Cet article contient une preuve complète du théorème du rang de Gabrielov, un résultat fondamental en géométrie analytique locale. Nous appuyant sur les travaux de Gabrielov et Tougeron, nous développons des techniques de géométrie formelle qui clarifient les parties difficiles de la preuve originale. Ces techniques ont un intérêt intrinsèque, comme l’illustre par exemple une nouvelle preuve très courte du théorème d’Abhyankar-Jung présentée ici. Nous donnons aussi de nouvelles extensions du théorème du rang en algèbre commutative (liées au théorème principal de Zariski et à la théorie de l’élimination).
This article contains a complete proof of Gabrielov’s rank theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we develop formal-geometric techniques which clarify the difficult parts of the original proof. These techniques are of independent interest, and we illustrate this by adding a new (very short) proof of the Abhyankar-Jung theorem. We include, furthermore, new extensions of the rank theorem (concerning the Zariski main theorem and elimination theory) to commutative algebra.
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Keywords: Local analytic geometry, formal power series, Weierstrass preparation theorem, rank of an analytic map, Abhyankar-Jung’s theorem
Mot clés : Géométrie analytique locale, séries formelles, théorème de préparation de Weierstrass, rang d’une application analytique, théorème d’Abhyankar-Jung
@article{JEP_2021__8__1329_0, author = {Belotto da Silva, Andr\'e and Curmi, Octave and Rond, Guillaume}, title = {A proof of {A.} {Gabrielov{\textquoteright}s} rank theorem}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {1329--1396}, publisher = {Ecole polytechnique}, volume = {8}, year = {2021}, doi = {10.5802/jep.173}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jep.173/} }
TY - JOUR AU - Belotto da Silva, André AU - Curmi, Octave AU - Rond, Guillaume TI - A proof of A. Gabrielov’s rank theorem JO - Journal de l’École polytechnique — Mathématiques PY - 2021 SP - 1329 EP - 1396 VL - 8 PB - Ecole polytechnique UR - http://www.numdam.org/articles/10.5802/jep.173/ DO - 10.5802/jep.173 LA - en ID - JEP_2021__8__1329_0 ER -
%0 Journal Article %A Belotto da Silva, André %A Curmi, Octave %A Rond, Guillaume %T A proof of A. Gabrielov’s rank theorem %J Journal de l’École polytechnique — Mathématiques %D 2021 %P 1329-1396 %V 8 %I Ecole polytechnique %U http://www.numdam.org/articles/10.5802/jep.173/ %R 10.5802/jep.173 %G en %F JEP_2021__8__1329_0
Belotto da Silva, André; Curmi, Octave; Rond, Guillaume. A proof of A. Gabrielov’s rank theorem. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1329-1396. doi : 10.5802/jep.173. http://www.numdam.org/articles/10.5802/jep.173/
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