51. CMB 2011 (vol 56 pp. 258)
 Chandoul, A.; Jellali, M.; Mkaouar, M.

The Smallest Pisot Element in the Field of Formal Power Series Over a Finite Field
Dufresnoy and Pisot characterized the smallest
Pisot number of degree $n \geq 3$ by giving explicitly its minimal
polynomial. In this paper, we translate Dufresnoy and Pisot's
result to the Laurent series case.
The
aim of this paper is to prove that the minimal polynomial
of the smallest Pisot element (SPE) of degree $n$ in the field of
formal power series over a finite field
is given by $P(Y)=Y^{n}\alpha XY^{n1}\alpha^n,$ where $\alpha$
is the least element of the finite field $\mathbb{F}_{q}\backslash\{0\}$
(as a finite total ordered set). We prove that the sequence of
SPEs of degree $n$ is decreasing and converges to $\alpha X.$
Finally, we show how to obtain explicit continued fraction
expansion of the smallest Pisot element over a finite field.
Keywords:Pisot element, continued fraction, Laurent series, finite fields Categories:11A55, 11D45, 11D72, 11J61, 11J66 

52. CMB 2011 (vol 56 pp. 500)
 Browning, T. D.

The LangWeil Estimate for Cubic Hypersurfaces
An improved estimate is provided for the number of $\mathbb{F}_q$rational points
on a geometrically irreducible, projective, cubic hypersurface that is
not equal to a cone.
Keywords:cubic hypersurface, rational points, finite fields Categories:11G25, 14G15 

53. 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 

54. CMB 2011 (vol 56 pp. 148)
55. CMB 2011 (vol 56 pp. 161)
 Rêgo, L. C.; Cintra, R. J.

An Extension of the Dirichlet Density for Sets of Gaussian Integers
Several measures for the density of sets of integers have been proposed,
such as the asymptotic density, the Schnirelmann density, and the Dirichlet density. There has been some work in the literature on extending some of these concepts of density to higher dimensional sets of integers. In this work, we propose an extension of the Dirichlet density for sets of Gaussian integers and
investigate some of its properties.
Keywords:Gaussian integers, Dirichlet density Categories:11B05, 11M99, 11N99 

56. CMB 2011 (vol 55 pp. 842)
57. CMB 2011 (vol 56 pp. 283)
 Coons, Michael

Transcendental Solutions of a Class of Minimal Functional Equations
We prove a result concerning power series
$f(z)\in\mathbb{C}[\mkern3mu[z]\mkern3mu]$
satisfying a functional equation of the form
$$
f(z^d)=\sum_{k=1}^n
\frac{A_k(z)}{B_k(z)}f(z)^k,
$$
where $A_k(z),B_k(z)\in
\mathbb{C}[z]$. In particular, we show that if $f(z)$ satisfies a
minimal functional equation of the above form with $n\geqslant 2$,
then $f(z)$ is necessarily transcendental. Towards a more complete
classification, the case $n=1$ is also considered.
Keywords:transcendence, generating functions, Mahlertype functional equation Categories:11B37, 11B83, , 11J91 

58. CMB 2011 (vol 56 pp. 70)
 Hrubeš, P.; Wigderson, A.; Yehudayoff, A.

An Asymptotic Bound on the Composition Number of Integer Sums of Squares Formulas
Let $\sigma_{\mathbb Z}(k)$ be the smallest $n$ such that there exists an
identity
\[
(x_1^2 + x_2^2 + \cdots + x_k^2) \cdot (y_1^2 + y_2^2 + \cdots + y_k^2)
= f_1^2 + f_2^2 + \cdots + f_n^2,
\]
with $f_1,\dots,f_n$ being polynomials with integer coefficients in
the variables $x_1,\dots,x_k$ and $y_1,\dots,y_k$. We prove that
$\sigma_{\mathbb Z}(k) \geq \Omega(k^{6/5})$.
Keywords:composition formulas, sums of squares, RadonHurwitz number Category:11E25 

59. 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 

60. CMB 2011 (vol 55 pp. 774)
 Mollin, R. A.; Srinivasan, A.

Pell Equations: NonPrincipal Lagrange Criteria and Central Norms
We provide a criterion for the central norm to be
any value in the simple continued fraction expansion of $\sqrt{D}$
for any nonsquare integer $D>1$. We also provide a simple criterion
for the solvability of the Pell equation $x^2Dy^2=1$ in terms of
congruence conditions modulo $D$.
Keywords:Pell's equation, continued fractions, central norms Categories:11D09, 11A55, 11R11, 11R29 

61. CMB 2011 (vol 55 pp. 435)
 Zelator, Konstantine

A Note on the Diophantine Equation $x^2 + y^6 = z^e$, $e \geq 4$
We consider the diophantine equation $x^2 + y^6 = z^e$, $e \geq 4$.
We show that, when $e$ is a multiple of $4$ or $6$, this equation
has no solutions in positive integers with $x$ and $y$ relatively prime.
As a corollary, we show
that there exists no primitive Pythagorean triangle one of whose
leglengths is a perfect cube, while the hypotenuse length is an
integer square.
Keywords:diophantine equation Category:11D 

62. CMB 2011 (vol 55 pp. 400)
 Sebbar, Abdellah; Sebbar, Ahmed

Eisenstein Series and Modular Differential Equations
The purpose of this paper is to solve various differential
equations having Eisenstein series as coefficients using various tools and techniques. The solutions
are given in terms of modular forms, modular functions, and
equivariant forms.
Keywords:differential equations, modular forms, Schwarz derivative, equivariant forms Categories:11F11, 34M05 

63. CMB 2011 (vol 55 pp. 26)
 Bertin, Marie José

A Mahler Measure of a $K3$ Surface Expressed as a Dirichlet $L$Series
We present another example of a $3$variable polynomial defining a $K3$hypersurface
and having a logarithmic Mahler measure expressed in terms of a Dirichlet
$L$series.
Keywords:modular Mahler measure, EisensteinKronecker series, $L$series of $K3$surfaces, $l$adic representations, LivnÃ© criterion, RankinCohen brackets Categories:11, 14D, 14J 

64. CMB 2011 (vol 55 pp. 67)
65. CMB 2011 (vol 55 pp. 193)
 Ulas, Maciej

Rational Points in Arithmetic Progressions on $y^2=x^n+k$
Let $C$ be a hyperelliptic curve given by the equation $y^2=f(x)$
for $f\in\mathbb{Z}[x]$ without multiple roots. We say that points
$P_{i}=(x_{i}, y_{i})\in C(\mathbb{Q})$ for $i=1,2,\dots, m$ are in
arithmetic progression if the numbers $x_{i}$ for $i=1,2,\dots, m$
are in arithmetic progression.
In this paper we show that there exists a polynomial $k\in\mathbb{Z}[t]$
with the property that on the elliptic curve $\mathcal{E}':
y^2=x^3+k(t)$ (defined over the field $\mathbb{Q}(t)$) we can find four
points in arithmetic progression that are independent in the group
of all $\mathbb{Q}(t)$rational points on the curve $\mathcal{E}'$. In
particular this result generalizes earlier results of Lee and
V\'{e}lez. We also show that if $n\in\mathbb{N}$ is odd,
then there are infinitely many $k$'s with the property that on
curves $y^2=x^n+k$ there are four rational points in arithmetic
progressions. In the case when $n$ is even we can find infinitely
many $k$'s such that on curves $y^2=x^n+k$ there are six rational
points in arithmetic progression.
Keywords:arithmetic progressions, elliptic curves, rational points on hyperelliptic curves Category:11G05 

66. 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 

67. CMB 2011 (vol 55 pp. 38)
 Butske, William

Endomorphisms of Two Dimensional Jacobians and Related Finite Algebras
Zarhin proves that if $C$ is the curve $y^2=f(x)$ where
$\textrm{Gal}_{\mathbb{Q}}(f(x))=S_n$ or $A_n$, then
${\textrm{End}}_{\overline{\mathbb{Q}}}(J)=\mathbb{Z}$. In seeking to examine his
result in the genus $g=2$ case supposing other Galois groups, we
calculate
$\textrm{End}_{\overline{\mathbb{Q}}}(J)\otimes_{\mathbb{Z}} \mathbb{F}_2$
for a genus $2$ curve where $f(x)$ is irreducible.
In particular, we show that unless the Galois group is $S_5$ or
$A_5$, the Galois group does not determine ${\textrm{End}}_{\overline{\mathbb{Q}}}(J)$.
Categories:11G10, 20C20 

68. CMB 2011 (vol 54 pp. 645)
 Flores, André Luiz; Interlando, J. Carmelo; Neto, Trajano Pires da Nóbrega

An Extension of Craig's Family of Lattices
Let $p$ be a prime, and let $\zeta_p$ be a primitive $p$th root of
unity. The lattices in Craig's family are $(p1)$dimensional and
are geometrical representations of the integral $\mathbb
Z[\zeta_p]$ideals $\langle 1\zeta_p \rangle^i$, where $i$ is a
positive integer. This lattice construction technique is a powerful
one. Indeed, in dimensions $p1$ where $149 \leq p \leq 3001$,
Craig's lattices are the densest packings known. Motivated by this,
we construct $(p1)(q1)$dimensional lattices from the integral
$\mathbb Z[\zeta _{pq}]$ideals $\langle 1\zeta_p \rangle^i \langle
1\zeta_q \rangle^j$, where $p$ and $q$ are distinct primes and $i$
and $j$ are positive integers. In terms of spherepacking density,
the new lattices and those in Craig's family have the same
asymptotic behavior. In conclusion, Craig's family is greatly
extended while preserving its spherepacking properties.
Keywords:geometry of numbers, lattice packing, Craig's lattices, quadratic forms, cyclotomic fields Categories:11H31, 11H55, 11H50, 11R18, 11R04 

69. CMB 2011 (vol 55 pp. 60)
 Coons, Michael

Extension of Some Theorems of W. Schwarz
In this paper, we prove that a nonzero power series $F(z)\in\mathbb{C}
[\mkern3mu[ z]\mkern3mu]
$
satisfying $$F(z^d)=F(z)+\frac{A(z)}{B(z)},$$ where $d\geq 2$, $A(z),B(z)\in\mathbb{C}[z]$
with $A(z)\neq 0$ and $\deg A(z),\deg B(z)
Keywords:functional equations, transcendence, power series Categories:11B37, 11J81 

70. CMB 2011 (vol 54 pp. 757)
 Sun, Qingfeng

Cancellation of Cusp Forms Coefficients over Beatty Sequences on $\textrm{GL}(m)$
Let $A(n_1,n_2,\dots,n_{m1})$
be the normalized Fourier coefficients of
a Maass cusp form on $\textrm{GL}(m)$.
In this paper, we study the cancellation of $A
(n_1,n_2,\dots,n_{m1})$ over Beatty sequences.
Keywords:Fourier coefficients, Maass cusp form on $\textrm{GL}(m)$, Beatty sequence Categories:11F30, 11M41, 11B83 

71. CMB 2011 (vol 54 pp. 739)
 Samuels, Charles L.

The Infimum in the Metric Mahler Measure
Dubickas and Smyth defined the metric Mahler measure on the
multiplicative group of nonzero algebraic numbers.
The definition involves taking an infimum over representations
of an algebraic number $\alpha$ by other
algebraic numbers. We verify their conjecture that the
infimum in its definition is always achieved, and we establish its
analog for the ultrametric Mahler measure.
Keywords:Weil height, Mahler measure, metric Mahler measure, Lehmer's problem Categories:11R04, 11R09 

72. 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 

73. CMB 2011 (vol 54 pp. 316)
 Mazhouda, Kamel

The SaddlePoint Method and the Li Coefficients
In this paper, we apply the saddlepoint method in conjunction with
the theory of the NÃ¶rlundRice integrals to derive precise
asymptotic formula for the generalized Li coefficients established
by Omar and Mazhouda.
Actually, for any function $F$ in the Selberg class
$\mathcal{S}$ and under the Generalized Riemann Hypothesis, we have
$$
\lambda_{F}(n)=\frac{d_{F}}{2}n\log n+c_{F}n+O(\sqrt{n}\log n),
$$
with
$$
c_{F}=\frac{d_{F}}{2}(\gamma1)+\frac{1}{2}\log(\lambda
Q_{F}^{2}),\ \lambda=\prod_{j=1}^{r}\lambda_{j}^{2\lambda_{j}},
$$
where $\gamma$ is the Euler's constant and the notation is as below.
Keywords:Selberg class, Saddlepoint method, Riemann Hypothesis, Li's criterion Categories:11M41, 11M06 

74. CMB 2011 (vol 54 pp. 330)
 Mouhib, A.

Sur la borne infÃ©rieure du rang du 2groupe de classes de certains corps multiquadratiques
Soient $p_1,p_2,p_3$ et $q$ des nombres premiers distincts tels que
$p_1\equiv p_2\equiv p_3\equiv q\equiv 1 \pmod{4}$, $k = \mathbf{Q}
(\sqrt{p_1}, \sqrt{p_2}, \sqrt{p_3}, \sqrt{q})$ et $\operatorname{Cl}_2(k)$ le
$2$groupe de classes de $k$. A. FrÃ¶hlich a
dÃ©montrÃ© que $\operatorname{Cl}_2(k)$ n'est jamais trivial. Dans cet article,
nous donnons une extension de ce rÃ©sultat, en dÃ©montrant que le
rang de $\operatorname{Cl}_2(k)$ est toujours supÃ©rieur ou Ã©gal Ã $2$. Nous
dÃ©montrons aussi, que la valeur $2$ est optimale pour une famille
infinie de corps $k$.
Keywords:class group, units, multiquadratic number fields Categories:11R29, 11R11 

75. CMB 2010 (vol 54 pp. 39)
 Chapman, S. T.; GarcíaSánchez, P. A.; Llena, D.; Marshall, J.

Elements in a Numerical Semigroup with Factorizations of the Same Length
Questions concerning the lengths of factorizations into irreducible
elements in numerical monoids
have gained much attention in the recent literature. In this note,
we show that a numerical monoid has an element with two different
irreducible factorizations of the same length if and only if its
embedding dimension is greater than
two. We find formulas in embedding dimension three for the smallest
element with two different irreducible factorizations of the same
length and the largest element whose different irreducible
factorizations all have distinct lengths. We show that these
formulas do not naturally extend to higher embedding
dimensions.
Keywords:numerical monoid, numerical semigroup, nonunique factorization Categories:20M14, 20D60, 11B75 
