1. CMB Online first
 Alaghmandan, Mahmood; Crann, Jason

Character density in central subalgebras of compact quantum groups
We investigate quantum group generalizations
of various density results from Fourier analysis on compact groups.
In particular, we establish the density of characters in the
space of fixed points of the conjugation action on $L^2(\mathbb{G})$, and
use this result to show the weak* density and norm density of
characters in $ZL^\infty(\mathbb{G})$ and $ZC(\mathbb{G})$, respectively. As a corollary,
we partially answer an open question of Woronowicz.
At the level of $L^1(\mathbb{G})$, we show that the center
$\mathcal{Z}(L^1(\mathbb{G}))$
is precisely the closed linear span of the quantum characters
for a large class of compact quantum groups, including arbitrary
compact Kac algebras. In the latter setting, we show, in addition,
that $\mathcal{Z}(L^1(\mathbb{G}))$ is a completely complemented
$\mathcal{Z}(L^1(\mathbb{G}))$submodule
of $L^1(\mathbb{G})$.
Keywords:compact quantum group, irreducible character Categories:43A20, 43A40, 46J40 

2. CMB 2016 (vol 59 pp. 326)
3. 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 

4. 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 

5. CMB 2013 (vol 57 pp. 125)
 Mlaiki, Nabil M.

Camina Triples
In this paper, we study Camina triples. Camina triples are a
generalization of Camina pairs. Camina pairs were first introduced
in 1978 by A .R. Camina.
Camina's work
was inspired by the study of Frobenius groups. We
show that if $(G,N,M)$ is a Camina triple, then either $G/N$ is a
$p$group, or $M$ is abelian, or $M$ has a nontrivial nilpotent or
Frobenius quotient.
Keywords:Camina triples, Camina pairs, nilpotent groups, vanishing off subgroup, irreducible characters, solvable groups Category:20D15 

6. 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 

7. CMB 2011 (vol 54 pp. 748)
 Shparlinski, Igor E.

On the Distribution of Irreducible Trinomials
We obtain new results about the number of trinomials $t^n + at + b$
with integer coefficients in a box $(a,b) \in [C, C+A] \times [D,
D+B]$ that are irreducible modulo a prime $p$. As a byproduct we
show that for any $p$ there are irreducible polynomials of height at
most $p^{1/2+o(1)}$, improving on the previous estimate of
$p^{2/3+o(1)}$ obtained by the author in 1989.
Keywords:irreducible trinomials, character sums Categories:11L40, 11T06 

8. CMB 2009 (vol 52 pp. 511)
9. CMB 2009 (vol 52 pp. 224)
 Ghiloni, Riccardo

Equations and Complexity for the DuboisEfroymson Dimension Theorem
Let $\R$ be a real closed field, let $X \subset \R^n$ be an
irreducible real algebraic set and let $Z$ be an algebraic subset of
$X$ of codimension $\geq 2$. Dubois and Efroymson proved the existence
of an irreducible algebraic subset of $X$ of codimension $1$
containing~$Z$. We improve this dimension theorem as follows. Indicate
by $\mu$ the minimum integer such that the ideal of polynomials in
$\R[x_1,\ldots,x_n]$ vanishing on $Z$ can be generated by polynomials
of degree $\leq \mu$. We prove the following two results:
\begin{inparaenum}[\rm(1)]
\item There
exists a polynomial $P \in \R[x_1,\ldots,x_n]$ of degree~$\leq \mu+1$
such that $X \cap P^{1}(0)$ is an irreducible algebraic subset of $X$
of codimension $1$ containing~$Z$.
\item Let $F$ be a polynomial in
$\R[x_1,\ldots,x_n]$ of degree~$d$ vanishing on $Z$. Suppose there
exists a nonsingular point $x$ of $X$ such that $F(x)=0$ and the
differential at $x$ of the restriction of $F$ to $X$ is nonzero. Then
there exists a polynomial $G \in \R[x_1,\ldots,x_n]$ of degree $\leq
\max\{d,\mu+1\}$ such that, for each $t \in (1,1) \setminus \{0\}$,
the set $\{x \in X \mid F(x)+tG(x)=0\}$ is an irreducible algebraic
subset of $X$ of codimension $1$ containing~$Z$.
\end{inparaenum} Result (1) and a
slightly different version of result~(2) are valid over any
algebraically closed field also.
Keywords:Irreducible algebraic subvarieties, complexity of algebraic varieties, Bertini's theorems Categories:14P05, 14P20 

10. CMB 2003 (vol 46 pp. 265)
 Oh, Seungsang

Reducing Spheres and Klein Bottles after Dehn Fillings
Let $M$ be a compact, connected, orientable, irreducible 3manifold with a
torus boundary. It is known that if two Dehn fillings on $M$ along the
boundary produce a reducible manifold and a manifold containing a Klein
bottle, then the distance between the filling slopes is at most three. This
paper gives a remarkably short proof of this result.
Keywords:Dehn filling, reducible, Klein bottle Category:57M50 
