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

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

MathSciNet review:
1317027

Full-text PDF

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]**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.**[NeR]**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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
13B10,
13B25,
14C40,
14H20

Retrieve articles in all journals with MSC (1991): 13B10, 13B25, 14C40, 14H20

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