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101. CMB 2001 (vol 44 pp. 223)

Marshall, M.
Extending the Archimedean Positivstellensatz to the Non-Compact Case
A generalization of Schm\"udgen's Positivstellensatz is given which holds for any basic closed semialgebraic set in $\mathbb{R}^n$ (compact or not). The proof is an extension of W\"ormann's proof.

Categories:12D15, 14P10, 44A60

102. 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 best-possible.

Categories:13C15, 13B25, 04A10, 14A05, 13M05

103. CMB 2000 (vol 43 pp. 304)

Darmon, Henri; Mestre, Jean-François
Courbes hyperelliptiques à multiplications réelles 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 au-dessus 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

104. 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

105. 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

106. CMB 2000 (vol 43 pp. 174)

Gantz, Christian; Steer, Brian
Stable Parabolic Bundles over Elliptic Surfaces and over Riemann Surfaces
We show that the use of orbifold bundles enables some questions to be reduced to the case of flat bundles. The identification of moduli spaces of certain parabolic bundles over elliptic surfaces is achieved using this method.

Categories:14J27, 32L07, 14H60, 14D20

107. 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 (non-emptyness, 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. $\card \bigl( V(E) \bigr) = 2$).


108. 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 $A-D-E$ singularities. We give two new characterizations of simple isolated line singularities.

Categories:32S25, 14B05

109. 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

110. CMB 1999 (vol 42 pp. 307)

Kapovich, Michael; Millson, John J.
On the Moduli Space of a Spherical Polygonal Linkage
We give a ``wall-crossing'' 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 side---the length of the last side is allowed to vary, the first $(n - 1)$ side-lengths 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 back-tracks and the winding number. We use our formula to determine the moduli spaces of all regular pentagonal spherical linkages.

Categories:14D20, 14P05

111. CMB 1999 (vol 42 pp. 354)

Marshall, Murray A.
A Real Holomorphy Ring without the Schmüdgen 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 $N+f, N-f \in T$ for some integer $N$, then $f \in \R$. This is in sharp contrast with the Schm\"udgen-W\"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

112. 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

113. CMB 1999 (vol 42 pp. 209)

Lanteri, Antonio; Maeda, Hidetoshi
Ample Vector Bundles of Curve Genus One
We investigate the pairs $(X,\cE)$ consisting of a smooth complex projective variety $X$ of dimension $n$ and an ample vector bundle $\cE$ of rank $n-1$ on $X$ such that $\cE$ has a section whose zero locus is a smooth elliptic curve.

Categories:14J60, 14F05, 14J40

114. 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

115. 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 Reduction-of-Degree}) {\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 Ott-Heinrich 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

116. 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

117. 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 ${\cal C}^\infty$ compact-open topology, by real algebraic morphisms.

Categories:14P05, 14P25

118. 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 $n-m$ generators, then $G$ is free of rank $n-m$.

Categories:20E05, 20C99, 14Q99
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