Embeddings of curves in the plane

Vladimir Shpilrain; Jie-Tai Yu

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arXiv · math/9809049

Embeddings of curves in the plane

Vladimir Shpilrain·Jie-Tai Yu

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Abstract

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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Vladimir Shpilrain, Jie-Tai Yu. 1998-09-10. Embeddings of curves in the plane. https://arxiv.org/abs/math/9809049

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