101. CMB 2001 (vol 44 pp. 257)
 Abánades, Miguel A.

Algebraic Homology For Real Hyperelliptic and Real Projective Ruled Surfaces
Let $X$ be a reduced nonsingular quasiprojective scheme over ${\mathbb
R}$ such that the set of real rational points $X({\mathbb R})$ is dense
in $X$ and compact. Then $X({\mathbb R})$ is a real algebraic variety.
Denote by $H_k^{\alg}(X({\mathbb R}), {\mathbb Z}/2)$ the group of
homology classes represented by Zariski closed $k$dimensional
subvarieties of $X({\mathbb R})$. In this note we show that $H_1^{\alg}
(X({\mathbb R}), {\mathbb Z}/2)$ is a proper subgroup of
$H_1(X({\mathbb R}), {\mathbb Z}/2)$ for a nonorientable hyperelliptic
surface $X$. We also determine all possible groups $H_1^{\alg}(X({\mathbb R}),
{\mathbb Z}/2)$ for a real ruled surface $X$ in connection with the previously
known description of all possible topological configurations of $X$.
Categories:14P05, 14P25 

102. CMB 2001 (vol 44 pp. 313)
 Reverter, Amadeu; Vila, Núria

Images of mod $p$ Galois Representations Associated to Elliptic Curves
We give an explicit recipe for the determination of the images
associated to the Galois action on $p$torsion points of elliptic
curves. We present a table listing the image for all the elliptic
curves defined over $\QQ$ without complex multiplication with
conductor less than 200 and for each prime number~$p$.
Keywords:Galois groups, elliptic curves, Galois representation, isogeny Categories:11R32, 11G05, 12F10, 14K02 

103. CMB 2001 (vol 44 pp. 223)
104. CMB 2000 (vol 43 pp. 312)
 Dobbs, David E.

On the Prime Ideals in a Commutative Ring
If $n$ and $m$ are positive integers, necessary and sufficient
conditions are given for the existence of a finite commutative ring $R$
with exactly $n$ elements and exactly $m$ prime ideals. Next,
assuming the Axiom of Choice, it is proved that if $R$ is a
commutative ring and $T$ is a commutative $R$algebra which is
generated by a set $I$, then each chain of prime ideals of $T$ lying
over the same prime ideal of $R$ has at most $2^{I}$ elements. A
polynomial ring example shows that the preceding result is
bestpossible.
Categories:13C15, 13B25, 04A10, 14A05, 13M05 

105. CMB 2000 (vol 43 pp. 304)
 Darmon, Henri; Mestre, JeanFrançois

Courbes hyperelliptiques Ã multiplications rÃ©elles et une construction de Shih
Soient $r$ et $p$ deux nombres premiers distincts, soit $K = \Q(\cos
\frac{2\pi}{r})$, et soit $\F$ le corps r\'esiduel de $K$ en une place
audessus de $p$. Lorsque l'image de $(2  2\cos \frac{2\pi}{r})$
dans $\F$ n'est pas un carr\'e, nous donnons une construction
g\'eom\'etrique d'une extension r\'eguliere de $K(t)$ de groupe de
Galois $\PSL_2 (\F)$. Cette extension correspond \`a un rev\^etement
de $\PP^1_{/K}$ de \og{} signature $(r,p,p)$ \fg{} au sens de [3,
sec.~6.3], et son existence est pr\'edite par le crit\`ere de
rigidit\'e de Belyi, Fried, Thompson et Matzat. Sa construction
s'obtient en tordant la representation galoisienne associ\'ee aux
points d'ordre $p$ d'une famille de vari\'et\'es ab\'eliennes \`a
multiplications r\'eelles par $K$ d\'ecouverte par Tautz, Top et
Verberkmoes [6]. Ces vari\'et\'es ab\'eliennes sont d\'efinies sur un
corps quadratique, et sont isog\`enes \`a leur conjugu\'e galoisien.
Notre construction g\'en\'eralise une m\'ethode de Shih [4], [5], que
l'on retrouve quand $r = 2$ et $r = 3$.
Let $r$ and $p$ be distinct prime numbers, let $K = \Q(\cos
\frac{2\pi}{r})$, and let $\F$ be the residue field of $K$ at a place
above $p$. When the image of $(2  2\cos \frac{2\pi}{r})$ in $\F$ is
not a square, we describe a geometric construction of a regular
extension of $K(t)$ with Galois group $\PSL_2 (\F)$. This extension
corresponds to a covering of $\PP^1_{/K}$ of ``signature $(r,p,p)$''
in the sense of [3, sec.~6.3], and its existence is predicted by the
rigidity criterion of Belyi, Fried, Thompson and Matzat. Its
construction is obtained by twisting the mod $p$ galois representation
attached to a family of abelian varieties with real multiplications by
$K$ discovered by Tautz, Top and Verberkmoes [6]. These abelian
varieties are defined in general over a quadratic field, and are
isogenous to their galois conjugate. Our construction generalises a
method of Shih [4], [5], which one recovers when $r = 2$ and $r = 3$.
Categories:11G30, 14H25 

106. CMB 2000 (vol 43 pp. 162)
 Foth, Philip

Moduli Spaces of Polygons and Punctured Riemann Spheres
The purpose of this note is to give a simple combinatorial
construction of the map from the canonically compactified moduli
spaces of punctured complex projective lines to the moduli spaces
$\P_r$ of polygons with fixed side lengths in the Euclidean space
$\E^3$. The advantage of this construction is that one can obtain a
complete set of linear relations among the cycles that generate
homology of $\P_r$. We also classify moduli spaces of pentagons.
Categories:14D20, 18G55, 14H10 

107. CMB 2000 (vol 43 pp. 239)
 Yu, Gang

On the Number of Divisors of the Quadratic Form $m^2+n^2$
For an integer $n$, let $d(n)$ denote the ordinary divisor function.
This paper studies the asymptotic behavior of the sum
$$
S(x) := \sum_{m\leq x, n\leq x} d(m^2 + n^2).
$$
It is proved in the paper that, as $x \to \infty$,
$$
S(x) := A_1 x^2 \log x + A_2 x^2 + O_\epsilon (x^{\frac32 +
\epsilon}),
$$
where $A_1$ and $A_2$ are certain constants and $\epsilon$ is any
fixed positive real number.
The result corrects a false formula given in a paper of Gafurov
concerning the same problem, and improves the error $O \bigl(
x^{\frac53} (\log x)^9 \bigr)$ claimed by Gafurov.
Keywords:divisor, large sieve, exponential sums Categories:11G05, 14H52 

108. CMB 2000 (vol 43 pp. 174)
109. CMB 2000 (vol 43 pp. 129)
 Ballico, E.

Maximal Subbundles of Rank 2 Vector Bundles on Projective Curves
Let $E$ be a stable rank 2 vector bundle on a smooth projective
curve $X$ and $V(E)$ be the set of all rank~1 subbundles of $E$
with maximal degree. Here we study the varieties (nonemptyness,
irreducibility and dimension) of all rank~2 stable vector bundles,
$E$, on $X$ with fixed $\deg(E)$ and $\deg(L)$, $L \in V(E)$ and
such that $\card \bigl( V(E) \bigr) \geq 2$ (resp. $\card \bigl(
V(E) \bigr) = 2$).
Category:14H60 

110. CMB 1999 (vol 42 pp. 499)
 Zaharia, Alexandru

Characterizations of Simple Isolated Line Singularities
A line singularity is a function germ $f\colon(\CC ^{n+1},0) \lra\CC$
with a smooth $1$dimensional critical set $\Sigma=\{(x,y)\in \CC\times
\CC^n \mid y=0\}$. An isolated line singularity is defined by the
condition that for every $x \neq 0$, the germ of $f$ at $(x,0)$ is
equivalent to $y_1^2 +\cdots+y_n ^2$. Simple isolated line
singularities were classified by Dirk Siersma and are analogous
of the famous $ADE$ singularities. We give two new
characterizations of simple isolated line singularities.
Categories:32S25, 14B05 

111. CMB 1999 (vol 42 pp. 445)
 Bochnak, J.; Kucharz, W.

Smooth Maps and Real Algebraic Morphisms
Let $X$ be a compact nonsingular real algebraic variety and let $Y$
be either the blowup of $\mathbb{P}^n(\mathbb{R})$ along a linear
subspace or a nonsingular hypersurface of $\mathbb{P}^m(\mathbb{R})
\times \mathbb{P}^n(\mathbb{R})$ of bidegree $(1,1)$. It is proved
that a $\mathcal{C}^\infty$ map $f \colon X \rightarrow Y$ can be
approximated by regular maps if and only if $f^* \bigl( H^1(Y,
\mathbb{Z}/2) \bigr) \subseteq H^1_{\alg} (X,\mathbb{Z}/2)$, where
$H^1_{\alg} (X,\mathbb{Z}/2)$ is the subgroup of $H^1 (X,
\mathbb{Z}/2)$ generated by the cohomology classes of algebraic
hypersurfaces in $X$. This follows from another result on maps
into generalized flag varieties.
Categories:14P05, 14P25 

112. CMB 1999 (vol 42 pp. 307)
 Kapovich, Michael; Millson, John J.

On the Moduli Space of a Spherical Polygonal Linkage
We give a ``wallcrossing'' formula for computing the topology of
the moduli space of a closed $n$gon linkage on $\mathbb{S}^2$.
We do this by determining the Morse theory of the function
$\rho_n$ on the moduli space of $n$gon linkages which is given by
the length of the last sidethe length of the last side is
allowed to vary, the first $(n  1)$ sidelengths are fixed. We
obtain a Morse function on the $(n  2)$torus with level sets
moduli spaces of $n$gon linkages. The critical points of $\rho_n$
are the linkages which are contained in a great circle. We give a
formula for the signature of the Hessian of $\rho_n$ at such a
linkage in terms of the number of backtracks and the winding
number. We use our formula to determine the moduli spaces of all
regular pentagonal spherical linkages.
Categories:14D20, 14P05 

113. CMB 1999 (vol 42 pp. 354)
 Marshall, Murray A.

A Real Holomorphy Ring without the SchmÃ¼dgen Property
A preordering $T$ is constructed in the polynomial ring $A = \R
[t_1,t_2, \dots]$ (countably many variables) with the following two
properties: (1)~~For each $f \in A$ there exists an integer $N$
such that $N \le f(P) \le N$ holds for all $P \in \Sper_T(A)$.
(2)~~For all $f \in A$, if $N+f, Nf \in T$ for some integer $N$,
then $f \in \R$. This is in sharp contrast with the
Schm\"udgenW\"ormann result that for any preordering $T$ in a
finitely generated $\R$algebra $A$, if property~(1) holds, then
for any $f \in A$, $f > 0$ on $\Sper_T(A) \Rightarrow f \in T$.
Also, adjoining to $A$ the square roots of the generators of $T$
yields a larger ring $C$ with these same two properties but with
$\Sigma C^2$ (the set of sums of squares) as the preordering.
Categories:12D15, 14P10, 44A60 

114. CMB 1999 (vol 42 pp. 263)
 Choie, Youngju; Lee, Min Ho

Mellin Transforms of Mixed Cusp Forms
We define generalized Mellin transforms of mixed cusp forms, show
their convergence, and prove that the function obtained by such a
Mellin transform of a mixed cusp form satisfies a certain
functional equation. We also prove that a mixed cusp form can be
identified with a holomorphic form of the highest degree on an
elliptic variety.
Categories:11F12, 11F66, 11M06, 14K05 

115. CMB 1999 (vol 42 pp. 209)
116. CMB 1999 (vol 42 pp. 78)
 González, Josep

Fermat Jacobians of Prime Degree over Finite Fields
We study the splitting of Fermat Jacobians of prime
degree $\ell$ over an algebraic closure of a finite field of
characteristic $p$ not equal to $\ell$. We prove that their
decomposition is determined by the residue degree of $p$ in the
cyclotomic field of the $\ell$th roots of unity. We provide a
numerical criterion that allows to compute the absolutely simple
subvarieties and their multiplicity in the Fermat Jacobian.
Categories:11G20, 14H40 

117. 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 

118. CMB 1998 (vol 41 pp. 267)
 Fukuma, Yoshiaki

On the nonemptiness of the adjoint linear system of polarized manifold
Let $(X,L)$ be a polarized manifold over the complex number field
with $\dim X=n$. In this paper, we consider a conjecture of
M.~C.~Beltrametti and A.~J.~Sommese and we obtain that this
conjecture is true if $n=3$ and $h^{0}(L)\geq 2$, or $\dim \Bs
L\leq 0$ for any $n\geq 3$. Moreover we can generalize the
result of Sommese.
Keywords:Polarized manifold, adjoint bundle Categories:14C20, 14J99 

119. CMB 1997 (vol 40 pp. 456)
 Kucharz, Wojciech; Rusek, Kamil

Approximation of smooth maps by real algebraic morphisms
Let $\Bbb G_{p,q}(\Bbb F)$ be the Grassmann space of all
$q$dimensional $\Bbb F$vector subspaces of $\Bbb F^{p}$, where $\Bbb F$
stands for $\Bbb R$, $\Bbb C$ or $\Bbb H$ (the quaternions). Here
$\Bbb G_{p,q}(\Bbb F)$ is regarded as a real algebraic variety. The paper
investigates which ${\cal C}^\infty$ maps from a nonsingular real algebraic
variety $X$ into $\Bbb G_{p,q}(\Bbb F)$ can be approximated, in the
${\cal C}^\infty$ compactopen topology, by real algebraic morphisms.
Categories:14P05, 14P25 

120. CMB 1997 (vol 40 pp. 352)
 Liriano, Sal

A New Proof of a Theorem of Magnus
Using naive algebraic geometric methods a new proof of the
following celebrated theorem of Magnus is given:
Let $G$ be a group with a presentation having $n$ generators and $m$
relations. If $G$ also has a presentation on $nm$ generators, then
$G$ is free of rank $nm$.
Categories:20E05, 20C99, 14Q99 
