A proof of A. Gabrielov’s rank theorem

André Belotto da Silva; Octave Curmi; Guillaume Rond

doi:10.5802/jep.173

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

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

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 :

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {A proof of {A.} {Gabrielov{\textquoteright}s} rank theorem},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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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, Tome 8 (2021), pp. 1329-1396. doi : 10.5802/jep.173. https://jep.centre-mersenne.org/articles/10.5802/jep.173/

Bibliographie
Cité par

[Abh58] S. S. Abhyankar - “On the ramification of algebraic functions. II. Unaffected equations for characteristic two”, Trans. Amer. Math. Soc. 89 (1958), p. 310-324 | DOI | MR | Zbl

[Abh64] S. S. Abhyankar - Local analytic geometry, Pure and Applied Math., vol. XIV, Academic Press, New York-London, 1964 | MR | Zbl

[ABM08] J. Adamus, E. Bierstone & P. D. Milman - “Uniform linear bound in Chevalley’s lemma”, Canad. J. Math. 60 (2008) no. 4, p. 721-733 | DOI | MR | Zbl

[AM70] S. S. Abhyankar & T. T. Moh - “A reduction theorem for divergent power series”, J. reine angew. Math. 241 (1970), p. 27-33 | MR | Zbl

[Art68] M. Artin - “On the solutions of analytic equations”, Invent. Math. 5 (1968), p. 277-291 | DOI | Zbl

[AvdP70] S. S. Abhyankar & M. van der Put - “Homomorphisms of analytic local rings”, J. reine angew. Math. 242 (1970), p. 26-60 | MR | Zbl

[BB19] A. Belotto da Silva & E. Bierstone - “Monomialization of a quasianalytic morphism”, 2019 | arXiv

[Bec77] J. Becker - “Exposé on a conjecture of Tougeron”, Ann. Inst. Fourier (Grenoble) 27 (1977) no. 4, p. 9-27 | DOI | Numdam | MR | Zbl

[BM82] E. Bierstone & P. D. Milman - “Composite differentiable functions”, Ann. of Math. (2) 116 (1982) no. 3, p. 541-558 | DOI | MR | Zbl

[BM87a] E. Bierstone & P. D. Milman - “Relations among analytic functions. I”, Ann. Inst. Fourier (Grenoble) 37 (1987) no. 1, p. 187-239 | DOI | Numdam | MR

[BM87b] E. Bierstone & P. D. Milman - “Relations among analytic functions. II”, Ann. Inst. Fourier (Grenoble) 37 (1987) no. 2, p. 49-77 | DOI | Numdam | MR

[BM00] E. Bierstone & P. D. Milman - “Subanalytic geometry”, in Model theory, algebra, and geometry, Math. Sci. Res. Inst. Publ., vol. 39, Cambridge Univ. Press, Cambridge, 2000, p. 151-172 | MR | Zbl

[BP18] E. Bierstone & A. Parusiński - “Global smoothing of a subanalytic set”, Duke Math. J. 167 (2018) no. 16, p. 3115-3128 | DOI | MR | Zbl

[BS83] E. Bierstone & G. W. Schwarz - “Continuous linear division and extension of $𝒞^{\infty}$ functions”, Duke Math. J. 50 (1983) no. 1, p. 233-271 | DOI | MR | Zbl

[BZ79] J. Becker & W. R. Zame - “Applications of functional analysis to the solution of power series equations”, Math. Ann. 243 (1979) no. 1, p. 37-54 | DOI | MR | Zbl

[CCD13] F. Cano, D. Cerveau & J. Déserti - Théorie élémentaire des feuilletages holomorphes singuliers, Collection Échelles, Belin, Paris, 2013

[Che43] C. Chevalley - “On the theory of local rings”, Ann. of Math. (2) 44 (1943), p. 690-708 | DOI | MR | Zbl

[Cho58] W. L. Chow - “On the theorem of Bertini for local domains”, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), p. 580-584 | DOI | MR | Zbl

[CJPR19] F.-J. Castro-Jiménez, D. Popescu & G. Rond - “Linear nested Artin approximation theorem for algebraic power series”, Manuscripta Math. 158 (2019) no. 1-2, p. 55-73 | DOI | MR | Zbl

[CM82] D. Cerveau & J.-F. Mattei - Formes intégrables holomorphes singulières, Astérisque, vol. 97, Société Mathématique de France, Paris, 1982 | Zbl

[dJP00] T. de Jong & G. Pfister - Local analytic geometry. Basic theory and applications, Advanced Lectures in Math., Friedr. Vieweg & Sohn, Braunschweig, 2000 | DOI | Zbl

[EH77] P. M. Eakin & G. A. Harris - “When $Φ (f)$ convergent implies $f$ is convergent”, Math. Ann. 229 (1977) no. 3, p. 201-210 | DOI | MR | Zbl

[Eis52] G. Eisenstein - “Über eine allgemeine Eigenschaft der Reihen-Entwicklungen aller Algebraischen Funktionen”, Bericht Königl. Preuss. Akad. d. Wiss. Zu Berlin (1852), p. 441-443

[Gab71] A. M. Gabrièlov - “The formal relations between analytic functions”, Funkcional. Anal. i Priložen. 5 (1971) no. 4, p. 64-65 | MR

[Gab73] A. M. Gabrièlov - “Formal relations among analytic functions”, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), p. 1056-1090 | MR

[Gel60] A. O. Gel’fond - Transcendental and algebraic numbers, Dover Publications, Inc., New York, 1960 | Zbl

[Gil69] R. Gilmer - “Integral dependence in power series rings”, J. Algebra 11 (1969), p. 488-502 | DOI | MR | Zbl

[GP00] P. D. González Pérez - “Singularités quasi-ordinaires toriques et polyèdre de Newton du discriminant”, Canad. J. Math. 52 (2000) no. 2, p. 348-368 | DOI | Zbl

[Gro61] A. Grothendieck - “Techniques de construction en géométrie analytique VI”, in Familles d’espaces complexes et fondements de la géométrie analytique, Séminaire Henri Cartan, vol. 13 no. 1, Secrétariat mathématique, Paris, 1960/61, Exp. no. 13

[Hir75] H. Hironaka - “Flattening theorem in complex-analytic geometry”, Amer. J. Math. 97 (1975), p. 503-547 | DOI | MR | Zbl

[Hir86] H. Hironaka - “Local analytic dimensions of a subanalytic set”, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986) no. 2, p. 73-75 | MR | Zbl

[Izu86] S. Izumi - “Gabrielov’s rank condition is equivalent to an inequality of reduced orders”, Math. Ann. 276 (1986) no. 1, p. 81-89 | DOI | MR | Zbl

[Izu89] S. Izumi - “The rank condition and convergence of formal functions”, Duke Math. J. 59 (1989) no. 1, p. 241-264 | DOI | MR | Zbl

[Jun08] H. W. E. Jung - “Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Veränderlichen $x, y$ in der Umgebung einer Stelle $x = a, y = b$ ”, J. reine angew. Math. 133 (1908), p. 289-314 | DOI | Zbl

[KV04] K. Kiyek & J. L. Vicente - “On the Jung-Abhyankar theorem”, Arch. Math. (Basel) 83 (2004) no. 2, p. 123-134 | DOI | MR | Zbl

[Lan88] S. Lang - Introduction to Arakelov theory, Springer-Verlag, New York, 1988 | DOI | Zbl

[Mah62] K. Mahler - “On some inequalities for polynomials in several variables”, J. London Math. Soc. 37 (1962), p. 341-344 | DOI | MR | Zbl

[Mal67] B. Malgrange - Ideals of differentiable functions, TIFR Studies in Math., vol. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967 | MR

[Mal77] B. Malgrange - “Frobenius avec singularités. II. Le cas général”, Invent. Math. 39 (1977) no. 1, p. 67-89 | DOI | Zbl

[Mat89] H. Matsumura - Commutative ring theory, Cambridge Studies in Advanced Math., vol. 8, Cambridge University Press, Cambridge, 1989 | MR | Zbl

[McD95] J. McDonald - “Fiber polytopes and fractional power series”, J. Pure Appl. Algebra 104 (1995) no. 2, p. 213-233 | DOI | MR | Zbl

[Mil78] P. D. Milman - “Analytic and polynomial homomorphisms of analytic rings”, Math. Ann. 232 (1978) no. 3, p. 247-253 | DOI | MR | Zbl

[MT76] R. Moussu & J.-C. Tougeron - “Fonctions composées analytiques et différentiables”, C. R. Acad. Sci. Paris Sér. A-B 282 (1976) no. 21, p. A1237-A1240 | Zbl

[Nag62] M. Nagata - Local rings, Interscience Tracts in Pure and Applied Math., vol. 13, Interscience Publishers, New York-London, 1962 | MR | Zbl

[Osg16] W. F. Osgood - “On functions of several complex variables”, Trans. Amer. Math. Soc. 17 (1916) no. 1, p. 1-8 | DOI | MR

[Paw89] W. Pawłucki - “On relations among analytic functions and geometry of subanalytic sets”, Bull. Polish Acad. Sci. Math. 37 (1989) no. 1-6, p. 117-125 (1990) | MR | Zbl

[Paw90] W. Pawłucki - Points de Nash des ensembles sous-analytiques, Mem. Amer. Math. Soc., vol. 84, no. 425, American Mathematical Society, Providence, RI, 1990 | DOI | MR | Zbl

[Paw92] W. Pawłucki - “On Gabrielov’s regularity condition for analytic mappings”, Duke Math. J. 65 (1992) no. 2, p. 299-311 | DOI | MR | Zbl

[PR12] A. Parusiński & G. Rond - “The Abhyankar-Jung theorem”, J. Algebra 365 (2012), p. 29-41 | DOI | MR | Zbl

[Rem57] R. Remmert - “Holomorphe und meromorphe Abbildungen komplexer Räume”, Math. Ann. 133 (1957), p. 328-370 | DOI | MR | Zbl

[Ron09] G. Rond - “Homomorphisms of local algebras in positive characteristic”, J. Algebra 322 (2009) no. 12, p. 4382-4407 | DOI | MR | Zbl

[Ron18] G. Rond - “Artin approximation”, J. Singul. 17 (2018), p. 108-192 | DOI | MR | Zbl

[Tam81] M. Tamm - “Subanalytic sets in the calculus of variation”, Acta Math. 146 (1981) no. 3-4, p. 167-199 | DOI | MR | Zbl

[Tar48] A. Tarski - A decision method for elementary algebra and geometry, RAND Corporation, Santa Monica, Calif., 1948 | Zbl

[Tou72] J.-C. Tougeron - Idéaux de fonctions différentiables, Ergeb. Math. Grenzgeb. (3), vol. 71, Springer-Verlag, Berlin-New York, 1972 | Zbl

[Tou76] J.-C. Tougeron - “Courbes analytiques sur un germe d’espace analytique et applications”, Ann. Inst. Fourier (Grenoble) 26 (1976) no. 2, p. 117-131 | DOI | MR | Zbl

[Tou90] J.-C. Tougeron - “Sur les racines d’un polynôme à coefficients séries formelles”, in Real analytic and algebraic geometry (Trento, 1988), Lect. Notes in Math., vol. 1420, Springer, Berlin, 1990, p. 325-363 | DOI | MR | Zbl

[Zar48] O. Zariski - “Analytical irreducibility of normal varieties”, Ann. of Math. (2) 49 (1948), p. 352-361 | DOI | MR | Zbl

[Zar50] O. Zariski - “Sur la normalité analytique des variétés normales”, Ann. Inst. Fourier (Grenoble) 2 (1950), p. 161-164 (1951) | DOI | Numdam | Zbl

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