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.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Notes
In fact, they consider the monoid \(\frac{1}{n}\Gamma _H\subset \mathbb {Q}^r\).
References
Abhyankar, S.S.: On the ramification of algebraic functions. Am. J. Math. 77, 575–592 (1955)
Assi, A.: The Frobenius vector of a free affine semigroup. J. Algebra Appl. 11(4), 1–10 (2012)
Bruce, J.W., Kirk, N.P., du Plessis, A.A.: Complete transversals and the classification of singularities. Nonlinearity 10(1), 253–275 (1997)
Gau, Y.-N.: Embedded topological classification of quasi-ordinary singularities. Mem. Am. Math. Soc. 388, 109–129 (1988)
Gibson, C.G.: Singular Points of Smooth Mappings, Research Notes in Mathematics, vol. 25. Pitman, London (1979)
González Pérez, P.D.: Quasi-Ordinary SingularitiesVia Toric Geometry Hypersurface. Thesis University of La Laguna (2000)
González Pérez, P.D.: The semigroup of a quasi-ordinary hypersurface. J. Inst. Math. Jussieu 2(3), 383–399 (2003)
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)
Hefez, A., Hernandes, M.E.: The analytic classification of plane branches. Bull. London Math. Soc. 43(2), 289–298 (2011)
Hefez, A., Hernandes, M.E., Rodrigues Hernandes, M.E.: The analytic classification of plane curves with two branches. Mathematische Zeitschrift 279, 509–520 (2015)
Lipman, J.: Quasi-Ordinary Singularities of Embedded Surfaces. Thesis, Harvard Univ (1965)
Lipman, J.: Topological invariants of quasi-ordinary singularities Mem. Am. Math. Soc. 388, 1–107 (1988)
Pol, D.: On the values of logarithmic residues along curves. Ann. Int. Fourier (Grenoble) 68(2), 725–766 (2018)
Popescu-Pampu, P.: On the analytical invariance of the semigroup of a quasi-ordinary hypersurface singularity. Duke Math. J. 124(1), 67–104 (2004)
Zariski, O.: Characterization of plane algebroid curves whose module of differentials has maximum torsion. Proc. Natl. Acad. Sci. USA 56, 781–786 (1966)
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)
Zariski, O.: The Moduli Problem for Plane Branches. University Lecture Series. AMS, Providence (2006)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hernandes, M.E., Panek, N.M.P. On the \({\mathcal {A}}\)-equivalence of quasi-ordinary parameterizations. Rev Mat Complut 32, 255–272 (2019). https://doi.org/10.1007/s13163-018-0276-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13163-018-0276-3