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26. 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 $P(Y)=Y^{n}-\alpha XY^{n-1}-\alpha^n,$ 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 fieldsCategories:11A55, 11D45, 11D72, 11J61, 11J66

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 equationCategories:11N60, 11B83, 11D09

28. CMB 2011 (vol 56 pp. 500)

Browning, T. D.
 The Lang--Weil 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 fieldsCategories:11G25, 14G15

29. CMB 2011 (vol 56 pp. 148)

Oukhaba, Hassan; Viguié, Stéphane
 On the Gras Conjecture for Imaginary Quadratic Fields In this paper we extend K. Rubin's methods to prove the Gras conjecture for abelian extensions of a given imaginary quadratic field $k$ and prime numbers $p$ that divide the number of roots of unity in $k$. Keywords:elliptic units, Stark units, Gras conjecture, Euler systemsCategories:11R27, 11R29, 11G16

30. CMB 2011 (vol 55 pp. 842)

Sairaiji, Fumio; Yamauchi, Takuya
 The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the Frey-Jarden Conjecture Frey and Jarden asked if any abelian variety over a number field $K$ has the infinite Mordell-Weil rank over the maximal abelian extension $K^{\operatorname{ab}}$. In this paper, we give an affirmative answer to their conjecture for the Jacobian variety of any smooth projective curve $C$ over $K$ such that $\sharp C(K^{\operatorname{ab}})=\infty$ and for any abelian variety of $\operatorname{GL}_2$-type with trivial character. Keywords:Mordell-Weil rank, Jacobian varieties, Frey-Jarden conjecture, abelian pointsCategories:11G05, 11D25, 14G25, 14K07

31. CMB 2011 (vol 56 pp. 283)

Coons, Michael
 Transcendental Solutions of a Class of Minimal Functional Equations We prove a result concerning power series $f(z)\in\mathbb{C}[\mkern-3mu[z]\mkern-3mu]$ 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, Mahler-type functional equationCategories:11B37, 11B83, , 11J91

32. 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 densityCategories:11B05, 11M99, 11N99

33. 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, Radon-Hurwitz numberCategory:11E25

34. 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 sumCategories:11L40, 14H52

35. CMB 2011 (vol 55 pp. 774)

Mollin, R. A.; Srinivasan, A.
 Pell Equations: Non-Principal 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 non-square integer $D>1$. We also provide a simple criterion for the solvability of the Pell equation $x^2-Dy^2=-1$ in terms of congruence conditions modulo $D$. Keywords:Pell's equation, continued fractions, central normsCategories:11D09, 11A55, 11R11, 11R29

36. 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 equationCategory:11D

37. 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 formsCategories:11F11, 34M05

38. 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, Eisenstein-Kronecker series, $L$-series of $K3$-surfaces, $l$-adic representations, LivnÃ© criterion, Rankin-Cohen bracketsCategories:11, 14D, 14J

39. CMB 2011 (vol 55 pp. 67)

Cummins, C. J.; Duncan, J. F.
 An $E_8$ Correspondence for Multiplicative Eta-Products We describe an $E_8$ correspondence for the multiplicative eta-products of weight at least $4$. Keywords:We describe an E8 correspondence for the multiplicative eta-products of weight at leastÂ 4.Categories:11F20, 11F12, 17B60

40. 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 curvesCategory:11G05

41. 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 by-product 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 sumsCategories:11L40, 11T06

42. 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

43. CMB 2011 (vol 55 pp. 60)

Coons, Michael
 Extension of Some Theorems of W. Schwarz In this paper, we prove that a non--zero power series $F(z)\in\mathbb{C} [\mkern-3mu[ z]\mkern-3mu]$ 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 $\deg A(z),\deg B(z) Keywords:functional equations, transcendence, power seriesCategories:11B37, 11J81 44. CMB 2011 (vol 54 pp. 645) Flores, André Luiz; Interlando, J. Carmelo; Neto, Trajano Pires da Nóbrega  An Extension of Craig's Family of Lattices Let$p$be a prime, and let$\zeta_p$be a primitive$p$-th root of unity. The lattices in Craig's family are$(p-1)$-dimensional and are geometrical representations of the integral$\mathbb Z[\zeta_p]$-ideals$\langle 1-\zeta_p \rangle^i$, where$i$is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions$p-1$where$149 \leq p \leq 3001$, Craig's lattices are the densest packings known. Motivated by this, we construct$(p-1)(q-1)$-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 sphere-packing 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 sphere-packing properties. Keywords:geometry of numbers, lattice packing, Craig's lattices, quadratic forms, cyclotomic fieldsCategories:11H31, 11H55, 11H50, 11R18, 11R04 45. CMB 2011 (vol 54 pp. 757) Sun, Qingfeng  Cancellation of Cusp Forms Coefficients over Beatty Sequences on$\textrm{GL}(m)$Let$A(n_1,n_2,\dots,n_{m-1})$be the normalized Fourier coefficients of a Maass cusp form on$\textrm{GL}(m)$. In this paper, we study the cancellation of$A (n_1,n_2,\dots,n_{m-1})$over Beatty sequences. Keywords:Fourier coefficients, Maass cusp form on$\textrm{GL}(m)$, Beatty sequenceCategories:11F30, 11M41, 11B83 46. 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 non-zero 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 problemCategories:11R04, 11R09 47. 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 polynomialCategories:11Y11, 68W20 48. CMB 2011 (vol 54 pp. 316) Mazhouda, Kamel  The Saddle-Point Method and the Li Coefficients In this paper, we apply the saddle-point method in conjunction with the theory of the NÃ¶rlund-Rice 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 $$c_{F}=\frac{d_{F}}{2}(\gamma-1)+\frac{1}{2}\log(\lambda Q_{F}^{2}),\ \lambda=\prod_{j=1}^{r}\lambda_{j}^{2\lambda_{j}},$$ where$\gamma$is the Euler's constant and the notation is as below. Keywords:Selberg class, Saddle-point method, Riemann Hypothesis, Li's criterionCategories:11M41, 11M06 49. CMB 2011 (vol 54 pp. 330) Mouhib, A.  Sur la borne infÃ©rieure du rang du 2-groupe 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 fieldsCategories:11R29, 11R11

50. CMB 2010 (vol 53 pp. 654)

Elliott, P. D. T. A.
 Variations on a Paper of ErdÅs and Heilbronn It is shown that an old direct argument of ErdÅs and Heilbronn may be elaborated to yield a result of the current inverse type. Categories:11L07, 11P70
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