the matter distribution. The presence of matter can therefore change
the scalar field’s vacuum state.

Figure 1:  The effective potential of a scalar field which interacts with a
massive object. The solid line (no interaction) shows a potential with two
equal minima at $\phi=\pm 1$. The potential and couplings are chosen as in
Eqs. (9) and (12) with $B=\Lambda=1$. The dashed line, $A=0.1$, shows a
potential with the true vacuum at $\phi=-1$ and the false vacuum at $\phi=1$.
For larger values of $A$ there is no false vacuum.

##  III Method

The equations of motion in Eq. (5) must be solved numerically. We replace the
derivatives with

$\displaystyle\partial_{t}^{2}\phi$ $\displaystyle=$
$\displaystyle\frac{\phi(t+\Delta t)-2\phi(t)+\phi(t-\Delta t)}{\Delta t^{2}}$
(13) $\displaystyle\partial_{z}^{2}\phi$ $\displaystyle=$
$\displaystyle\frac{\phi(z+\Delta z)-2\phi(z)+\phi(z-\Delta z)}{\Delta z^{2}}$
(14) $\displaystyle\partial_{r}^{2}\phi$ $\displaystyle=$
$\displaystyle\frac{\phi(r+\Delta r)-2\phi(r)+\phi(r-\Delta r)}{\Delta r^{2}}$
(15) $\displaystyle\partial_{r}\phi$ $\displaystyle=$
$\displaystyle\frac{\phi(r+\Delta r)-\phi(r-\Delta r)}{2\Delta r},$ (16)

where $\phi(t=0)$, $\phi(t=\Delta t)$, $v_{O}(t=0)$ and $x_{0}(t=0)$ are given
as initial conditions. The evolution of the field and object is obtained by
iterations. We choose $\Delta r=\Delta z=3\Delta t=10^{-2}$ to preserve the
stability of the numerical method. The numerical range is $0\leq r\leq 30$ and
$-30\leq z\leq 30$. The singularity at $r=0$ is avoided by replacing

$\lim_{r\rightarrow 0}\frac{1}{r}\partial_{r}\phi=\lim_{r\rightarrow
0}\partial_{r}^{2}\phi.$ (17)

The integral term in Eq. (5) is calculated using the Simpson method.

Figure 2: Distortion of the wall in cylindrical coordinates $(r,z)$, during
the collision with a massive object moving in the $z$-direction. The color
code represents the values of the scalar filed. The initial conditions are
chosen to be $m=10^{4}$, $R=1.5$, $v_{O}=0.1$, $A=0.5$ and $x_{0}=-3.5$. At
$t=0$, the domain wall is located at $z=0$ without any distortion. At $t=20$,
the wall is pushed by the object to the right around $r=0$ and $z=0$. From
$t=40$ to $t=80$, the distortion becomes clearer. At $t=90$, the object starts
to separate from the wall, where a significant bump can be observed. From
$t=94$ to $t=100$, the object is completely separated from the wall, and a
separate closed region with $\phi=\Lambda$ (new vacuum) is created. This is a
bubble of the scalar field formed around the massive object. The wall bounces
back to its equilibrium configuration. There are domain wall waves generated
in this process, but they cannot be clearly seen in these plots because of the
small amplitudes.

##  IV Results

We discuss two distinct cases. The first case is an object crossing the domain
wall from one vacuum to another. The second case is an object triggering the
vacuum decay by generating a bubble of the local vacuum state.

###  IV.1 Collision between a domain wall and a classical object

We consider the scalar potential in Eq. (9) with $B=1$ and $\Lambda=1$. The
scalar field generates a domain wall structure as in Eq. (10). In the absence
of the interaction, i.e. $A=0$, Fig. 1 shows that the potential has two equal
minima, which are at $\phi=\pm\Lambda$. Initially, the domain wall is at rest,
located at $z_{i}=0$. The $t=0$ plot in Fig. 2 shows the scalar field with the
domain wall structure. There is an obvious transition at $z=0$. The vacuum
state at $z<0$ ($z>0$) is $\phi=-\Lambda$ ($\phi=\Lambda$). Initially the
object is at $z_{O}=-3.5$ and moves to the right with velocity $v_{O}$. The
crossing starts when the object reaches the wall. The wall is pushed to the
right and it is distorted (segments from $t=0$ to $t=90$ in Fig. 2). The wall
is then pushed farther, and if the object has enough energy it crosses the
wall, while the wall bounces back to its original place (segments from $t=90$
to $t=100$ in Fig. 2). If the object does not have enough energy, the object
is reflected back (as described with $m<5000$ case in Fig. 3). In this
process, the object changes its mass and velocity and its energy is applied to
distorting the wall and generating waves, which cannot be seen directly in the
plots in Fig. 2, because of their small amplitude. These waves can be seen in
Fig. 9 and 10. We note that we can chose the values for $R$ and $m$
independently since gravity is not included in the action. In other words, an
object with $R=1$ and $m=10^{5}$ is not necessarily within its own
Schwarzschild radius.

Fig. 4 shows how the velocity of an object changes in the process. Very light
objects are reflected completely, which resembles the classical collision
between a very light and very heavy object. As the object becomes more
massive, it distorts the wall more and more. The object’s energy is applied
toward the wall’s distortion and wave dispersion. The reflected object’s speed
is lower than the initial speed. An object with a mass in a medium range may
gain some distortion energy back when the wall is bounced back to its balanced
position (e.g. $m=10^{3}$ case). In this case the object loses less energy
than in some of the lower mass cases, since the wall acts like a spring. In
contrast, very massive objects just pass through the wall and do not recover
any energy back from the distorted wall, so they lose more energy than less
massive objects.

Fig. 5 shows how the energy of an object changes during the collision. The
energy is calculated according to

$E_{O}=M/\sqrt{1-v_{O}^{2}},$ (18)

where $v_{O}$ is the velocity of the object. At the beginning, the object
gains some energy, because the value of $\phi$ is increasing and the effective
mass, $M$, is increasing along with it. Then this gain is not sufficient to
compensate for the loss, and more and more energy is released to the wall
distortion and waves. At the end, the lost energy is higher than the energy
gained from the wall. This lost energy does not come back to the object.

The velocity of the object plays twofold role during the crossing. First, the
velocity increases available kinetic energy, and second, it causes the Lorentz
contraction which also changes the interaction between the object and wall.
Fig. 6 shows that only objects with high enough velocity can cross the wall.

Fig. 7 shows how the motion of the object varies as a function of its radius.
If the radius is small, the wall resistance is smaller, and the object easier
penetrates the wall. Objects of larger radius induce greater distortion during
the crossing, and in turn more energy is lost.

Figure 3:  The path of the object during its interaction with the wall. On the
y-axis we have $z_{O}$, which represents the location of the center of the
object at some moment in time. The parameters are chosen to be $A=0.5$,
initial $v_{O}=0.1$, $R=1.5$, while the values for the object’s mass $m$ are
marked in the plot. If the object is light, its momentum and energy are low,
so it bounces back from the wall. As it becomes more massive, the wall bends
more and more, up until the point where energy and momentum are high enough so
that the wall cannot stop it any more, and the object passes through the wall.
Figure 4:  Evolution of the velocity of the object, $v_{O}$, during the
collision with the wall as a function of time. The parameters are chosen to be
$A=0.5$, initial $v_{O}=0.1$, $R=1.5$, while the values for the object’s mass
$m$ are marked in the plot. A very light object is bounced back (velocity is
reversed) without any energy loss nor gain. Medium mass objects are reflected
by the wall, but they also gain some wall distortion energy back during the
collision. In these cases the wall acts like a spring. Therefore their
velocity can be larger than in some of the lighter cases. Very massive objects
pass through the wall and cannot gain the distortion energy back, so they lose
the highest amount of energy.  Figure 5:  The relative change in energy of the
object, $\Delta E/E_{i}$, in the collision process with the wall as a function
of time. The parameters are chosen to be $A=0.5$, initial $v_{O}=0.1$,
$R=1.5$, while the values for the object’s mass $m$ are marked in the plot. As
an object is approaching the wall its energy is increasing. Since the value of
the scalar field at the object’s location is increasing during the approach,
the effective mass of the object, $M$, is increasing. Thus the energy of the
object is increasing. During and after the interaction, the energy is
decreasing since the velocity is decreasing, and also some of the energy is
used to deform the wall. For medium mass cases, the object regains some energy
when the wall bounces back (e.g. for $M=10^{3}$), so energy is increasing
again at late times.  Figure 6: The path of the object in the collision
process with the wall as a function of its velocity. On the y-axis we have
$z_{O}$, which represents the location of the center of the object at some
moment in time. The parameters are chosen to be $A=5\times 10^{-4}$, $m=1$,
$R=1.5$, while the initial values for initial $v_{O}$ are marked in the plot.
For low velocities the object cannot cross the wall. For high enough
velocities the object has enough energy to pass through the wall.  Figure 7:
The path of the object in the collision process with the wall as a function of
its radius. On the y-axis we have $z_{O}$, which represents the location of
the center of the object at some moment in time. The parameters are chosen to
be $m=5000$. $A=0.5$. initial $v_{O}=0.1$, while the values for $R$ are marked
in the plot. If the radius of the object is small the resistance of the wall
is small, and it is easier to go through the wall. Objects of larger radii
create greater distortion of the wall and lose more energy.

###  IV.2 Vacuum decay in the presence of a massive object

The presence of matter can change the scalar field effective potential, as we
have seen in Fig. 1. The true vacuum can be modified by the matter
distribution and become a false vacuum locally. The opposite is also true, a
region of false vacuum can be converted into a true vacuum state. In other
words, the presence of matter can trigger a vacuum decay. To study this
effect, we add an extra term in the scalar filed potential which breaks the
$\phi\rightarrow-\phi$ symmetry and introduces the difference in the vacua

$V=B(\phi^{2}-\Lambda^{2})^{2}+a(\phi^{3}-3\Lambda^{2}\phi-2\Lambda^{3}).$
(19)

This potential is plotted in Fig. 8. We see that for $a=-0.2$ ($a=0.2$),
$\phi=\Lambda$($\phi=-\Lambda$) is the false vacuum expectation value, while
$\phi=-\Lambda$($\phi=\Lambda$) is the true vacuum expectation value.

Consider the region which is in the true vacuum ($a=0.2$ and $\phi=\Lambda$).
The presence of the massive object will perturb the space and modifies the
scalar field potential. Fig. 9 shows that the scalar field is in the
$\phi=\Lambda$ state at the beginning. The scalar field inside the object
evolves toward the lower values, because in this region the true vacuum is no
longer at $\phi=\Lambda$. Outside of the object the field stays in the true
vacuum at $\phi=\Lambda$. At the same time the waves are generated which
propagate away and take energy away from the location. The object mostly
changes the vacuum state in its own neighborhood (within the bubble which is
formed around it), while the energy from the vacuum decay is released. This is
a clear demonstration that a massive object can affect the vacuum state around
it.

Consider now the opposite situation, i.e. the region of space which is in the
false vacuum ($a=-0.2$ and $\phi=\Lambda$) at the beginning. The presence of
the massive object again perturbs the space and modifies the scalar field
potential. Fig. 10 shows that the scalar field is in the $\phi=\Lambda$ state
at the beginning, but it starts to evolve toward the lower values inside the
object. The crucial difference from the previous case is that after the
transition starts, the region converted into the true vacuum grows. In other
words, the true vacuum bubble expands, which means that a massive object can
trigger (or catalyze) the vacuum decay. At the end of the process the whole
space could be converted into the true vacuum. Whether this will happen or not
depends on whether there is enough energy to support the the domain wall
bubble expansion. Fig. 11 shows that if the strength of the interaction
(determined by the parameter $A$) is low, then the true vacuum will be created
only locally around the object. This happens because the scalar field still
sits in the region where the potential is higher than the false vacuum outside
the matter, though inside the matter the scalar field may reach its true
minimum. In that case, the bubble needs some extra energy source to keep
expanding.

In other words, an expanding domain wall bubble must generate more energy than
that released from the difference in vacua, or the scalar field cannot freely
settle down into the true vacuum in the region outside of the matter. The
condition for the false vacuum decay to release enough energy to support an
expanding bubble is

$4\pi R_{b}^{2}\frac{\Delta\phi^{2}}{L}\lessapprox\frac{4\pi}{3}R^{3}\Delta
V,$ (20)

where $L$ is the domain wall width, $R_{b}$ is the domain wall bubble radius,
while $\Delta V$ is the potential difference between the false and true
vacuum. The left side represents the surface energy of the domain wall bubble,
while the right side is the volume energy released by the vacuum decay. If the
volume of the matter distribution is large enough, then a large enough region
populated by the scalar field will be pushed to its true vacuum, and once it
passes the potential barrier outside the matter distribution, the false vacuum
decay will become spontaneous. Figs. 11 and 12 demonstrate how larger massive
objects can trigger a successful vacuum decay easier than smaller ones. Higher
values of the coupling parameter, $A$, can also trigger vacuum decay easier
(as shown in Figs. 10 and 11), because more energy is generated in the
process.

Figure 8: The potential of a scalar field in the presence of a massive object
as given in Eq. (19). We set $\Lambda=B=1$. The solid line, $a=0$, shows the
potential with two equal minima at $\phi=\pm 1$. The doted line, $a=-0.2$,
shows the potential with a false vacuum at $\phi=1$ and a true vacuum at
$\phi=-1$. The dashed line, $a=0.2$, shows a potential with a false vacuum at
$\phi=-1$ and a true vacuum at $\phi=1$.  Figure 9:  A massive object
affecting the vacuum in its neighborhood. We set initial $v_{O}=0$,$A=20$,
$R=1.5$, $\Lambda=1$, $B=1$, $m=10^{10}$, $a=0.2$, $r=0$. The whole space was
initially occupied by $\phi=\Lambda$. The scalar field inside the object
evolves toward the lower values, because in this region the true vacuum is no
longer at $\phi=\Lambda$. Outside of the object the field stays in the true
vacuum at $\phi=\Lambda$. In this process, the waves are generated which
propagate away and take energy away from the location. The scalar field
potential is modified only locally (in the bubble) around the object.  Figure
10:  A massive object triggering a false vacuum decay. We set initial
$v_{O}=0$, $A=20$, $R=1.5$, $\Lambda=1$, $B=1$, $m=10^{10}$, $a=-0.2$, $r=0$.
The whole space was initially occupied by $\phi=\Lambda$. The scalar field
starts to evolve toward the lower values (true minimum) inside the object. The
crucial difference from the previous case in Fig. 9 is that after the
transition starts, the region converted into the true vacuum grows. In other
words, the true vacuum bubble expands, which means that a massive object can
trigger (or catalyze) the vacuum decay.  Figure 11: Not all of the bubbles of
the true vacuum are able to expand. If there is no enough energy to support
the expansion of the bubble (e.g. if the strength of the interaction between
the object and the field is low), the vacuum decay process cannot be
completed. We set here initial $v_{O}=0$, $A=2$, $R=1.5$, $m=10^{10}$,
$a=-0.2$, $r=0$. Note that the value of the parameter $A$ is much lower than
in Fig. 10.  Figure 12:  If the massive object is large enough, the bubble of
true vacuum will expand. We set here initial $v_{O}=0$, $A=2$, $R=3$,
$m=10^{10}$, $a=-0.2$, $r=0$.

##  V Conclusions

On the quantum level, a microscopic particle can either tunnel through or be
reflected by a domain wall. However, a large classical object cannot tunnel
through a domain wall, so it can either pass through, if it has enough energy,
or bounce back, if it is not energetic enough. Obviously, a classical object
encountering a domain wall must be treated differently from a microscopic
particle.

In our analysis, we considered a very general Lagrangian for a massive
classical object interacting with a domain wall. By solving the equations of
motion, we showed that the domain wall gets distorted as the classical object
encounters it. The degree of distortion depends on the object’s mass, velocity
and size, and the strength of the interaction between the object and the wall.
Some of the energy of the object is dissipated away by exciting waves on the
wall. To cross the wall, the object must have enough energy to overcome the
wall distortion and energy dissipation. Otherwise, the object will rebound and
gain some energy back, since the wall will act as a spring in that case.

Our results imply that if Earth (or any other planet), or a satellite in
Earth’s orbit, crosses the domain wall, it will lose (or under certain
conditions gain) energy and momentum, and change its mass. One may then use
this fact to put a constraint on theories that allow for the existence of
domain walls by studying orbits of planets or satellites.

A sub-decimeter position accuracy and a sub mm/s velocity accuracy can be
achieved in a ground-based reduced dynamic orbit determination using dual-
frequency carrier phase measurements along with precise GPS ephemeris products
and auxiliary environmental information. Here we adopt 0.5mm/s as the relevant
velocity precision Montenbruck . When the satellite mass changes when it
passes through the wall, its velocity will change too. If the interaction is
weak, we may estimate the mass difference from energy conservation

$\frac{M_{f}}{1-v_{f}^{2}}\lessapprox\frac{M_{i}}{1-v_{i}^{2}},$ (21)

where the subscripts $f$ and $i$ label the final and initial states
respectively. Since the satellite’s velocity is of the order of $10$km/s, the
satellite’s mass change is

$\frac{\delta M}{M}\lessapprox v\delta v\approx 5\times 10^{-17}.$ (22)

We note that this is an underestimate of the effect because the domain wall
will also be distorted which will in turn also cause energy and momentum
change of the satellite. Therefore, the existence of any domain wall that
causes satellite mass distortion greater than this is already excluded. This
method is different from using atomic clocks to constrain domain wall crossing
Derevianko:2013oaa ; Roberts:2017hla .

On the other hand, it is known that velocities of some satellites deviate from
theoretical predictions when they pass their perigee. This is known as the
flyby anomaly Sun et al. (2017). Such an effect can easily be explained by the
satellite crossing a domain wall. In Fig. 2 we saw that a closed domain wall
(bubble) can be formed when a massive object like Earth passes through the
domain wall. A boundary of the bubble is also a domain wall of the same kind.
Depending on the exact radius and shape of the bubble, a satellite’s orbit
might or might not intersect the bubble. This might explain why the anomaly
does not show up consistently for all the satellites all the time. Since the
typical magnitude of the flyby anomaly is $\delta v\sim 10mm/s$, the relative
mass change of $\frac{\delta M}{M}\sim 10^{-15}$ would be sufficient to
explain the anomaly.

We also showed than a massive object can trigger vacuum decay by creating a
bubble of true vacuum around it. However, to create a critical bubble which is
able to expand, enough energy must be released in this process. The released
energy depends on the strength of the coupling between the massive object and
the scalar field, and also on the size of the object. A similar effect was
described in Burda:2015isa  where black holes trigger the vacuum decay.

###### Acknowledgements.

D.C Dai is supported by the National Natural Science Foundation of China
(Grant No. 11775140). D. M. is supported in part by the US Department of
Energy (under grant DE-SC0020262) and by the Julian Schwinger Foundation. D.S.
is partially supported by the US National Science Foundation, under Grant No.
PHY-2014021. We thank Yu Sang for very useful suggestions.

## References

  * (1) T. W. B. Kibble, J. Phys. A 9, 1387-1398 (1976) doi:10.1088/0305-4470/9/8/029
  * (2) A. Vilenkin and A. E. Everett, Phys. Rev. Lett. 48, 1867-1870 (1982) doi:10.1103/PhysRevLett.48.1867
  * (3) A. Vilenkin, Phys. Rept. 121, 263-315 (1985) doi:10.1016/0370-1573(85)90033-X
  * Zeldovich et al. (1975) Zeldovich, I. B., Kobzarev, I. I., & Okun, L. B. 1975, Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, 67, 3
  * (5) A. Vilenkin, Phys. Rev. D 23, 852-857 (1981) doi:10.1103/PhysRevD.23.852
  * (6) G. B. Gelmini, M. Gleiser and E. W. Kolb, Phys. Rev. D 39, 1558 (1989) doi:10.1103/PhysRevD.39.1558
  * (7) S. E. Larsson, S. Sarkar and P. L. White, Phys. Rev. D 55, 5129-5135 (1997) doi:10.1103/PhysRevD.55.5129 [arXiv:hep-ph/9608319 [hep-ph]].
  * (8) G. R. Dvali and G. Senjanovic, Phys. Rev. Lett. 74, 5178-5181 (1995) doi:10.1103/PhysRevLett.74.5178 [arXiv:hep-ph/9501387 [hep-ph]].
  * (9) D. Stojkovic, K. Freese and G. D. Starkman, Phys. Rev. D 72, 045012 (2005) doi:10.1103/PhysRevD.72.045012 [arXiv:hep-ph/0505026 [hep-ph]].
  * (10) D. Stojkovic and K. Freese, Phys. Lett. B 606, 251-257 (2005) doi:10.1016/j.physletb.2004.12.019 [arXiv:hep-ph/0403248 [hep-ph]].
  * (11) D. C. Dai, D. Minic and D. Stojkovic, Eur. Phys. J. C 80, no.12, 1103 (2020) doi:10.1140/epjc/s10052-020-08698-x [arXiv:2010.03947 [gr-qc]].
  * (12) R. L. McNally and T. Zelevinsky, Eur. Phys. J. D 74, no.4, 61 (2020) doi:10.1140/epjd/e2020-100632-0 [arXiv:1912.06703 [astro-ph.IM]].
  * (13) B. M. Roberts, P. Delva, A. Al-Masoudi, A. Amy-Klein, C. Baerentsen, C. F. A. Baynham, E. Benkler, S. Bilicki, S. Bize and W. Bowden, et al. New J. Phys. 22, no.9, 093010 (2020) doi:10.1088/1367-2630/abaace [arXiv:1907.02661 [astro-ph.CO]].
  * (14) Y. V. Stadnik and V. V. Flambaum, Phys. Rev. Lett. 113, no.15, 151301 (2014) doi:10.1103/PhysRevLett.113.151301 [arXiv:1405.5337 [hep-ph]].
  * (15) M. Pospelov, S. Pustelny, M. P. Ledbetter, D. F. Jackson Kimball, W. Gawlik and D. Budker, Phys. Rev. Lett. 110, no.2, 021803 (2013) doi:10.1103/PhysRevLett.110.021803 [arXiv:1205.6260 [hep-ph]].
  * Pustelny et al. (2013) Pustelny, S., Kimball, D. F. J., Pankow, C., et al. ann. phys. 525, 659 (2013), arXiv:1303.5524.
  * (17) D. F. Jackson Kimball, D. Budker, J. Eby, M. Pospelov, S. Pustelny, T. Scholtes, Y. V. Stadnik, A. Weis and A. Wickenbrock, Phys. Rev. D 97, no.4, 043002 (2018) doi:10.1103/PhysRevD.97.043002 [arXiv:1710.04323 [physics.atom-ph]].
  * (18) S. Afach, B. C. Buchler, D. Budker, C. Dailey, A. Derevianko, V. Dumont, N. L. Figueroa, I. Gerhardt, Z. D. Grujić and H. Guo, et al. [arXiv:2102.13379 [astro-ph.CO]].
  * (19) A. Derevianko and M. Pospelov, Nature Phys. 10, 933 (2014) doi:10.1038/nphys3137 [arXiv:1311.1244 [physics.atom-ph]].
  * (20) B. M. Roberts, G. Blewitt, C. Dailey, M. Murphy, M. Pospelov, A. Rollings, J. Sherman, W. Williams and A. Derevianko, Nature Commun. 8, no.1, 1195 (2017) doi:10.1038/s41467-017-01440-4 [arXiv:1704.06844 [hep-ph]].
  * (21) T. Kalaydzhyan and N. Yu, Phys. Rev. D 96, no.7, 075007 (2017) doi:10.1103/PhysRevD.96.075007 [arXiv:1705.05833 [hep-ph]].
  * (22) E. D. Hall, R. X. Adhikari, V. V. Frolov, H. Müller, M. Pospelov and R. X. Adhikari, Phys. Rev. D 98, no.8, 083019 (2018) doi:10.1103/PhysRevD.98.083019 [arXiv:1605.01103 [gr-qc]].
  * (23) J. Jaeckel, S. Schenk and M. Spannowsky, [arXiv:2004.13724 [astro-ph.CO]].
  * (24) H. Grote and Y. V. Stadnik, Phys. Rev. Res. 1, no.3, 033187 (2019) doi:10.1103/PhysRevResearch.1.033187 [arXiv:1906.06193 [astro-ph.IM]].
  * (25) R. Essig, J. A. Jaros, W. Wester, P. Hansson Adrian, S. Andreas, T. Averett, O. Baker, B. Batell, M. Battaglieri and J. Beacham, et al. [arXiv:1311.0029 [hep-ph]].
  * (26) D. Stojkovic, JHEP 09, 061 (2004) doi:10.1088/1126-6708/2004/09/061 [arXiv:gr-qc/0409038 [gr-qc]].
  * (27) V. P. Frolov, M. Snajdr and D. Stojkovic, Phys. Rev. D 68, 044002 (2003) doi:10.1103/PhysRevD.68.044002 [arXiv:gr-qc/0304083 [gr-qc]].
  * (28) V. P. Frolov, D. V. Fursaev and D. Stojkovic, JHEP 06, 057 (2004) doi:10.1088/1126-6708/2004/06/057 [arXiv:gr-qc/0403002 [gr-qc]].
  * (29) V. P. Frolov, D. V. Fursaev and D. Stojkovic, Class. Quant. Grav. 21, 3483-3498 (2004) doi:10.1088/0264-9381/21/14/011 [arXiv:gr-qc/0403054 [gr-qc]].
  * (30) V. P. Frolov, A. L. Larsen and M. Christensen, Phys. Rev. D 59, 125008 (1999) doi:10.1103/PhysRevD.59.125008 [arXiv:hep-th/9811148 [hep-th]
  * (31) M. Christensen, V. P. Frolov and A. L. Larsen, Phys. Rev. D 58, 085008 (1998) doi:10.1103/PhysRevD.58.085008 [arXiv:hep-th/9803158 [hep-th]].
  * (32) O. Montenbruck, P. Ramos-Bosch, GPS Solut 12, 187 (2008). https://doi.org/10.1007/s10291-007-0080-x
  * Sun et al. (2017) Sun, X., Han, C., & Chen, P. 2017, Aerospace Science and Technology, Vol. 67, 2017, pp.228-236, arXiv:1704.02094. See also, J. D. Anderson, J. K. Campbell and M. M. Nieto, New Astron. 12, 383-397 (2007) doi:10.1016/j.newast.2006.11.004 [arXiv:astro-ph/0608087 [astro-ph]]; S. L. Adler, Phys. Rev. D 79, 023505 (2009) doi:10.1103/PhysRevD.79.023505 [arXiv:0805.2895 [astro-ph]].
  * (34) P. Burda, R. Gregory and I. Moss, Phys. Rev. Lett. 115, 071303 (2015) doi:10.1103/PhysRevLett.115.071303 [arXiv:1501.04937 [hep-th]].