A proof of A. Gabrielov’s rank theorem

Belotto da Silva, André; Curmi, Octave; Rond, Guillaume

doi:10.5802/jep.173

A proof of A. Gabrielov’s rank theorem
[Théorème du rang de Gabrielov]

Belotto da Silva, André ¹ ; Curmi, Octave ¹ ; Rond, Guillaume ¹

¹ Université Aix-Marseille, Institut de Mathématiques de Marseille (UMR CNRS 7373), Centre de Mathématiques et Informatique 39 rue F. Joliot Curie, 13013 Marseille, France

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1329-1396.

Résumé
Abstract

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.

Reçu le : 2020-09-02
Accepté le : 2021-06-17
Publié le : 2021-07-28

DOI : 10.5802/jep.173

Classification : 13J05, 32B05, 12J10, 13A18, 13B35, 14B05, 14B20, 30C10, 32A22, 32S45
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

Affiliations des auteurs :

Belotto da Silva, André ¹ ; Curmi, Octave ¹ ; Rond, Guillaume ¹

¹ Université Aix-Marseille, Institut de Mathématiques de Marseille (UMR CNRS 7373), Centre de Mathématiques et Informatique 39 rue F. Joliot Curie, 13013 Marseille, France

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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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