Implicit function theorem for locally blow-analytic functions  [ Théorème des fonctions implicites pour les fonctions localement "blow-analytiques" ]
Annales de l'institut Fourier, Tome 51 (2001) no. 4 , p. 1089-1100
doi : 10.5802/aif.1846
URL stable : http://www.numdam.org/item?id=AIF_2001__51_4_1089_0

Classification:  58C05,  58K99,  14Pxx
Mots clés: blow-analytique, arc-analytique
Nous montrons le théorème des fonctions implicites pour les fonctions localement "blow- analytiques" et, comme application de l'utilisation des homéomorphismes "blow- analytiques", nous donnons une méthode très facile pour résoudre les singularités des courbes analytiques.
In this paper we prove the implicit function theorem for locally blow-analytic functions, and as an interesting application of using blow-analytic homeomorphisms, we describe a very easy way to resolve singularities of analytic curves.

### Bibliographie

[1] C. Camacho; P. Sad Invariant varieties through singularities of holomorphic vector fields, Annals of Math., 115 (1982) MR 657239 | Zbl 0503.32007

[1] E. Bierstone; P. D. Milman Arc-analytic functions, Invent. Math., 101 (1990), p. 411 -424 Article  MR 1062969 | Zbl 0723.32005

[2] T. Fukui; S. Koike; T.-C. Kuo Blow-analytic equisingularities, properties, problems and progress, Real analytic and algebraic singularities, Longman (Pitman Research Notes in Mathematics Series) 381 (1997), p. 8 -29 Zbl 0954.26012

[3] T. Fukui; T.-C. Kuo; L. Paunescu Constructing blow-analytic homeomorphisms (Submitted)

[4] H. Hironaka Introduction to real-analytic sets and real-analytic maps, Quaderni dei Gruppi di Ricerca Matematica del Consiglio Nazionale delle Ricerche, Istituto Matematico "L. Tonelli" dell'Università di Pisa (1973)

[5] M. Kobayashi; T.-C. Kuo On Blow-analytic equivalence of embedded curve singularities, Real analytic and algebraic singularities, Longman (Pitman Research Notes in Mathematics Series) 381 (1997), p. 30 -37 Zbl 0899.32002

[6] T.-C. Kuo The modified analytic trivialization of singularities, J. Math. Soc. Japan, 32 (1980), p. 605 -614 Article  MR 589100 | Zbl 0509.58007

[7] T.-C. Kuo On classification of real singularities, Invent. Math., 82 (1985), p. 257 -262 Article  MR 809714 | Zbl 0587.32018

[8] T.-C. Kuo; J. N. Ward A Theorem on almost analytic equisingularity, J. Math. Soc. Japan, 33 (1981), p. 471 -484 Article  MR 620284 | Zbl 0476.58004

[9] K. Kurdyka Ensembles semi-algébriques symétriques par arcs, Math. Ann., 282 (1988), p. 445 -462 Article  MR 967023 | Zbl 0686.14027

[10] K. Kurdyka Points réguliers d'un sous-analytique, Ann. Inst. Fourier, Grenoble, 38 (1988) no. 1, p. 133 -156 Article  Numdam | MR 949002 | Zbl 0619.32007

[11] A. Parusi#X0144;Ski Subanalytic Functions, Trans. of the A.M.S, 344 (1994) no. 2, p. 583 -595 Article  Zbl 0819.32006

[12] A. Parusi#X0144;Ski Lipschitz stratification of subanalytic sets, Ann. École Norm. Sup., tome 27 (1994) no. 6, p. 661 -696 Numdam | MR 1307677 | Zbl 0819.32007

[13] L. Paunescu An example of blow-analytic homeomorphism, Real analytic and algebraic singularities, Longman (Pitman Research Notes in Mathematics Series) 381 (1997), p. 62 -63 Zbl 0896.58012