26. 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 

27. 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 

28. 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 

29. 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 

30. CMB 2011 (vol 55 pp. 271)
31. 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 

32. 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 

33. 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 

34. 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 

35. 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 

36. CMB 2009 (vol 52 pp. 511)
37. CMB 2009 (vol 52 pp. 535)
 Daigle, Daniel; Kaliman, Shulim

A Note on Locally Nilpotent Derivations\\ and Variables of $k[X,Y,Z]$
We strengthen certain results
concerning actions of $(\Comp,+)$ on $\Comp^{3}$
and embeddings of $\Comp^{2}$ in $\Comp^{3}$,
and show that these results are in fact valid
over any field of characteristic zero.
Keywords:locally nilpotent derivations, group actions, polynomial automorphisms, variable, affine space Categories:14R10, 14R20, 14R25, 13N15 

38. CMB 2009 (vol 52 pp. 95)
 Miranian, L.

Matrix Valued Orthogonal Polynomials on the Unit Circle: Some Extensions of the Classical Theory
In the work presented below the classical subject of orthogonal
polynomials on the unit
circle is discussed in the matrix setting. An explicit matrix
representation of the matrix valued orthogonal polynomials in terms of
the moments of the measure is presented. Classical recurrence
relations are revisited using the matrix representation of the
polynomials. The matrix expressions for the kernel polynomials and the
ChristoffelDarboux formulas are presented for the first time.
Keywords:Matrix valued orthogonal polynomials, unit circle, Schur complements, recurrence relations, kernel polynomials, ChristoffelDarboux Category:42C99 

39. CMB 2008 (vol 51 pp. 584)
 Purbhoo, Kevin; Willigenburg, Stephanie van

On Tensor Products of Polynomial Representations
We determine the necessary and sufficient combinatorial
conditions for which the tensor product of two irreducible polynomial
representations of $\GL(n,\mathbb{C})$ is isomorphic to another.
As a consequence we discover families of LittlewoodRichardson
coefficients that are nonzero, and a condition on Schur nonnegativity.
Keywords:polynomial representation, symmetric function, LittlewoodRichardson coefficient, Schur nonnegative Categories:05E05, 05E10, 20C30 

40. CMB 2005 (vol 48 pp. 547)
 Fehér, L. M.; Némethi, A.; Rimányi, R.

Degeneracy of 2Forms and 3Forms
We study some global aspects of differential complex 2forms and 3forms
on complex manifolds.
We compute the cohomology classes represented by the sets of points
on a manifold where such a form degenerates in various senses,
together with other similar cohomological obstructions.
Based on these results and a formula for projective
representations, we calculate the degree of the projectivization
of certain orbits of the representation $\Lambda^k\C^n$.
Keywords:Classes of degeneracy loci, 2forms, 3forms, Thom polynomials, global singularity theory Categories:14N10, 57R45 

41. CMB 2002 (vol 45 pp. 231)
 Hironaka, Eriko

Erratum:~~The Lehmer Polynomial and Pretzel Links
Erratum to {\it The Lehmer Polynomial and Pretzel Links},
Canad. J. Math. {\bf 44}(2001), 440451.
Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups Categories:57M05, 57M25, 11R04, 11R27 

42. CMB 2002 (vol 45 pp. 272)
 Neusel, Mara D.

The Transfer in the Invariant Theory of Modular Permutation Representations II
In this note we show that the image of the transfer for permutation
representations of finite groups is generated by the transfers of
special monomials. This leads to a description of the image of the
transfer of the alternating groups. We also determine the height of
these ideals.
Keywords:polynomial invariants of finite groups, permutation representation, transfer Category:13A50 

43. CMB 2001 (vol 44 pp. 440)
 Hironaka, Eriko

The Lehmer Polynomial and Pretzel Links
In this paper we find a formula for the Alexander polynomial
$\Delta_{p_1,\dots,p_k} (x)$ of pretzel knots and links with
$(p_1,\dots,p_k, \nega 1)$ twists, where $k$ is odd and
$p_1,\dots,p_k$ are positive integers. The polynomial $\Delta_{2,3,7}
(x)$ is the wellknown Lehmer polynomial, which is conjectured to have
the smallest Mahler measure among all monic integer polynomials. We
confirm that $\Delta_{2,3,7} (x)$ has the smallest Mahler measure among
the polynomials arising as $\Delta_{p_1,\dots,p_k} (x)$.
Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups Categories:57M05, 57M25, 11R04, 11R27 

44. CMB 2001 (vol 44 pp. 337)
 Vinet, Luc; Zhedanov, Alexei

Spectral Transformations of the Laurent Biorthogonal Polynomials, II. Pastro Polynomials
We continue to study the simplest closure conditions for chains of
spectral transformations of the Laurent biorthogonal polynomials
($\LBP$). It is shown that the 11periodic $q$closure condition
leads to the $\LBP$ introduced by Pastro. We introduce classes of
semiclassical and LaguerreHahn $\LBP$ associated to generic closure
conditions of the chain of spectral transformations.
Keywords:Laurent orthogonal polynomials, Pastro polynomials, spectral transformations Category:33D45 

45. CMB 1999 (vol 42 pp. 125)
 Smith, Larry

Modular Vector Invariants of Cyclic Permutation Representations
Vector invariants of finite groups (see the introduction for an
explanation of the terminology) have often been used to illustrate the
difficulties of invariant theory in the modular case: see,
\eg., \cite{Ber}, \cite{norway}, \cite{fossum}, \cite{MmeB},
\cite{poly} and \cite{survey}. It is therefore all the more
surprising that the {\it unpleasant} properties of these invariants
may be derived from two unexpected, and remarkable, {\it nice}
properties: namely for vector permutation invariants of the cyclic
group $\mathbb{Z}/p$ of prime order in characteristic $p$ the
image of the transfer homomorphism $\Tr^{\mathbb{Z}/p} \colon
\mathbb{F}[V] \lra \mathbb{F}[V]^{\mathbb{Z}/p}$ is a prime ideal,
and the quotient algebra $\mathbb{F}[V]^{\mathbb{Z}/p}/ \Im
(\Tr^{\mathbb{Z}/p})$ is a polynomial algebra on the top Chern
classes of the action.
Keywords:polynomial invariants of finite groups Category:13A50 

46. CMB 1998 (vol 41 pp. 442)
 Chamberland, Marc; Meisters, Gary

A Mountain Pass to the Jacobian Conjecture.
This paper presents an approach to injectivity theorems via the
Mountain Pass Lemma and raises an open question. The main result
of this paper (Theorem~1.1) is proved by means of the Mountain Pass
Lemma and states that if the eigenvalues of $F' (\x)F' (\x)^{T}$
are uniformly bounded away from zero for $\x \in \hbox{\Bbbvii
R}^{n}$, where $F \colon \hbox{\Bbbvii R}^n \rightarrow
\hbox{\Bbbvii R}^n$ is a class $\cC^{1}$ map, then $F$ is
injective. This was discovered in a joint attempt by the authors
to prove a stronger result conjectured by the first author: Namely,
that a sufficient condition for injectivity of class $\cC^{1}$ maps
$F$ of $\hbox{\Bbbvii R}^n$ into itself is that all the eigenvalues
of $F'(\x)$ are bounded away from zero on $\hbox{\Bbbvii
R}^n$. This is stated as Conjecture~2.1. If true, it would imply
(via {\it ReductionofDegree}) {\it injectivity of polynomial
maps} $F \colon \hbox{\Bbbvii R}^n \rightarrow \hbox{\Bbbvii R}^n$
{\it satisfying the hypothesis}, $\det F'(\x) \equiv 1$, of the
celebrated Jacobian Conjecture (JC) of OttHeinrich Keller. The
paper ends with several examples to illustrate a variety of cases
and known counterexamples to some natural questions.
Keywords:Injectivity, ${\cal C}^1$maps, polynomial maps, Jacobian Conjecture, Mountain Pass Categories:14A25, 14E09 

47. CMB 1997 (vol 40 pp. 47)
 Hartl, Manfred

A universal coefficient decomposition for subgroups induced by submodules of group algebras
Dimension subgroups and Lie dimension subgroups are known to satisfy a
`universal coefficient decomposition', {\it i.e.} their value with respect to
an arbitrary coefficient ring can be described in terms of their values with
respect to the `universal' coefficient rings given by the cyclic groups of
infinite and prime power order. Here this fact is generalized to much more
general types of induced subgroups, notably covering Fox subgroups and
relative dimension subgroups with respect to group algebra filtrations
induced by arbitrary $N$series, as well as certain common generalisations
of these which occur in the study of the former. This result relies on an
extension of the principal universal coefficient decomposition theorem on
polynomial ideals (due to Passi, Parmenter and Seghal), to all additive
subgroups of group rings. This is possible by using homological instead
of ring theoretical methods.
Keywords:induced subgroups, group algebras, Fox subgroups, relative dimension, subgroups, polynomial ideals Categories:20C07, 16A27 
