Uniqueness of plane embeddings of special curves

Authors:
Shreeram S. Abhyankar and 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 | References | Similar Articles | Additional Information

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.

**[Ab1]**Shreeram S. Abhyankar,*On the semigroup of a meromorphic curve. I*, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) Kinokuniya Book Store, Tokyo, 1978, pp. 249–414. MR**578864****[Ab2]**S. S. Abhyankar,*Lectures on expansion techniques in algebraic geometry*, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 57, Tata Institute of Fundamental Research, Bombay, 1977. Notes by Balwant Singh. MR**542446****[Ab3]**Shreeram S. Abhyankar,*Irreducibility criterion for germs of analytic functions of two complex variables*, Adv. Math.**74**(1989), no. 2, 190–257. MR**997097**, 10.1016/0001-8708(89)90009-1**[AbS]**Shreeram S. Abhyankar and Balwant Singh,*Embeddings of certain curves in the affine plane*, Amer. J. Math.**100**(1978), no. 1, 99–175. MR**0498566****[LZ1]**M. G. Zaĭdenberg and V. Ya. Lin,*An irreducible, simply connected algebraic curve in 𝐶² is equivalent to a quasihomogeneous curve*, Dokl. Akad. Nauk SSSR**271**(1983), no. 5, 1048–1052 (Russian). MR**722017****[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.**[NeR]**Walter Neumann and Lee Rudolph,*Corrigendum: “Unfoldings in knot theory”*, Math. Ann.**282**(1988), no. 2, 349–351. MR**963022**, 10.1007/BF01456981**[SaS]**Avinash Sathaye and Jon Stenerson,*Plane polynomial curves*, Algebraic geometry and its applications (West Lafayette, IN, 1990), Springer, New York, 1994, pp. 121–142. MR**1272025**

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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:
https://doi.org/10.1090/S0002-9939-96-03254-6

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

Article copyright:
© Copyright 1996
American Mathematical Society