Embeddings of curves in the plane

Let K[x,y] be the polynomial algebra in two variables over a field K of cha racteristic 0. In this paper, we contribute toward a classification of two- variable polynomials by classifying (up to an automorphism of K[x, yl) poly nomials of the form ax(n) + by(m) + Sigma(im+jn less than or equal to mn) c (ij)x(i)y(j), a, b, c(ij) is an element of K (i.e., polynomials whose Newto n polygon is either a triangle or a line segment). Our classification has s everal applications to the study of embeddings of algebraic curves in the p lane. In particular, we show that for any k greater than or equal to 2, the re 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 can decide "almost" j ust by inspection whether or not a polynomial fiber {p(x, y) = 0} is an irr educible simply connected curve, (C) 1999 Academic Press.