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101. 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 $N+f, N-f \in T$ 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

102. 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

103. 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

104. 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

105. 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 PassCategories:14A25, 14E09

106. 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 bundleCategories:14C20, 14J99

107. 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

108. 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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