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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Topological classification of irreducible plane curve singularities in terms of Weierstrass polynomials
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by Chunghyuk Kang PDF
Proc. Amer. Math. Soc. 123 (1995), 1363-1371 Request permission

Abstract:

Let $f(z,y)$ be analytically irreducible at 0 and $f(0) = 0$. Then the plane curve singularity defined by f has the same topological type as the curve defined by ${f_{k + 1}}$ for some $k \geq 0$ where ${f_1} = {z^a} + {y^b},{f_2} = {f_1}^{{n_{21}}} + {y^{{m_{11}}}}{z^{{m_{12}}}},{f_3} = {f_2}^{{n_{31}}} + {f_1}^{{n_{22}}}{y^{{m_{21}}}}{z^{{m_{22}}}}, \ldots$ are defined by induction on k with distinct numerical conditions topologically invariant. Moreover, we give an easy alternate proof of Zariski’s topological classification theorem of irreducible plane curve singularities.
References
  • Oscar Zariski, Studies in equisingularity. I. Equivalent singularities of plane algebroid curves, Amer. J. Math. 87 (1965), 507–536. MR 177985, DOI 10.2307/2373019
  • Oscar Zariski, Studies in equisingularity. III. Saturation of local rings and equisingularity, Amer. J. Math. 90 (1968), 961–1023. MR 237493, DOI 10.2307/2373492
  • Henry B. Laufer, Normal two-dimensional singularities, Annals of Mathematics Studies, No. 71, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. MR 0320365
  • Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. MR 0434929
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 123 (1995), 1363-1371
  • MSC: Primary 14H20; Secondary 14B05
  • DOI: https://doi.org/10.1090/S0002-9939-1995-1291777-2
  • MathSciNet review: 1291777