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Embeddings of Curves in the Plane

✍ Scribed by Vladimir Shpilrain; Jie-Tai Yu

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 88 KB
Volume: 217
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.1998.7811

No coin nor oath required. For personal study only.

✦ Synopsis

w

x Let K x, y be the polynomial algebra in two variables over a field K of characteristic 0. In this paper, we contribute toward a classification of two-variable Ž w x. polynomials by classifying up to an automorphism of K x, y polynomials of the

e., polynomials whose New-

. ton polygon is either a triangle or a line segment . Our classification has several applications to the study of embeddings of algebraic curves in the plane. In particular, we show that for any k G 2, there is an irreducible curve with one place at infinity which has at least k equivalent embeddings in C 2 . Also, upon combining our method with a well-known theorem of Zaidenberg and Lin, we show that one Ä Ž . can decide ''almost'' just by inspection whether or not a polynomial fiber p x, y 4 s 0 is an irreducible simply connected curve.

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