A proof of A. Gabrielov’s rank theorem
[Théorème du rang de Gabrielov]
Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1329-1396.

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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DOI : https://doi.org/10.5802/jep.173
Classification : 13J05,  32B05,  12J10,  13A18,  13B35,  14B05,  14B20,  30C10,  32A22,  32S45
Mots clés : Géométrie analytique locale, séries formelles, théorème de préparation de Weierstrass, rang d’une application analytique, théorème d’Abhyankar-Jung
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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/

[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 | Article | MR 98749 | Zbl 0093.04501

[Abh64] S. S. Abhyankar - Local analytic geometry, Pure and Applied Math., vol. XIV, Academic Press, New York-London, 1964 | MR 175897 | Zbl 0205.50401

[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 | Article | MR 2423454 | Zbl 1152.13018

[AM70] S. S. Abhyankar & T. T. Moh - “A reduction theorem for divergent power series”, J. reine angew. Math. 241 (1970), p. 27-33 | MR 259158 | Zbl 0191.04403

[Art68] M. Artin - “On the solutions of analytic equations”, Invent. Math. 5 (1968), p. 277-291 | Article | Zbl 0172.05301

[AvdP70] S. S. Abhyankar & M. van der Put - “Homomorphisms of analytic local rings”, J. reine angew. Math. 242 (1970), p. 26-60 | MR 260729 | Zbl 0193.00501

[BB19] A. Belotto da Silva & E. Bierstone - “Monomialization of a quasianalytic morphism”, 2019 | arXiv:1907.09502

[Bec77] J. Becker - “Exposé on a conjecture of Tougeron”, Ann. Inst. Fourier (Grenoble) 27 (1977) no. 4, p. 9-27 | Article | Numdam | MR 491700 | Zbl 0337.14002

[BM82] E. Bierstone & P. D. Milman - “Composite differentiable functions”, Ann. of Math. (2) 116 (1982) no. 3, p. 541-558 | Article | MR 678480 | Zbl 0519.58003

[BM87a] E. Bierstone & P. D. Milman - “Relations among analytic functions. I”, Ann. Inst. Fourier (Grenoble) 37 (1987) no. 1, p. 187-239 | Article | Numdam | MR 894566

[BM87b] E. Bierstone & P. D. Milman - “Relations among analytic functions. II”, Ann. Inst. Fourier (Grenoble) 37 (1987) no. 2, p. 49-77 | Article | Numdam | MR 898931

[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 1773706 | Zbl 0970.32008

[BP18] E. Bierstone & A. Parusiński - “Global smoothing of a subanalytic set”, Duke Math. J. 167 (2018) no. 16, p. 3115-3128 | Article | MR 3870082 | Zbl 1411.32007

[BS83] E. Bierstone & G. W. Schwarz - “Continuous linear division and extension of 𝒞 functions”, Duke Math. J. 50 (1983) no. 1, p. 233-271 | Article | MR 700140 | Zbl 0521.32008

[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 | Article | MR 543093 | Zbl 0413.13015

[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 | Article | MR 9603 | Zbl 0060.06908

[Cho58] W. L. Chow - “On the theorem of Bertini for local domains”, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), p. 580-584 | Article | MR 96667 | Zbl 0099.16002

[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 | Article | MR 3895748 | Zbl 1403.13014

[CM82] D. Cerveau & J.-F. Mattei - Formes intégrables holomorphes singulières, Astérisque, vol. 97, Société Mathématique de France, Paris, 1982 | Zbl 0545.32006

[dJP00] T. de Jong & G. Pfister - Local analytic geometry. Basic theory and applications, Advanced Lectures in Math., Friedr. Vieweg & Sohn, Braunschweig, 2000 | Article | Zbl 0959.32011

[EH77] P. M. Eakin & G. A. Harris - “When Φ(f) convergent implies f is convergent”, Math. Ann. 229 (1977) no. 3, p. 201-210 | Article | MR 444651 | Zbl 0355.13010

[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 302930

[Gab73] A. M. Gabrièlov - “Formal relations among analytic functions”, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), p. 1056-1090 | MR 346184

[Gel60] A. O. Gel’fond - Transcendental and algebraic numbers, Dover Publications, Inc., New York, 1960 | Zbl 0090.26103

[Gil69] R. Gilmer - “Integral dependence in power series rings”, J. Algebra 11 (1969), p. 488-502 | Article | MR 234950 | Zbl 0185.09803

[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 | Article | Zbl 0970.14027

[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 | Article | MR 393556 | Zbl 0307.32011

[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 839813 | Zbl 0589.32015

[Izu86] S. Izumi - “Gabrielov’s rank condition is equivalent to an inequality of reduced orders”, Math. Ann. 276 (1986) no. 1, p. 81-89 | Article | MR 863708 | Zbl 0612.32013

[Izu89] S. Izumi - “The rank condition and convergence of formal functions”, Duke Math. J. 59 (1989) no. 1, p. 241-264 | Article | MR 1016886 | Zbl 0688.32008

[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 | Article | Zbl 39.0493.01

[KV04] K. Kiyek & J. L. Vicente - “On the Jung-Abhyankar theorem”, Arch. Math. (Basel) 83 (2004) no. 2, p. 123-134 | Article | MR 2104940 | Zbl 1085.13009

[Lan88] S. Lang - Introduction to Arakelov theory, Springer-Verlag, New York, 1988 | Article | Zbl 0667.14001

[Mah62] K. Mahler - “On some inequalities for polynomials in several variables”, J. London Math. Soc. 37 (1962), p. 341-344 | Article | MR 138593 | Zbl 0105.06301

[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 212575

[Mal77] B. Malgrange - “Frobenius avec singularités. II. Le cas général”, Invent. Math. 39 (1977) no. 1, p. 67-89 | Article | Zbl 0375.32012

[Mat89] H. Matsumura - Commutative ring theory, Cambridge Studies in Advanced Math., vol. 8, Cambridge University Press, Cambridge, 1989 | MR 1011461 | Zbl 0666.13002

[McD95] J. McDonald - “Fiber polytopes and fractional power series”, J. Pure Appl. Algebra 104 (1995) no. 2, p. 213-233 | Article | MR 1360177 | Zbl 0842.52009

[Mil78] P. D. Milman - “Analytic and polynomial homomorphisms of analytic rings”, Math. Ann. 232 (1978) no. 3, p. 247-253 | Article | MR 492359 | Zbl 0357.32005

[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 0334.32012

[Nag62] M. Nagata - Local rings, Interscience Tracts in Pure and Applied Math., vol. 13, Interscience Publishers, New York-London, 1962 | MR 155856 | Zbl 0123.03402

[Osg16] W. F. Osgood - “On functions of several complex variables”, Trans. Amer. Math. Soc. 17 (1916) no. 1, p. 1-8 | Article | MR 1501027

[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 1101459 | Zbl 0769.32003

[Paw90] W. Pawłucki - Points de Nash des ensembles sous-analytiques, Mem. Amer. Math. Soc., vol. 84, no. 425, American Mathematical Society, Providence, RI, 1990 | Article | MR 1042040 | Zbl 0694.32002

[Paw92] W. Pawłucki - “On Gabrielov’s regularity condition for analytic mappings”, Duke Math. J. 65 (1992) no. 2, p. 299-311 | Article | MR 1150588 | Zbl 0773.32009

[PR12] A. Parusiński & G. Rond - “The Abhyankar-Jung theorem”, J. Algebra 365 (2012), p. 29-41 | Article | MR 2928451 | Zbl 1268.13008

[Rem57] R. Remmert - “Holomorphe und meromorphe Abbildungen komplexer Räume”, Math. Ann. 133 (1957), p. 328-370 | Article | MR 92996 | Zbl 0079.10201

[Ron09] G. Rond - “Homomorphisms of local algebras in positive characteristic”, J. Algebra 322 (2009) no. 12, p. 4382-4407 | Article | MR 2558869 | Zbl 1189.13022

[Ron18] G. Rond - “Artin approximation”, J. Singul. 17 (2018), p. 108-192 | Article | MR 3813133 | Zbl 1396.13001

[Tam81] M. Tamm - “Subanalytic sets in the calculus of variation”, Acta Math. 146 (1981) no. 3-4, p. 167-199 | Article | MR 611382 | Zbl 0478.58010

[Tar48] A. Tarski - A decision method for elementary algebra and geometry, RAND Corporation, Santa Monica, Calif., 1948 | Zbl 0035.00602

[Tou72] J.-C. Tougeron - Idéaux de fonctions différentiables, Ergeb. Math. Grenzgeb. (3), vol. 71, Springer-Verlag, Berlin-New York, 1972 | Zbl 0251.58001

[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 | Article | MR 414919 | Zbl 0318.32005

[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 | Article | MR 1051220 | Zbl 0698.13008

[Zar48] O. Zariski - “Analytical irreducibility of normal varieties”, Ann. of Math. (2) 49 (1948), p. 352-361 | Article | MR 24158 | Zbl 0037.22701

[Zar50] O. Zariski - “Sur la normalité analytique des variétés normales”, Ann. Inst. Fourier (Grenoble) 2 (1950), p. 161-164 (1951) | Article | Numdam | Zbl 0044.26601

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