Embeddings of Curves in the Plane

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 polynomials by classifying (up to an automorphism of K[x, y]) polynomials of the form axn + bym + ∑im + jn ≤ mncijxiyj, a, b, cij K (i.e., 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 ≥ 2, there is an irreducible curve with one place at infinity which has at least k equivalent embeddings in C2. 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 {p(x, y) = 0} is an irreducible simply connected curve.