Uniqueness of plane embeddings of special curves
Authors:
Shreeram S. Abhyankar and Avinash Sathaye
Journal:
Proc. Amer. Math. Soc. 124 (1996), 10611069
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 LinZaidenberg 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
(83h:14020)
 [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
(80m:14016)
 [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
(90h:32018), http://dx.doi.org/10.1016/00018708(89)900091
 [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
(58 #16663)
 [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 (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), 66263.
 [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
(89j:57017b), http://dx.doi.org/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
(95a:14032)
 [Ab1]
 S. S. Abhyankar, On the semigroup of a meromorphic curve (Part I), Proceedings of the International (Kyoto) Symposium on Algebraic Geometry (1977), 249414. 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), 100257. MR 90h:32018
 [AbS]
 S. S. Abhyankar and B. Singh, Embeddings of certain curves in the affine plane, Amer. Jour. Math. 100 (1978), 99175. 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), 10481052 = 200204. 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), 66263.
 [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), 409439 and 349351. MR 89j:57017b
 [SaS]
 A. Sathaye and J. Stenerson, Plane Polynomial Curves, Algebraic Geometry and Applications (1994), 121142. 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:
http://dx.doi.org/10.1090/S0002993996032546
PII:
S 00029939(96)032546
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
