1. CMB Online first
 Ying, Zhiling; Koşan, Tamer; Zhou, Yiqiang

Rings in which every element is a sum of two tripotents
Let $R$ be a ring. The following results are proved: $(1)$ every
element of $R$ is a sum of an idempotent and a tripotent that
commute iff $R$ has the identity $x^6=x^4$ iff $R\cong R_1\times
R_2$, where $R_1/J(R_1)$ is Boolean with $U(R_1)$ a group of
exponent $2$ and $R_2$ is zero or a subdirect product of $\mathbb
Z_3$'s; $(2)$ every element of $R$ is either a sum or a difference
of two commuting idempotents iff $R\cong R_1\times R_2$, where
$R_1/J(R_1)$ is Boolean with $J(R_1)=0$ or $J(R_1)=\{0,2\}$,
and $R_2$ is zero or a subdirect product of $\mathbb Z_3$'s;
$(3)$ every element of $R$ is a sum of two commuting tripotents
iff $R\cong R_1\times R_2\times R_3$, where $R_1/J(R_1)$ is Boolean
with $U(R_1)$ a group of exponent $2$, $R_2$ is zero or a subdirect
product of $\mathbb Z_3$'s, and $R_3$ is zero or a subdirect
product of $\mathbb Z_5$'s.
Keywords:idempotent, tripotent, Boolean ring, polynomial identity $x^3=x$, polynomial identity $x^6=x^4$, polynomial identity $x^8=x^4$ Categories:16S50, 16U60, 16U90 

2. CMB 2016 (vol 59 pp. 340)
 Kȩpczyk, Marek

A Note on Algebras that are Sums of Two Subalgebras
We study an associative algebra $A$ over an arbitrary field,
that is
a sum of two subalgebras $B$ and $C$ (i.e. $A=B+C$). We show
that if $B$ is a right or left Artinian $PI$ algebra and $C$
is a $PI$ algebra, then $A$ is a $PI$ algebra. Additionally we
generalize this result for semiprime algebras $A$.
Consider the class of
all semisimple finite dimensional algebras $A=B+C$ for some
subalgebras $B$ and $C$ which satisfy given polynomial identities
$f=0$ and $g=0$, respectively.
We prove that all algebras in this class satisfy a common polynomial
identity.
Keywords:rings with polynomial identities, prime rings Categories:16N40, 16R10, , 16S36, 16W60, 16R20 

3. CMB 2015 (vol 58 pp. 818)
 Llibre, Jaume; Zhang, Xiang

On the Limit Cycles of Linear Differential Systems with Homogeneous Nonlinearities
We consider the class of polynomial differential systems of the
form
$\dot x= \lambda xy+P_n(x,y)$, $\dot y=x+\lambda y+ Q_n(x,y),$ where
$P_n$ and $Q_n$ are homogeneous polynomials of degree $n$. For
this
class of differential systems we summarize the known results
for the
existence of limit cycles, and we provide new results for their
nonexistence and existence.
Keywords:polynomial differential system, limit cycles, differential equations on the cylinder Categories:34C35, 34D30 

4. CMB 2015 (vol 58 pp. 704)
 Benamar, H.; Chandoul, A.; Mkaouar, M.

On the Continued Fraction Expansion of Fixed Period in Finite Fields
The Chowla conjecture
states that,
if $t$ is any given
positive integer, there are infinitely many prime positive
integers $N$ such that $\operatorname{Per} (\sqrt{N})=t$, where
$\operatorname{Per} (\sqrt{N})$
is the period length of the continued fraction expansion for
$\sqrt{N}$.
C. Friesen proved
that, for any $k\in \mathbb{N}$, there are infinitely many
squarefree integers $N$, where the continued fraction expansion
of $\sqrt{N}$ has a fixed period. In this paper, we describe all
polynomials $Q\in \mathbb{F}_q[X] $ for which the continued fraction
expansion of $\sqrt {Q}$ has a fixed period, also we give a
lower
bound of the number of monic, nonsquares polynomials $Q$ such
that $\deg Q= 2d$ and $ Per \sqrt {Q}=t$.
Keywords:continued fractions, polynomials, formal power series Categories:11A55, 13J05 

5. CMB 2015 (vol 58 pp. 877)
 Zaatra, Mohamed

Generating Some Symmetric Semiclassical Orthogonal Polynomials
We show that if $v$ is a regular semiclassical form
(linear functional), then the symmetric form $u$ defined by the
relation
$x^{2}\sigma u = \lambda v$,
where $(\sigma f)(x)=f(x^{2})$ and the odd
moments of $u$ are $0$, is also
regular and semiclassical form for every
complex $\lambda $ except for a discrete set of numbers depending
on $v$. We give explicitly the threeterm recurrence relation
and the
structure relation coefficients of the orthogonal polynomials
sequence associated with $u$ and the class of the form $u$ knowing
that of $v$. We conclude with an illustrative example.
Keywords:orthogonal polynomials, quadratic decomposition, semiclassical forms, structure relation Categories:33C45, 42C05 

6. CMB 2015 (vol 59 pp. 159)
 MacColl, Joseph

Rotors in Khovanov Homology
Anstee, Przytycki, and Rolfsen introduced the idea of rotants,
pairs of links related by a generalised form of link mutation.
We exhibit infinitely many pairs of rotants which can be distinguished
by Khovanov homology, but not by the Jones polynomial.
Keywords:geometric topology, knot theory, rotants, khovanov homology, jones polynomial Categories:57M27, 57M25 

7. CMB 2015 (vol 58 pp. 423)
 Yamagishi, Masakazu

Resultants of Chebyshev Polynomials: The First, Second, Third, and Fourth Kinds
We give an explicit formula for the resultant of Chebyshev polynomials of the
first, second, third, and fourth kinds.
We also compute the resultant of modified cyclotomic polynomials.
Keywords:resultant, Chebyshev polynomial, cyclotomic polynomial Categories:11R09, 11R18, 12E10, 33C45 

8. CMB 2015 (vol 58 pp. 225)
 Aghigh, Kamal; Nikseresht, Azadeh

Characterizing Distinguished Pairs by Using Liftings of Irreducible Polynomials
Let $v$ be a henselian valuation of any rank of a field
$K$ and $\overline{v}$ be the unique extension of $v$ to a
fixed algebraic closure $\overline{K}$ of $K$. In 2005, it was studied properties
of those pairs $(\theta,\alpha)$ of elements of $\overline{K}$
with $[K(\theta): K]\gt [K(\alpha): K]$ where $\alpha$ is an element
of smallest degree over $K$ such that
$$
\overline{v}(\theta\alpha)=\sup\{\overline{v}(\theta\beta)
\ \beta\in \overline{K}, \ [K(\beta): K]\lt [K(\theta): K]\}.
$$
Such pairs are referred to as distinguished pairs.
We use the concept of liftings of irreducible polynomials to give a
different characterization of distinguished pairs.
Keywords:valued fields, nonArchimedean valued fields, irreducible polynomials Categories:12J10, 12J25, 12E05 

9. CMB 2014 (vol 57 pp. 609)
 NasrIsfahani, Alireza

Jacobson Radicals of Skew Polynomial Rings of Derivation Type
We provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive, when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation
type.
Keywords:skew polynomial rings, Jacobson radical, derivation Categories:16S36, 16N20 

10. CMB 2014 (vol 57 pp. 538)
 Ide, Joshua; Jones, Lenny

Infinite Families of $A_4$Sextic Polynomials
In this article we develop a test to determine whether a sextic
polynomial that is irreducible over $\mathbb{Q}$ has Galois group isomorphic
to the alternating group $A_4$. This test does not involve the
computation of resolvents, and we use this test to construct several
infinite families of such polynomials.
Keywords:Galois group, sextic polynomial, inverse Galois theory, irreducible polynomial Categories:12F10, 12F12, 11R32, 11R09 

11. CMB 2012 (vol 56 pp. 759)
 Issa, Zahraa; Lalín, Matilde

A Generalization of a Theorem of Boyd and Lawton
The Mahler measure of a nonzero $n$variable polynomial $P$ is the integral of
$\logP$ on the unit $n$torus. A result of Boyd and Lawton says that
the Mahler measure of a multivariate polynomial is the limit of Mahler
measures of univariate polynomials. We prove the analogous
result for different extensions of Mahler measure such as generalized
Mahler measure (integrating the maximum of $\logP$ for possibly
different $P$'s),
multiple Mahler measure (involving products of $\logP$ for possibly
different $P$'s), and higher Mahler measure (involving $\log^kP$).
Keywords:Mahler measure, polynomial Categories:11R06, 11R09 

12. CMB 2012 (vol 56 pp. 844)
 Shparlinski, Igor E.

On the Average Number of SquareFree Values of Polynomials
We obtain an asymptotic formula for the number
of squarefree integers in $N$ consecutive values
of polynomials on average over integral
polynomials of degree at most $k$ and of
height at most $H$, where $H \ge N^{k1+\varepsilon}$
for some fixed $\varepsilon\gt 0$.
Individual results of this kind for polynomials of degree $k \gt 3$,
due to A. Granville (1998),
are only known under the $ABC$conjecture.
Keywords:polynomials, squarefree numbers Category:11N32 

13. CMB 2012 (vol 56 pp. 602)
14. CMB 2012 (vol 56 pp. 769)
15. CMB 2012 (vol 56 pp. 584)
 Liau, PaoKuei; Liu, ChengKai

On Automorphisms and Commutativity in Semiprime Rings
Let $R$ be a semiprime ring with center
$Z(R)$. For $x,y\in R$, we denote by $[x,y]=xyyx$ the commutator of
$x$ and $y$. If $\sigma$ is a nonidentity automorphism of $R$ such
that
$$
\Big[\big[\dots\big[[\sigma(x^{n_0}),x^{n_1}],x^{n_2}\big],\dots\big],x^{n_k}\Big]=0
$$
for all $x \in R$, where $n_{0},n_{1},n_{2},\dots,n_{k}$ are fixed
positive integers, then there exists a map $\mu\colon R\rightarrow Z(R)$
such that $\sigma(x)=x+\mu(x)$ for all $x\in R$. In particular, when
$R$ is a prime ring, $R$ is commutative.
Keywords:automorphism, generalized polynomial identity (GPI) Categories:16N60, 16W20, 16R50 

16. CMB 2011 (vol 56 pp. 510)
 Dubickas, Artūras

Linear Forms in Monic Integer Polynomials
We prove a necessary and sufficient condition on the list of
nonzero integers $u_1,\dots,u_k$, $k \geq 2$, under which a monic
polynomial $f \in \mathbb{Z}[x]$ is expressible by a linear form
$u_1f_1+\dots+u_kf_k$ in monic polynomials $f_1,\dots,f_k \in
\mathbb{Z}[x]$. This condition is independent of $f$. We also show that if
this condition holds, then the monic polynomials $f_1,\dots,f_k$
can be chosen to be irreducible in $\mathbb{Z}[x]$.
Keywords:irreducible polynomial, height, linear form in polynomials, Eisenstein's criterion Categories:11R09, 11C08, 11B83 

17. CMB 2011 (vol 56 pp. 251)
 Borwein, Peter; Choi, Stephen K. K.; Ganguli, Himadri

Sign Changes of the Liouville Function on Quadratics
Let $\lambda (n)$ denote the Liouville function. Complementary to the prime number theorem, Chowla conjectured
that
\begin{equation*}
\label{a.1}
\sum_{n\le x} \lambda (f(n)) =o(x)\tag{$*$}
\end{equation*}
for any polynomial $f(x)$ with integer coefficients which is not of
form $bg(x)^2$.
When $f(x)=x$, $(*)$ is equivalent to the prime number theorem.
Chowla's conjecture has been proved for linear functions,
but for degree
greater than 1, the conjecture seems
to be extremely hard and remains wide open.
One can consider a weaker form
of Chowla's conjecture.
Conjecture 1.
[Cassaigne et al.]
If $f(x) \in \mathbb{Z} [x]$ and is not in the form of $bg^2(x)$
for some $g(x)\in \mathbb{Z}[x]$, then $\lambda (f(n))$
changes sign infinitely often.
Clearly, Chowla's conjecture implies Conjecture 1.
Although weaker,
Conjecture 1 is still wide open for polynomials of degree $\gt 1$.
In this article, we study Conjecture 1 for
quadratic polynomials. One of our main theorems is the following.
Theorem 1
Let $f(x) = ax^2+bx +c $ with $a\gt 0$ and $l$
be a positive integer such that $al$ is
not a perfect square. If the
equation $f(n)=lm^2 $ has one solution
$(n_0,m_0) \in \mathbb{Z}^2$, then it has infinitely
many positive solutions $(n,m) \in \mathbb{N}^2$.
As a direct consequence of Theorem 1, we prove the following.
Theorem 2
Let $f(x)=ax^2+bx+c$ with $a \in \mathbb{N}$ and $b,c \in \mathbb{Z}$. Let
\[
A_0=\Bigl[\frac{b+(D+1)/2}{2a}\Bigr]+1.
\]
Then either the binary sequence $\{ \lambda (f(n)) \}_{n=A_0}^\infty$ is
a constant sequence or it changes sign infinitely often.
Some partial results of Conjecture 1 for quadratic polynomials are also proved using Theorem 1.
Keywords:Liouville function, Chowla's conjecture, prime number theorem, binary sequences, changes sign infinitely often, quadratic polynomials, Pell equation Categories:11N60, 11B83, 11D09 

18. CMB 2011 (vol 56 pp. 194)
 Stefánsson, Úlfar F.

On the Smallest and Largest Zeros of MÃ¼ntzLegendre Polynomials
MÃ¼ntzLegendre
polynomials $L_n(\Lambda;x)$ associated with a
sequence $\Lambda=\{\lambda_k\}$ are obtained by orthogonalizing the
system $(x^{\lambda_0}, x^{\lambda_1}, x^{\lambda_2}, \dots)$ in
$L_2[0,1]$ with respect to the Legendre weight. If the $\lambda_k$'s
are distinct, it is well known that $L_n(\Lambda;x)$ has exactly $n$
zeros $l_{n,n}\lt l_{n1,n}\lt \cdots \lt l_{2,n}\lt l_{1,n}$ on $(0,1)$.
First we prove the following global bound for the smallest zero,
$$
\exp\biggl(4\sum_{j=0}^n \frac{1}{2\lambda_j+1}\biggr) \lt l_{n,n}.
$$
An important consequence is that if the associated MÃ¼ntz space is
nondense in $L_2[0,1]$, then
$$
\inf_{n}x_{n,n}\geq
\exp\biggl({4\sum_{j=0}^{\infty} \frac{1}{2\lambda_j+1}}\biggr)\gt 0,
$$
so
the elements $L_n(\Lambda;x)$ have no zeros close to 0.
Furthermore, we determine the asymptotic behavior of the largest zeros; for $k$ fixed,
$$
\lim_{n\rightarrow\infty} \vert \log l_{k,n}\vert \sum_{j=0}^n
(2\lambda_j+1)= \Bigl(\frac{j_k}{2}\Bigr)^2,
$$
where $j_k$ denotes the $k$th zero of the Bessel function $J_0$.
Keywords:MÃ¼ntz polynomials, MÃ¼ntzLegendre polynomials Categories:42C05, 42C99, 41A60, 30B50 

19. CMB 2011 (vol 55 pp. 850)
 Shparlinski, Igor E.; Stange, Katherine E.

Character Sums with Division Polynomials
We obtain nontrivial estimates of quadratic character sums of division polynomials $\Psi_n(P)$, $n=1,2, \dots$, evaluated at a given point $P$ on an elliptic curve over a finite field of $q$ elements. Our bounds are nontrivial if the order of $P$ is at least $q^{1/2 + \varepsilon}$ for some fixed $\varepsilon > 0$. This work is motivated by an open question about statistical indistinguishability of some cryptographically relevant sequences that was recently brought up by K. Lauter and the second author.
Keywords:division polynomial, character sum Categories:11L40, 14H52 

20. CMB 2011 (vol 55 pp. 271)
21. CMB 2011 (vol 55 pp. 249)
 Chang, DerChen; Li, Bao Qin

Description of Entire Solutions of Eiconal Type Equations
The paper describes entire solutions to the eiconal type nonlinear partial differential
equations, which include the eiconal equations $(X_1(u))^2+(X_2(u))^2=1$ as special cases,
where
$X_1=p_1{\partial}/{\partial z_1}+p_2{\partial}/{\partial z_2}$,
$X_2=p_3{\partial}/{\partial z_1}+p_4{\partial}/{\partial z_2}$
are linearly independent operators with $p_j$ being arbitrary
polynomials in $\mathbf{C}^2$.
Keywords:entire solution, eiconal equation, polynomial, transcendental function Categories:32A15, 35F20 

22. CMB 2011 (vol 54 pp. 288)
 Jacobs, David P.; Rayes, Mohamed O.; Trevisan, Vilmar

The Resultant of Chebyshev Polynomials
Let $T_{n}$ denote the $n$th
Chebyshev polynomial of the first kind,
and let $U_{n}$ denote the $n$th
Chebyshev polynomial of the second kind.
We give an explicit formula for the resultant
$\operatorname{res}( T_{m}, T_{n} )$.
Similarly, we give a formula for
$\operatorname{res}( U_{m}, U_{n} )$.
Keywords:resultant, Chebyshev polynomial Categories:11Y11, 68W20 

23. CMB 2011 (vol 54 pp. 217)
 Chen, William Y. C.; Wang, Larry X. W.; Yang, Arthur L. B.

Recurrence Relations for Strongly $q$LogConvex Polynomials
We consider a class of
strongly $q$logconvex polynomials based on a triangular recurrence
relation with linear coefficients, and we show that the Bell
polynomials, the Bessel polynomials, the Ramanujan polynomials and
the Dowling polynomials are strongly $q$logconvex. We also prove
that the Bessel transformation preserves logconvexity.
Keywords:logconcavity, $q$logconvexity, strong $q$logconvexity, Bell polynomials, Bessel polynomials, Ramanujan polynomials, Dowling polynomials Categories:05A20, 05E99 

24. CMB 2010 (vol 53 pp. 223)
 Chuang, ChenLian; Lee, TsiuKwen

Density of Polynomial Maps
Let $R$ be a dense subring of $\operatorname{End}(_DV)$, where $V$ is a left vector space over a division ring $D$. If $\dim{_DV}=\infty$, then the range of any nonzero polynomial $f(X_1,\dots,X_m)$ on $R$ is dense in $\operatorname{End}(_DV)$. As an application, let $R$ be a prime ring without nonzero nil onesided ideals and $0\ne a\in R$. If $af(x_1,\dots,x_m)^{n(x_i)}=0$ for all $x_1,\dots,x_m\in R$, where $n(x_i)$ is a positive integer depending on $x_1,\dots,x_m$, then $f(X_1,\dots,X_m)$ is a polynomial identity of $R$ unless $R$ is a finite matrix ring over a finite field.
Keywords:density, polynomial, endomorphism ring, PI Categories:16D60, 16S50 

25. CMB 2009 (vol 53 pp. 321)
 Lee, TsiuKwen; Zhou, Yiqiang

A Theorem on UnitRegular Rings
Let $R$ be a unitregular ring and let $\sigma $ be an endomorphism of
$R$ such that $\sigma (e)=e$ for all $e^2=e\in R$ and let $n\ge 0$. It
is proved that every element of $R[x \mathinner;\sigma]/(x^{n+1})$ is
equivalent to an element of the form $e_0+e_1x+\dots +e_nx^n$, where
the $e_i$ are orthogonal idempotents of $R$. As an application, it is
proved that $R[x \mathinner; \sigma ]/(x^{n+1})$ is left morphic for each
$n\ge 0$.
Keywords:morphic rings, unitregular rings, skew polynomial rings Categories:16E50, 16U99, 16S70, 16S35 
