The rational solutions with as denominators powers of to the elliptic diophantine equation are determined. An idea of Yuri Bilu is applied, which avoids Thue and Thue-Mahler equations, and deduces four-term (-) unit equations with special properties, that are solved by linear forms in real and -adic logarithms.
We degenerate Cox–Nagata rings to toric algebras by means of sagbi bases induced by configurations over the rational function field. For del Pezzo surfaces, this degeneration implies the Batyrev–Popov conjecture that these rings are presented by ideals of quadrics. For the blow-up of projective -space at points, sagbi bases of Cox–Nagata rings establish a link between the Verlinde formula and phylogenetic algebraic geometry, and we use this to answer questions due to D’Cruz–Iarrobino and Buczyńska–Wiśniewski....
The aim of this short survey is to give a quick introduction to the Salvetti complex as a tool for the study of the cohomology of Artin groups. In particular we show how a spectral sequence induced by a filtration on the complex provides a very natural and useful method to study recursively the cohomology of Artin groups, simplifying many computations. In the last section some examples of applications are presented.
Nous obtenons une version explicite de la théorie de Bruhat-Tits pour les groupes exceptionnels de type sur un corps local. Nous décrivons chaque construction concrètement en termes de réseaux : l’immeuble, les appartements, la structure simpliciale, les schémas en groupes associés. Les appendices traitent de l’analogie avec les espaces symétriques réels et des espaces symétriques associés à réel et complexe.
Nous obtenons une version explicite de la théorie de Bruhat-Tits pour les groupes exceptionnels des type ou sur un corps local. Nous décrivons chaque construction concrètement en termes de réseaux : l’immeuble, les appartements, la structure simpliciale, les schémas en groupes associés.