A proof of A. Gabrielov’s rank theorem
Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 1329-1396.

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.

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).

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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
André Belotto da Silva 1; Octave Curmi 1; Guillaume Rond 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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André Belotto da Silva; Octave Curmi; Guillaume Rond. A proof of A. Gabrielov’s rank theorem. Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 1329-1396. doi : 10.5802/jep.173. https://jep.centre-mersenne.org/articles/10.5802/jep.173/

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