Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo


IMPACT FACTOR 2018: 0.726
5-year IMPACT FACTOR: 0.869

CiteScore 2018: 0.90

SCImago Journal Rank (SJR) 2018: 0.323
Source Normalized Impact per Paper (SNIP) 2018: 0.821

Mathematical Citation Quotient (MCQ) 2018: 0.34

ICV 2018: 152.31

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 12, Issue 5

Issues

Volume 13 (2015)

A bound for the Milnor number of plane curve singularities

Arkadiusz Płoski
  • Department of Mathematics, Kielce University of Technology, Al. Tysiąclecia Państwa Polskiego 7, 25-314, Kielce, Poland
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2014-02-15 | DOI: https://doi.org/10.2478/s11533-013-0378-6

Abstract

Let f = 0 be a plane algebraic curve of degree d > 1 with an isolated singular point at 0 ∈ ℂ2. We show that the Milnor number μ0(f) is less than or equal to (d−1)2 − [d/2], unless f = 0 is a set of d concurrent lines passing through 0, and characterize the curves f = 0 for which μ0(f) = (d−1)2 − [d/2].

Keywords: Milnor number; Plane algebraic curve

MSC: 14B05; 14N99

  • [1] Cassou-Noguès P., Płoski A., Invariants of plane curve singularities and Newton diagrams, Uni. Iagel. Acta Math., 2011, 49, 9–34 Google Scholar

  • [2] Fulton W., Algebraic Curves, Adv. Book Classics, Addison-Wesley, Redwood City, 1989 Google Scholar

  • [3] Garcia Barroso E.R., Płoski A., An approach to plane algebroid branches, preprint available at http://arxiv.org/abs/1208.0913 Google Scholar

  • [4] Greuel G.-M., Lossen C., Shustin E., Plane curves of minimal degree with prescribed singularities, Invent. Math., 1998, 133(3), 539–580 http://dx.doi.org/10.1007/s002220050254CrossrefGoogle Scholar

  • [5] Gusein-Zade S.M., Nekhoroshev N.N., Singularities of type A k on plane curves of a chosen degree, Funct. Anal. Appl., 2000, 34(3), 214–215 http://dx.doi.org/10.1007/BF02482412CrossrefGoogle Scholar

  • [6] Gwozdziewicz J., Płoski A., Formulae for the singularities at infinity of plane algebraic curves, Univ. Iagel. Acta Math., 2001, 39, 109–133 Google Scholar

  • [7] Huh J., Milnor numbers of projective hypersurfaces with isolated singularities, preprint available at http://arxiv.org/abs/1210.2690 Web of ScienceGoogle Scholar

  • [8] Teissier B., Resolution Simultanée I, II, In: Séminaire sur les Singularités des Surfaces, Lecture Notes in Math., 777, Springer, Berlin, 1980, 71–146 http://dx.doi.org/10.1007/BFb0085880CrossrefGoogle Scholar

  • [9] Wall C.T.C., Singular Points of Plane Curves, London Math. Soc. Stud. Texts, 63, Cambridge University Press, Cambridge, 2004 693 Google Scholar

About the article

Published Online: 2014-02-15

Published in Print: 2014-05-01


Citation Information: Open Mathematics, Volume 12, Issue 5, Pages 688–693, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-013-0378-6.

Export Citation

© 2014 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in