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Uniqueness of plane embeddings of special curves
Author(s):
Shreeram
S.
Abhyankar;
Avinash
Sathaye
Journal:
Proc. Amer. Math. Soc.
124
(1996),
1061-1069.
MSC (1991):
Primary 13B10, 13B25, 14C40, 14H20
MathSciNet review:
1317027
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Abstract:
For a family of special affine plane curves, it is shown that their embeddings in the affine plane are unique up to automorphisms of the affine plane. Examples are also given for which the embedding is not unique. We also discuss the Lin-Zaidenberg estimate of the number of singular points of an irreducible curve in terms of its rank. Formulas concerning the rank of the curve lead to an alternate simpler version of the proof of the Epimorphism Theorem.
References:
- [Ab1]
- S. S. Abhyankar, On the semigroup of a meromorphic curve (Part I), Proceedings of the International (Kyoto) Symposium on Algebraic Geometry (1977), 249-414. MR 83h:14020
- [Ab2]
- S. S. Abhyankar, Expansion Techniques in Algebraic Geometry, Tata Institute of Fundamental Research, 1977. MR 80m:14016
- [Ab3]
- S. S. Abhyankar, Irreducibility criterion for germs of analytic functions of two complex variables, Advances in Mathematics 74(2) (1989), 100-257. MR 90h:32018
- [AbS]
- S. S. Abhyankar and B. Singh, Embeddings of certain curves in the affine plane, Amer. Jour. Math. 100 (1978), 99-175. MR 58:16663
- [LZ1]
- V. Lin and M. Zaidenberg, An irreducible simply connected algebraic curve in
 is equivalent to a quasihomogeneous curve, Dokl. Akad. Nauk SSSR = Soviet Math Dokl. 271 = 28 (1983), 1048-1052 = 200-204. MR 85i:14018 - [LZ2]
- V. Lin and M. Zaidenberg, On the number of singular points of a plane affine algebraic curve, Springer Lecture Notes in Mathematics 1043 (1984), 662-63.
- [LZ3]
- V. Lin and M. Zaidenberg, On the number of singular points of a plane affine algebraic curve, Springer Lecture Notes in Mathematics 1574 (1994), 479.
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- W. Neumann and L. Rudolph, Unfoldings in knot theory (and Corrigendum), Math. Ann. 278 and 282 (1987 and 1988), 409-439 and 349-351. MR 89j:57017b
- [SaS]
- A. Sathaye and J. Stenerson, Plane Polynomial Curves, Algebraic Geometry and Applications (1994), 121-142. MR 95a:14032
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Additional Information:
Shreeram
S.
Abhyankar
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email:
ram@cs.purdue.edu
Avinash
Sathaye
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email:
sohum@math.uky.edu
DOI:
10.1090/S0002-9939-96-03254-6
PII:
S 0002-9939(96)03254-6
Received by editor(s):
October 24, 1994
Additional Notes:
This work was partly supported by NSF grant DMS 91--01424 and NSA grant MDA 904--95--H--1008.
Communicated by:
Wolmer V. Vasconcelos
Copyright of article:
Copyright
1996,
American Mathematical Society
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