Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Puiseux parametric equations of analytic sets


Author: Fuensanta Aroca
Journal: Proc. Amer. Math. Soc. 132 (2004), 3035-3045
MSC (2000): Primary 32S05, 32B10; Secondary 14M25
Published electronically: June 2, 2004
MathSciNet review: 2063125
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the existence of local Puiseux-type parameterizations of complex analytic sets via Laurent series convergent on wedges. We describe the wedges in terms of the Newton polyhedron of a function vanishing on the discriminant locus of a projection. The existence of a local parameterization of quasi-ordinary singularities of complex analytic sets of any codimension will come as a consequence of our main result.


References [Enhancements On Off] (What's this?)

  • 1. Shreeram Abhyankar, On the ramification of algebraic functions, Amer. J. Math. 77 (1955), 575–592. MR 0071851
  • 2. Shreeram Shankar Abhyankar, Local analytic geometry, Pure and Applied Mathematics, Vol. XIV, Academic Press, New York-London, 1964. MR 0175897
  • 3. F. Aroca, Métodos algebraicos en ecuaciones diferenciales ordinarias en el campo complejo, Tesis Doctoral, Universidad de Valladolid, 2000.
  • 4. F. Aroca and J. Cano, Formal solutions of linear PDEs and convex polyhedra, J. Symbolic Comput. 32 (2001), no. 6, 717–737. Effective methods in rings of differential operators. MR 1866713, 10.1006/jsco.2001.0492
  • 5. José M. Aroca, Heisuke Hironaka, and José L. Vicente, Desingularization theorems, Memorias de Matemática del Instituto “Jorge Juan” [Mathematical Memoirs of the Jorge Juan Institute], vol. 30, Consejo Superior de Investigaciones Científicas, Madrid, 1977. MR 480502
  • 6. Alexander D. Bruno, Local methods in nonlinear differential equations, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1989. Part I. The local method of nonlinear analysis of differential equations. Part II. The sets of analyticity of a normalizing transformation; Translated from the Russian by William Hovingh and Courtney S. Coleman; With an introduction by Stephen Wiggins. MR 993771
  • 7. E. M. Chirka, Complex analytic sets, Mathematics and its Applications (Soviet Series), vol. 46, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by R. A. M. Hoksbergen. MR 1111477
  • 8. William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037
  • 9. I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417
  • 10. P. D. González Pérez, Singularités quasi-ordinaires toriques et polyèdre de Newton du discriminant, Canad. J. Math. 52 (2000), no. 2, 348–368 (French, with French summary). MR 1755782, 10.4153/CJM-2000-016-8
  • 11. Heisuke Hironaka, Introduction to the theory of infinitely near singular points, Consejo Superior de Investigaciones Científicas, Madrid, 1974. Memorias de Matematica del Instituto “Jorge Juan”, No. 28. MR 0399505
  • 12. Henry B. Laufer, Normal two-dimensional singularities, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. Annals of Mathematics Studies, No. 71. MR 0320365
  • 13. Ignacio Luengo, A new proof of the Jung-Abhyankar theorem, J. Algebra 85 (1983), no. 2, 399–409. MR 725092, 10.1016/0021-8693(83)90104-7
  • 14. John McDonald, Fiber polytopes and fractional power series, J. Pure Appl. Algebra 104 (1995), no. 2, 213–233. MR 1360177, 10.1016/0022-4049(94)00129-5
  • 15. William S. Massey, A basic course in algebraic topology, Graduate Texts in Mathematics, vol. 127, Springer-Verlag, New York, 1991. MR 1095046
  • 16. Mutsumi Saito, Bernd Sturmfels, and Nobuki Takayama, Gröbner deformations of hypergeometric differential equations, Algorithms and Computation in Mathematics, vol. 6, Springer-Verlag, Berlin, 2000. MR 1734566
  • 17. Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258
  • 18. R. J. Walker, Reduction of singularities of an algebraic surface, Ann. Math. (2) 36 (1935), 336-365.
  • 19. M.-Angeles Zurro, The Abhyankar-Jung theorem revisited, J. Pure Appl. Algebra 90 (1993), no. 3, 275–282. MR 1255715, 10.1016/0022-4049(93)90045-U

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32S05, 32B10, 14M25

Retrieve articles in all journals with MSC (2000): 32S05, 32B10, 14M25


Additional Information

Fuensanta Aroca
Affiliation: Instituto de Matematicas UNAM (Unidad Cuernavaca), Apartado Postal 273-3, Administración de Correos 3, CP 62251, Cuernavaca, Morelos, Mexico
Address at time of publication: Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, 13560-970 São Carlos SP, Brazil
Email: fuen@matcuer.unam.mx, fuen@icmc.usp.br

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07337-X
Keywords: Parameterization, wedges, quasi-ordinary singularities
Received by editor(s): February 6, 2002
Received by editor(s) in revised form: May 19, 2003
Published electronically: June 2, 2004
Additional Notes: The author was supported first by Post-doctoral Grant of TMR Project Singularidades de Ecuaciones Diferenciales y Foliaciones at the University of Lisbon, and then by UNAM at Instituto de Matemáticas-Cuernavaca (Mexico)
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2004 American Mathematical Society