Acta Mathematica Vietnamica

, Volume 42, Issue 2, pp 237–247 | Cite as

On Innermost Circles of the Sets of Singular Values for Generic Deformations of Isolated Singularities

  • Kazumasa Inaba
  • Masaharu IshikawaEmail author
  • Masayuki Kawashima
  • Nguyen Tat Thang


We will show that for each k≠1, there exists an isolated singularity of a real analytic map from \(\mathbb {R}^{4}\) to \(\mathbb {R}^{2}\) which admits a real analytic deformation such that the set of singular values of the deformed map has a simple, innermost component with k outward cusps and no inward cusps. Conversely, such a singularity does not exist if k=1.


Stable map Excellent map Critical value Higher differential Mixed polynomial 

Mathematics Subject Classification (2010)

Primary 57R45 Secondary 58C27 14B05 



The second author is partially supported by the Grant-in-Aid for Scientific Research (C), JSPS KAKENHI Grant Number 25400078 and the Kawai Fund for Mathematical Sciences.

The fourth author is partially supported by the Vietnam institute for Advance study in Mathematics (VIASM).

The second and fourth authors are partially supported by the JSPS Postdoctoral Fellowship for Foreign Researchers’ Grant-in-Aid 25/03014.


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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2017

Authors and Affiliations

  • Kazumasa Inaba
    • 1
  • Masaharu Ishikawa
    • 1
    Email author
  • Masayuki Kawashima
    • 2
  • Nguyen Tat Thang
    • 3
  1. 1.Mathematical InstituteTohoku UniversitySendaiJapan
  2. 2.Department of Information ScienceOkayama University of ScienceKitakuJapan
  3. 3.Institute of Mathematics, Vietnam Academy of Science and TechnologyCau GiayVietnam

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