Embeddings of curves in the plane

Details Embeddings of curves in the plane Embeddings of curves in the plane

1998-09-10

arxiv.org/abs/math/9809049v1

Mathematics

Algebraic Geometry

11 pages

Scientific paper

In this paper, we contribute toward a classification of two-variable polynomials by classifying (up to an automorphism of $C^2$) polynomials whose Newton 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 $k \ge 2$, there is an irreducible curve with one place at infinity, which has at least $k$ inequivalent embeddings in $C^2$. Also, upon combining our method with a well-known theorem of Zaidenberg and Lin, we show that one can decide "almost" just by inspection whether or not a polynomial fiber is an irreducible simply connected curve.

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