51. CMB 2006 (vol 49 pp. 438)
 Mercer, Idris David

Unimodular Roots of\\ Special Littlewood Polynomials
We call $\alpha(z) = a_0 + a_1 z + \dots + a_{n1} z^{n1}$ a Littlewood
polynomial if $a_j = \pm 1$ for all $j$. We call $\alpha(z)$ selfreciprocal
if $\alpha(z) = z^{n1}\alpha(1/z)$, and call $\alpha(z)$ skewsymmetric if
$n = 2m+1$ and $a_{m+j} = (1)^j a_{mj}$ for all $j$. It has been observed
that Littlewood polynomials with particularly high minimum modulus on
the unit
circle in $\bC$ tend to be skewsymmetric. In this paper, we prove that a
skewsymmetric Littlewood polynomial cannot have any zeros on the unit circle,
as well as providing a new proof of the known result that a selfreciprocal
Littlewood polynomial must have a zero on the unit circle.
Categories:26C10, 30C15, 42A05 

52. CMB 2006 (vol 49 pp. 381)
53. CMB 2005 (vol 48 pp. 580)
 Kot, Piotr

Exceptional Sets in Hartogs Domains
Assume that $\Omega$ is a Hartogs domain in $\mathbb{C}^{1+n}$,
defined as $\Omega=\{(z,w)\in\mathbb{C}^{1+n}:z<\mu(w),w\in H\}$, where $H$ is an open set in
$\mathbb{C}^{n}$ and $\mu$ is a continuous function with positive values in $H$ such that $\ln\mu$
is a strongly plurisubharmonic function in $H$. Let $\Omega_{w}=\Omega\cap(\mathbb{C}\times\{w\})$.
For a given set $E$ contained in $H$ of the type $G_{\delta}$ we construct a holomorphic function
$f\in\mathbb{O}(\Omega)$ such that
\[
E=\Bigl\{ w\in\mathbb{C}^{n}:\int_{\Omega_{w}}f(\cdot\,,w)^{2}\,d\mathfrak{L}^{2}=\infty\Bigr\}.
\]
Keywords:boundary behaviour of holomorphic functions,, exceptional sets Category:30B30 

54. CMB 2005 (vol 48 pp. 409)
55. CMB 2004 (vol 47 pp. 152)
56. CMB 2004 (vol 47 pp. 17)
 Gorkin, Pamela; Mortini, Raymond

Universal Singular Inner Functions
We show that there exists a singular inner function $S$ which is
universal for noneuclidean translates; that is one for which the set
$\{S(\frac{z+z_n}{1+\bar z_nz}):n\in\mathbb{N}\}$ is locally uniformly dense
in the set of all zerofree holomorphic functions in $\mathbb{D}$ bounded by
one.
Category:30D50 

57. CMB 2003 (vol 46 pp. 559)
 Marco, Nicolas; Massaneda, Xavier

On Density Conditions for Interpolation in the Ball
In this paper we study interpolating sequences for two related spaces of
holomorphic functions in the unit ball of $\C^n$, $n>1$. We first give density
conditions for a sequence to be interpolating for the class $A^{\infty}$ of
holomorphic functions with polynomial growth. The sufficient condition is
formally identical to the characterizing condition in dimension $1$, whereas the
necessary one goes along the lines of the results given by Li and Taylor for
some spaces of entire functions. In the second part of the paper we show that a
density condition, which for $n=1$ coincides with the characterizing condition
given by Seip, is sufficient for interpolation in the (weighted) Bergman space.
Categories:32A36, 32A38, 30E05 

58. CMB 2003 (vol 46 pp. 95)
 Gauthier, P. M.

Cercles de remplissage for the Riemann Zeta Function
The celebrated theorem of Picard asserts that each nonconstant entire
function assumes every value infinitely often, with at most one
exception. The Riemann zeta function has this Picard behaviour in a
sequence of discs lying in the critical band and whose diameters tend
to zero. According to the Riemann hypothesis, the value zero would be
this (unique) exceptional value.
Keywords:cercles de remplissage, Riemann zeta function Category:30 

59. CMB 2002 (vol 45 pp. 265)
 Nawrocki, Marek

On the Smirnov Class Defined by the Maximal Function
H.~O.~Kim has shown that contrary to the case of
$H^p$space, the Smirnov class $M$ defined by the radial maximal
function is essentially smaller than the classical Smirnov class
of the disk. In the paper we show that these two classes have the
same corresponding locally convex structure, {\it i.e.} they have the
same dual spaces and the same Fr\'echet envelopes. We describe a
general form of a continuous linear functional on $M$ and
multiplier from $M$ into $H^p$, $0 < p \leq \infty$.
Keywords:Smirnov class, maximal radial function, multipliers, dual space, FrÃ©chet envelope Categories:46E10, 30A78, 30A76 

60. CMB 2002 (vol 45 pp. 36)
61. CMB 2002 (vol 45 pp. 89)
 Grant, David

On Gunning's Prime Form in Genus $2$
Using a classical generalization of Jacobi's derivative formula, we
give an explicit expression for Gunning's prime form in genus 2 in
terms of theta functions and their derivatives.
Categories:14K25, 30F10 

62. CMB 2002 (vol 45 pp. 154)
 Weitsman, Allen

On the Poisson Integral of Step Functions and Minimal Surfaces
Applications of minimal surface methods are made to obtain information
about univalent harmonic mappings. In the case where the mapping arises
as the Poisson integral of a step function, lower bounds for the number
of zeros of the dilatation are obtained in terms of the geometry of the
image.
Keywords:harmonic mappings, dilatation, minimal surfaces Categories:30C62, 31A05, 31A20, 49Q05 

63. CMB 2001 (vol 44 pp. 420)
64. CMB 2000 (vol 43 pp. 183)
65. CMB 2000 (vol 43 pp. 105)
66. CMB 2000 (vol 43 pp. 115)
 Schmutz Schaller, Paul

Perfect NonExtremal Riemann Surfaces
An infinite family of perfect, nonextremal Riemann surfaces
is constructed, the first examples of this type of surfaces.
The examples are based on normal subgroups of the modular group
$\PSL(2,{\sf Z})$ of level $6$. They provide nonEuclidean
analogues to the existence of perfect, nonextremal positive
definite quadratic forms. The analogy uses the function {\it syst\/}
which associates to every Riemann surface $M$ the length of a systole,
which is a shortest closed geodesic of $M$.
Categories:11H99, 11F06, 30F45 

67. CMB 1999 (vol 42 pp. 139)
68. CMB 1999 (vol 42 pp. 3)
 Beauzamy, Bernard

How the Roots of a Polynomial Vary with Its Coefficients: A Local Quantitative Result
A wellknown result, due to Ostrowski, states that if $\Vert PQ
\Vert_2< \varepsilon$, then the roots $(x_j)$ of $P$ and $(y_j)$ of
$Q$ satisfy $x_j y_j\le C n \varepsilon^{1/n}$, where $n$ is the
degree of $P$ and $Q$. Though there are cases where this estimate
is sharp, it can still be made more precise in general, in two
ways: first by using Bombieri's norm instead of the classical $l_1$
or $l_2$ norms, and second by taking into account the multiplicity
of each root. For instance, if $x$ is a simple root of $P$, we show
that $xy< C \varepsilon$ instead of $\varepsilon^{1/n}$. The
proof uses the properties of Bombieri's scalar product and Walsh
Contraction Principle.
Category:30C10 

69. CMB 1998 (vol 41 pp. 473)
 Müller, Jürgen; Wengenroth, Jochen

Separating singularities of holomorphic functions
We present a short proof for a classical result on separating
singularities of holomorphic functions. The proof is based on the
open mapping theorem and the fusion lemma of Roth, which is a basic
tool in complex approximation theory. The same method yields
similar separation results for other classes of functions.
Categories:30E99, 30E10 
