Hironaka's concept of a characteristic polyhedron of a singularity has been one of the most powerful and fruitful ideas of the last decades in singularity theory. In fact, since then, combinatorics have become a major tool in many important results. However, this seminal concept is still not enough to cope with some effective problems: for instance, giving a bound on the maximum number of blowing-ups to be performed on a surface before its multiplicity decreases. This short Note shows why such a bounding is not possible, with the original definitions.
Le concept, introduit par Hironaka, du polyèdre caractéristique d'une singularité a été une des idées les plus puissantes et profitables des dernières décennies dans la théorie des singularités. En fait, depuis son apparition les combinatoires sont devenues un outil central pour plusieurs résultats importants dans ce domaine. Pourtant, ce concept séminal n'est pas encore suffisant pour gérer quelques problèmes effectifs : par exemple, trouver une borne supérieure pour le nombre d'éclatements qu'on peut appliquer à une surface sans faire descendre sa multiplicité. Dans cette brève Note on montre pourquoi l'obtention d'une telle borne n'est pas possible, au moins avec les définitions originales.
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Ramon Piedra 1; José M. Tornero 1
@article{CRMATH_2007__344_5_309_0, author = {Ramon Piedra and Jos\'e M. Tornero}, title = {Hironaka's characteristic polygon and effective resolution of surfaces}, journal = {Comptes Rendus. Math\'ematique}, pages = {309--312}, publisher = {Elsevier}, volume = {344}, number = {5}, year = {2007}, doi = {10.1016/j.crma.2007.01.021}, language = {en}, }
Ramon Piedra; José M. Tornero. Hironaka's characteristic polygon and effective resolution of surfaces. Comptes Rendus. Mathématique, Volume 344 (2007) no. 5, pp. 309-312. doi : 10.1016/j.crma.2007.01.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.01.021/
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