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Revista Matemática Complutense

, Volume 32, Issue 1, pp 255–272 | Cite as

On the \({\mathcal {A}}\)-equivalence of quasi-ordinary parameterizations

  • M. E. HernandesEmail author
  • N. M. P. Panek
Article
  • 30 Downloads

Abstract

We study the analytic equivalence of quasi-ordinary hypersurfaces in \({\mathbb {C}}^{r+1}\) by means of its normalized quasi-ordinary parameterization. In this context, two quasi-ordinary hypersurfaces are analytic equivalent if and only if their normalized quasi-ordinary parameterizations are \({\mathcal {A}}\)-equivalent. We introduce the set \(\Lambda _{H}^{\mathcal {D}}\subset {\mathbb {N}}^{r}\) associated to Kähler r-forms that generalizes an important analytic invariant of plane branches and allows us to identify terms in a normalized quasi-ordinary parameterization that can be eliminable by an element of \({\mathcal {A}}\)-group.

Keywords

Quasi-ordinary hypersurface Kähler r-forms Jacobian ideal 

Mathematics Subject Classification

14B05 (primary) 32S25 (secondary) 

Notes

Acknowledgements

We thank the referee for the observations made that helped us to improve the redaction of this paper and mainly for pointing out an important correction in Definition 2.3. The first author was partially supported by CNPq and the second one by CAPES.

References

  1. 1.
    Abhyankar, S.S.: On the ramification of algebraic functions. Am. J. Math. 77, 575–592 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Assi, A.: The Frobenius vector of a free affine semigroup. J. Algebra Appl. 11(4), 1–10 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bruce, J.W., Kirk, N.P., du Plessis, A.A.: Complete transversals and the classification of singularities. Nonlinearity 10(1), 253–275 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gau, Y.-N.: Embedded topological classification of quasi-ordinary singularities. Mem. Am. Math. Soc. 388, 109–129 (1988)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Gibson, C.G.: Singular Points of Smooth Mappings, Research Notes in Mathematics, vol. 25. Pitman, London (1979)Google Scholar
  6. 6.
    González Pérez, P.D.: Quasi-Ordinary SingularitiesVia Toric Geometry Hypersurface. Thesis University of La Laguna (2000)Google Scholar
  7. 7.
    González Pérez, P.D.: The semigroup of a quasi-ordinary hypersurface. J. Inst. Math. Jussieu 2(3), 383–399 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jung, H.W.E.: Darstellung der Funktionen eines algebraischen Korpers zweier unaghangigen Veranderlichen \(x, y\) in der Umgegung einer Stelle \(x=a, y=b\). J. Reine. Angew. Math. 133, 289–314 (1908)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Hefez, A., Hernandes, M.E.: The analytic classification of plane branches. Bull. London Math. Soc. 43(2), 289–298 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hefez, A., Hernandes, M.E., Rodrigues Hernandes, M.E.: The analytic classification of plane curves with two branches. Mathematische Zeitschrift 279, 509–520 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lipman, J.: Quasi-Ordinary Singularities of Embedded Surfaces. Thesis, Harvard Univ (1965)Google Scholar
  12. 12.
    Lipman, J.: Topological invariants of quasi-ordinary singularities Mem. Am. Math. Soc. 388, 1–107 (1988)zbMATHGoogle Scholar
  13. 13.
    Pol, D.: On the values of logarithmic residues along curves. Ann. Int. Fourier (Grenoble) 68(2), 725–766 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Popescu-Pampu, P.: On the analytical invariance of the semigroup of a quasi-ordinary hypersurface singularity. Duke Math. J. 124(1), 67–104 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Zariski, O.: Characterization of plane algebroid curves whose module of differentials has maximum torsion. Proc. Natl. Acad. Sci. USA 56, 781–786 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zariski, O.: Exceptional singularities of an algebroid surface and their reduction. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 43, 135-146 (1967)Google Scholar
  17. 17.
    Zariski, O.: The Moduli Problem for Plane Branches. University Lecture Series. AMS, Providence (2006)CrossRefzbMATHGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2018

Authors and Affiliations

  1. 1.Universidade Estadual de MaringáMaringáBrazil
  2. 2.Universidade Estadual do Oeste do ParanáFoz do IguaçuBrazil

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