1. CJM 2016 (vol 69 pp. 807)
 Günther, Christian; Schmidt, KaiUwe

$L^q$ Norms of Fekete and Related Polynomials
A Littlewood polynomial is a polynomial in $\mathbb{C}[z]$ having all
of its coefficients in $\{1,1\}$. There are various old unsolved
problems, mostly due to Littlewood and ErdÅs, that ask for
Littlewood polynomials that provide a good approximation to a
function that is constant on the complex unit circle, and in
particular have small $L^q$ norm on the complex unit circle.
We consider the Fekete polynomials
\[
f_p(z)=\sum_{j=1}^{p1}(j\,\,p)\,z^j,
\]
where $p$ is an odd prime and $(\,\cdot\,\,p)$ is the Legendre
symbol (so that $z^{1}f_p(z)$ is a Littlewood polynomial). We
give explicit and recursive formulas for the limit of the ratio
of $L^q$ and $L^2$ norm of $f_p$ when $q$ is an even positive
integer and $p\to\infty$. To our knowledge, these are the first
results that give these limiting values for specific sequences
of nontrivial Littlewood polynomials and infinitely many $q$.
Similar results are given for polynomials obtained by cyclically
permuting the coefficients of Fekete polynomials and for Littlewood
polynomials whose coefficients are obtained from additive characters
of finite fields. These results vastly generalise earlier results
on the $L^4$ norm of these polynomials.
Keywords:character polynomial, Fekete polynomial, $L^q$ norm, Littlewood polynomial Categories:11B83, 42A05, 30C10 

2. CJM 2016 (vol 68 pp. 1159)
 Yattselev, Maxim L.

Strong Asymptotics of HermitePadÃ© Approximants for Angelesco Systems
In this work type II HermitePadÃ© approximants for a vector
of Cauchy transforms of smooth Jacobitype densities are considered.
It is assumed that densities are supported on mutually disjoint
intervals (an Angelesco system with complex weights). The formulae
of strong asymptotics are derived for any ray sequence of multiindices.
Keywords:HermitePadÃ© approximation, multiple orthogonal polynomials, nonHermitian orthogonality, strong asymptotics, matrix RiemannHilbert approach Categories:42C05, 41A20, 41A21 

3. CJM 2015 (vol 69 pp. 408)
 Klep, Igor; Špenko, Špela

Free Function Theory Through Matrix Invariants
This paper concerns free function theory. Free maps are free
analogs of analytic functions in several complex variables,
and are defined in terms of freely noncommuting variables.
A function of $g$ noncommuting variables is a function on $g$tuples
of square matrices of all sizes that respects direct sums and
simultaneous conjugation.
Examples of such maps include noncommutative polynomials, noncommutative
rational functions and convergent noncommutative power series.
In sharp contrast to the existing literature in free analysis, this article
investigates free maps \emph{with involution} 
free analogs of real analytic functions.
To
get a grip on these,
techniques and tools from invariant theory are developed and
applied to free analysis. Here is a sample of the results obtained.
A characterization of polynomial free maps via properties of
their finitedimensional slices is presented and then used to
establish power series expansions for analytic free maps about
scalar and nonscalar points; the latter are series of generalized
polynomials for which an invarianttheoretic characterization
is given.
Furthermore,
an inverse and implicit function theorem for free maps with
involution is obtained.
Finally, with a selection of carefully chosen examples
it is shown that
free maps with involution
do not exhibit strong rigidity properties
enjoyed by their involutionfree
counterparts.
Keywords:free algebra, free analysis, invariant theory, polynomial identities, trace identities, concomitants, analytic maps, inverse function theorem, generalized polynomials Categories:16R30, 32A05, 46L52, 15A24, 47A56, 15A24, 46G20 

4. CJM 2015 (vol 68 pp. 3)
 Boden, Hans Ulysses; Curtis, Cynthia L

The SL$(2, C)$ Casson Invariant for Knots and the $\hat{A}$polynomial
In this paper, we extend the definition of the ${SL(2, {\mathbb C})}$ Casson
invariant
to arbitrary knots $K$ in integral homology 3spheres and relate
it to the $m$degree of the $\widehat{A}$polynomial of $K$. We
prove a product formula for the $\widehat{A}$polynomial of the connected
sum $K_1 \# K_2$ of two knots in $S^3$ and deduce additivity
of ${SL(2, {\mathbb C})}$ Casson knot invariant under connected sum for a large
class of knots in $S^3$. We also present an example of a nontrivial
knot $K$ in $S^3$ with trivial $\widehat{A}$polynomial and trivial
${SL(2, {\mathbb C})}$ Casson knot invariant, showing that neither of these invariants
detect the unknot.
Keywords:Knots, 3manifolds, character variety, Casson invariant, $A$polynomial Categories:57M27, 57M25, 57M05 

5. CJM 2015 (vol 68 pp. 150)
 Stavrova, Anastasia

Nonstable $K_1$functors of Multiloop Groups
Let $k$ be a field of characteristic 0. Let $G$ be a reductive
group over the ring of Laurent polynomials
$R=k[x_1^{\pm 1},...,x_n^{\pm 1}]$. Assume that $G$ contains
a maximal $R$torus, and
that every semisimple normal subgroup of $G$ contains a twodimensional
split torus $\mathbf{G}_m^2$.
We show that the natural map of nonstable $K_1$functors, also
called Whitehead groups,
$K_1^G(R)\to K_1^G\bigl( k((x_1))...((x_n)) \bigr)$ is injective,
and an isomorphism if $G$ is semisimple.
As an application, we provide a way to compute the difference
between the
full automorphism group of a Lie torus (in the sense of YoshiiNeher)
and the subgroup generated by
exponential automorphisms.
Keywords:loop reductive group, nonstable $K_1$functor, Whitehead group, Laurent polynomials, Lie torus Categories:20G35, 19B99, 17B67 

6. CJM 2015 (vol 67 pp. 1046)
 Dubickas, Arturas; Sha, Min; Shparlinski, Igor

Explicit Form of Cassels' $p$adic Embedding Theorem for Number Fields
In this paper, we mainly give a general explicit form of Cassels'
$p$adic embedding theorem for number fields. We also give its
refined form in the case of cyclotomic fields. As a byproduct,
given an irreducible polynomial $f$ over $\mathbb{Z}$, we give a general
unconditional upper bound for the smallest prime number $p$ such
that $f$ has a simple root modulo $p$.
Keywords:number field, $p$adic embedding, height, polynomial, cyclotomic field Categories:11R04, 11S85, 11G50, 11R09, 11R18 

7. CJM 2014 (vol 67 pp. 1024)
 Ashraf, Samia; Azam, Haniya; Berceanu, Barbu

Representation Stability of Power Sets and Square Free Polynomials
The symmetric group $\mathcal{S}_n$ acts on the power
set $\mathcal{P}(n)$ and also on the set of
square free polynomials in $n$ variables. These
two related representations are analyzed from the stability point
of view. An application is given for the action of the symmetric
group on the cohomology of the pure braid group.
Keywords:symmetric group modules, square free polynomials, representation stability, Arnold algebra Categories:20C30, 13A50, 20F36, 55R80 

8. CJM 2014 (vol 66 pp. 902)
9. CJM 2014 (vol 67 pp. 507)
 Borwein, Peter; Choi, Stephen; Ferguson, Ron; Jankauskas, Jonas

On Littlewood Polynomials with Prescribed Number of Zeros Inside the Unit Disk
We investigate the numbers of complex zeros of Littlewood polynomials
$p(z)$ (polynomials with coefficients $\{1, 1\}$) inside or
on the unit circle $z=1$, denoted by $N(p)$ and $U(p)$, respectively.
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. We
obtain explicit formulas for $N(p)$, $U(p)$ for polynomials $p(z)$
of these types. We show that, if $n+1$ is a prime number, then
for each integer $k$, $0 \leq k \leq n1$, there exists a Littlewood
polynomial $p(z)$ of degree $n$ with $N(p)=k$ and $U(p)=0$. Furthermore,
we describe some cases when the ratios $N(p)/n$ and $U(p)/n$
have limits as $n \to \infty$ and find the corresponding limit
values.
Keywords:Littlewood polynomials, zeros, complex roots Categories:11R06, 11R09, 11C08 

10. CJM 2013 (vol 66 pp. 525)
11. CJM 2013 (vol 66 pp. 874)
 Levandovskyy, Viktor; Shepler, Anne V.

Quantum Drinfeld Hecke Algebras
We consider finite groups acting on
quantum (or skew) polynomial rings. Deformations of the
semidirect product of the quantum polynomial ring with the acting group
extend symplectic reflection algebras and graded Hecke algebras
to the quantum setting over a field
of arbitrary characteristic.
We give necessary and sufficient conditions for such algebras to satisfy a
PoincarÃ©BirkhoffWitt property using the theory of noncommutative
GrÃ¶bner bases.
We include applications to the case of abelian groups
and the case of groups acting on coordinate rings of quantum planes.
In addition, we classify graded automorphisms of the coordinate ring of quantum 3space. In characteristic zero, Hochschild cohomology
gives an elegant description of the PBW conditions.
Keywords:skew polynomial rings, noncommutative GrÃ¶bner bases, graded Hecke algebras, symplectic reflection algebras, Hochschild cohomology Categories:16S36, 16S35, 16S80, 16W20, 16Z05, 16E40 

12. CJM 2013 (vol 66 pp. 625)
 Giambruno, Antonio; Mattina, Daniela La; Zaicev, Mikhail

Classifying the Minimal Varieties of Polynomial Growth
Let $\mathcal{V}$ be a variety of associative algebras generated by
an algebra with $1$ over a field of characteristic zero. This
paper is devoted to the classification of the varieties
$\mathcal{V}$ which are minimal of polynomial growth (i.e., their
sequence of codimensions growth like $n^k$ but any proper subvariety
grows like $n^t$ with $t\lt k$). These varieties are the building
blocks of general varieties of polynomial growth.
It turns out that for $k\le 4$ there are only a finite number of
varieties of polynomial growth $n^k$, but for each $k \gt 4$, the
number of minimal varieties is at least $F$, the cardinality of
the base field and we give a recipe of how to construct them.
Keywords:Tideal, polynomial identity, codimension, polynomial growth, Categories:16R10, 16P90 

13. CJM 2012 (vol 65 pp. 863)
 JosuatVergès, Matthieu

Cumulants of the $q$semicircular Law, Tutte Polynomials, and Heaps
The $q$semicircular distribution is a probability law that
interpolates between the Gaussian law and the semicircular law. There
is a combinatorial interpretation of its moments in terms of matchings
where $q$ follows the number of crossings, whereas for the free
cumulants one has to restrict the enumeration to connected matchings.
The purpose of this article is to describe combinatorial properties of
the classical cumulants. We show that like the free cumulants, they
are obtained by an enumeration of connected matchings, the weight
being now an evaluation of the Tutte polynomial of a socalled
crossing graph. The case $q=0$ of these cumulants was studied by
Lassalle using symmetric functions and hypergeometric series. We show
that the underlying combinatorics is explained through the theory of
heaps, which is Viennot's geometric interpretation of the
CartierFoata monoid. This method also gives a general formula for
the cumulants in terms of free cumulants.
Keywords:moments, cumulants, matchings, Tutte polynomials, heaps Categories:05A18, 05C31, 46L54 

14. CJM 2012 (vol 65 pp. 600)
 Kroó, A.; Lubinsky, D. S.

Christoffel Functions and Universality in the Bulk for Multivariate Orthogonal Polynomials
We establish asymptotics for Christoffel functions associated with
multivariate orthogonal polynomials. The underlying measures are assumed to
be regular on a suitable domain  in particular this is true if they are
positive a.e. on a compact set that admits analytic parametrization. As a
consequence, we obtain asymptotics for Christoffel functions for measures on
the ball and simplex, under far more general conditions than previously
known. As another consequence, we establish universality type limits in the
bulk in a variety of settings.
Keywords:orthogonal polynomials, random matrices, unitary ensembles, correlation functions, Christoffel functions Categories:42C05, 42C99, 42B05, 60B20 

15. CJM 2012 (vol 65 pp. 171)
 Lyall, Neil; Magyar, Ákos

Optimal Polynomial Recurrence
Let $P\in\mathbb Z[n]$ with $P(0)=0$ and $\varepsilon\gt 0$.
We show, using Fourier analytic techniques, that if $N\geq
\exp\exp(C\varepsilon^{1}\log\varepsilon^{1})$ and
$A\subseteq\{1,\dots,N\}$, then there must exist $n\in\mathbb N$ such that
\[\frac{A\cap (A+P(n))}{N}\gt \left(\frac{A}{N}\right)^2\varepsilon.\]
In addition to this we also show, using the same Fourier analytic
methods, that if $A\subseteq\mathbb N$, then the set of
$\varepsilon$optimal return times
\[R(A,P,\varepsilon)=\left\{n\in \mathbb N
\,:\,\delta(A\cap(A+P(n)))\gt \delta(A)^2\varepsilon\right\}\]
is syndetic for every $\varepsilon\gt 0$. Moreover, we show that
$R(A,P,\varepsilon)$ is dense in every sufficiently long interval, in particular we show that
there exists an $L=L(\varepsilon,P,A)$ such that
\[\leftR(A,P,\varepsilon)\cap I\right
\geq c(\varepsilon,P)I\]
for all intervals $I$ of natural numbers with $I\geq L$ and
$c(\varepsilon,P)=\exp\exp(C\,\varepsilon^{1}\log\varepsilon^{1})$.
Keywords:Sarkozy, syndetic, polynomial return times Category:11B30 

16. CJM 2012 (vol 64 pp. 318)
 Ingram, Patrick

Cubic Polynomials with Periodic Cycles of a Specified Multiplier
We consider cubic polynomials $f(z)=z^3+az+b$ defined over
$\mathbb{C}(\lambda)$, with a marked point of period $N$ and multiplier
$\lambda$. In the case $N=1$, there are infinitely many such objects,
and in the case $N\geq 3$, only finitely many (subject to a mild
assumption). The case $N=2$ has particularly rich structure, and we
are able to describe all such cubic polynomials defined over the field
$\bigcup_{n\geq 1}\mathbb{C}(\lambda^{1/n})$.
Keywords:cubic polynomials, periodic points, holomorphic dynamics Category:37P35 

17. CJM 2011 (vol 64 pp. 1036)
 Koh, Doowon; Shen, ChunYen

Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields
In this paper we study the extension problem, the
averaging problem, and the generalized ErdÅsFalconer distance
problem associated with arbitrary homogeneous varieties in three
dimensional vector spaces over finite fields. In the case when the
varieties do not contain any plane passing through the origin, we
obtain the best possible results on the aforementioned three problems. In
particular, our result on the extension problem modestly generalizes
the result by Mockenhaupt and Tao who studied the particular conical
extension problem. In addition, investigating the Fourier decay on
homogeneous varieties enables us to give complete mapping properties
of averaging operators. Moreover, we improve the size condition on a
set such that the cardinality of its distance set is nontrivial.
Keywords:extension problems, averaging operator, finite fields, ErdÅsFalconer distance problems, homogeneous polynomial Categories:42B05, 11T24, 52C17 

18. CJM 2011 (vol 64 pp. 481)
 Chamorro, Diego

Some Functional Inequalities on Polynomial Volume Growth Lie Groups
In this article we study some Sobolevtype inequalities on polynomial volume growth Lie groups.
We show in particular that improved Sobolev inequalities can be extended to this general framework
without the use of the LittlewoodPaley decomposition.
Keywords:Sobolev inequalities, polynomial volume growth Lie groups Category:22E30 

19. CJM 2010 (vol 63 pp. 413)
 Konvalinka, Matjaž; Skandera, Mark

Generating Functions for Hecke Algebra Characters
Certain polynomials in $n^2$ variables that serve as generating
functions for symmetric group characters are sometimes called
($S_n$) character immanants.
We point out a close connection between the identities of
LittlewoodMerrisWatkins
and GouldenJackson, which relate $S_n$ character immanants
to the determinant, the permanent and MacMahon's Master Theorem.
From these results we obtain a generalization
of Muir's identity.
Working with the quantum polynomial ring and the Hecke algebra
$H_n(q)$, we define quantum immanants that are generating
functions for Hecke algebra characters.
We then prove quantum analogs of the LittlewoodMerrisWatkins identities
and selected GouldenJackson identities
that relate $H_n(q)$ character immanants to
the quantum determinant, quantum permanent, and quantum Master Theorem
of GaroufalidisL\^eZeilberger.
We also obtain a generalization of Zhang's quantization of Muir's
identity.
Keywords:determinant, permanent, immanant, Hecke algebra character, quantum polynomial ring Categories:15A15, 20C08, 81R50 

20. CJM 2010 (vol 63 pp. 200)
 Rahman, Mizan

An Explicit Polynomial Expression for a $q$Analogue of the 9$j$ Symbols
Using standard transformation and summation formulas for basic
hypergeometric series we obtain an explicit polynomial form of the
$q$analogue of the 9$j$ symbols, introduced by the author in a
recent publication. We also consider a limiting case in which the
9$j$ symbol factors into two Hahn polynomials. The same
factorization occurs in another limit case of the corresponding
$q$analogue.
Keywords:6$j$ and 9$j$ symbols, $q$analogues, balanced and verywellpoised basic hypergeometric series, orthonormal polynomials in one and two variables, Racah and $q$Racah polynomials and their extensions Categories:33D45, 33D50 

21. CJM 2010 (vol 63 pp. 181)
 Ismail, Mourad E. H.; Obermaier, Josef

Characterizations of Continuous and Discrete $q$Ultraspherical Polynomials
We characterize the continuous $q$ultraspherical polynomials in
terms of the special form of the coefficients in the expansion
$\mathcal{D}_q P_n(x)$ in the basis $\{P_n(x)\}$, $\mathcal{D}_q$
being the AskeyWilson divided difference operator. The polynomials
are assumed to be symmetric, and the connection coefficients
are multiples of the reciprocal of the square of the $L^2$ norm of
the polynomials. A similar characterization is given for the discrete
$q$ultraspherical polynomials. A new proof of the evaluation of
the connection coefficients for big $q$Jacobi polynomials is given.
Keywords:continuous $q$ultraspherical polynomials, big $q$Jacobi polynomials, discrete $q$ultra\spherical polynomials, AskeyWilson operator, $q$difference operator, recursion coefficients Categories:33D45, 42C05 

22. CJM 2010 (vol 62 pp. 1058)
23. CJM 2010 (vol 62 pp. 261)
24. CJM 2009 (vol 61 pp. 351)
 Graham, William; Hunziker, Markus

Multiplication of Polynomials on Hermitian Symmetric spaces and LittlewoodRichardson Coefficients
Let $K$ be a complex reductive algebraic group and $V$ a
representation of $K$. Let $S$ denote the ring of polynomials on
$V$. Assume that the action of $K$ on $S$ is multiplicityfree. If
$\lambda$ denotes the isomorphism class of an irreducible
representation of $K$, let $\rho_\lambda\from K \rightarrow
GL(V_{\lambda})$ denote the corresponding irreducible representation
and $S_\lambda$ the $\lambda$isotypic component of $S$. Write
$S_\lambda \cdot S_\mu$ for the subspace of $S$ spanned by products of
$S_\lambda$ and $S_\mu$. If $V_\nu$ occurs as an irreducible
constituent of $V_\lambda\otimes V_\mu$, is it true that
$S_\nu\subseteq S_\lambda\cdot S_\mu$? In this paper, the authors
investigate this question for representations arising in the context
of Hermitian symmetric pairs. It is shown that the answer is yes in
some cases and, using an earlier result of Ruitenburg, that in the
remaining classical cases, the answer is yes provided that a
conjecture of Stanley on the multiplication of Jack polynomials is
true. It is also shown how the conjecture connects multiplication in
the ring $S$ to the usual LittlewoodRichardson rule.
Keywords:Hermitian symmetric spaces, multiplicity free actions, LittlewoodRichardson coefficients, Jack polynomials Categories:14L30, 22E46 

25. CJM 2008 (vol 60 pp. 1240)
 Beliakova, Anna; Wehrli, Stephan

Categorification of the Colored Jones Polynomial and Rasmussen Invariant of Links
We define a family of formal Khovanov brackets
of a colored link depending on two parameters.
The isomorphism classes of these brackets are
invariants of framed colored links.
The BarNatan functors applied to these brackets
produce Khovanov and Lee homology theories categorifying the colored
Jones polynomial. Further,
we study conditions under which
framed colored link cobordisms induce chain transformations between
our formal brackets. We conjecture that
for special choice of parameters, Khovanov and Lee homology theories
of colored links are functorial (up to sign).
Finally, we extend the Rasmussen invariant to links and give examples
where this invariant is a stronger obstruction to sliceness
than the multivariable LevineTristram signature.
Keywords:Khovanov homology, colored Jones polynomial, slice genus, movie moves, framed cobordism Categories:57M25, 57M27, 18G60 
