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1. CJM 2007 (vol 59 pp. 1069)

Reydy, Carine
 Quotients jacobiens : une approche algÃ©brique Le diagramme d'Eisenbud et Neumann d'un germe est un arbre qui repr\'esente ce germe et permet d'en calculer les invariants. On donne une d\'emonstration alg\'ebrique d'un r\'esultat caract\'erisant l'ensemble des quotients jacobiens d'un germe d'application $(f,g)$ \a partir du diagramme d'Eisenbud et Neumann de $fg$. Keywords:SingularitÃ©, jacobien, quotient jacobien, polygone de NewtonCategories:14B05, 32S05, 32S50

2. CJM 2005 (vol 57 pp. 844)

Williams, Gordon
 Petrie Schemes Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Gr\"unbaum--Dress polyhedra, possess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes. Keywords:Petrie polygon, polyhedron, polytope, abstract polytope, incidence complex, regular polytope, Coxeter groupCategories:52B15, 52B05

3. CJM 2003 (vol 55 pp. 533)

Edo, Eric
 Automorphismes modÃ©rÃ©s de l'espace affine Le probl\eme de Jung-Nagata ({\it cf.}\ [J], [N]) consiste \`a savoir s'il existe des automorphismes de $k[x,y,z]$ qui ne sont pas mod\'er\'es. Nous proposons une approche nouvelle de cette question, fond\'ee sur l'utilisation de la th\'eorie des automates et du polygone de Newton. Cette approche permet notamment de g\'en\'eraliser de fa\c con significative les r\'esultats de [A]. The Jung-Nagata's problem ({\it cf.}\ [J], [N]) asks if there exists non-tame (or wild) automorphisms of $k[x,y,z]$. We give a new way to attack this question, based on the automata theory and the Newton polygon. This new approch allows us to generalize significantly the results of [A]. Keywords:tame automorphisms, automata, Newton polygonCategory:14R10

4. CJM 1998 (vol 50 pp. 581)

Kamiyama, Yasuhiko
 The homology of singular polygon spaces Let $M_n$ be the variety of spatial polygons $P= (a_1, a_2, \dots, a_n)$ whose sides are vectors $a_i \in \text{\bf R}^3$ of length $\vert a_i \vert=1 \; (1 \leq i \leq n),$ up to motion in $\text{\bf R}^3.$ It is known that for odd $n$, $M_n$ is a smooth manifold, while for even $n$, $M_n$ has cone-like singular points. For odd $n$, the rational homology of $M_n$ was determined by Kirwan and Klyachko [6], [9]. The purpose of this paper is to determine the rational homology of $M_n$ for even $n$. For even $n$, let ${\tilde M}_n$ be the manifold obtained from $M_n$ by the resolution of the singularities. Then we also determine the integral homology of ${\tilde M}_n$. Keywords:singular polygon space, homologyCategories:14D20, 57N65

5. CJM 1997 (vol 49 pp. 1162)

Ku, Hsu-Tung; Ku, Mei-Chin; Zhang, Xin-Min
 Isoperimetric inequalities on surfaces of constant curvature In this paper we introduce the concepts of hyperbolic and elliptic areas and prove uncountably many new geometric isoperimetric inequalities on the surfaces of constant curvature. Keywords:Gaussian curvature, Gauss-Bonnet theorem, polygon, pseudo-polygon, pseudo-perimeter, hyperbolic surface, Heron's formula, analytic and geometric isoperimetric inequalitiesCategories:51M10, 51M25, 52A40, 53C20