Jacobian of Meromorphic Curves*

The contact structure of two meromorphic curves gives a factorization of their jacobian. Section 1: Introduction Let J(F,G) = J(X,Y )(F,G) be the jacobian of F = F (X,Y ) and G = G(X,Y ) with respect to X and Y , i.e., let J(F,G) = FXGY − FY GX where subscripts denote partial derivatives. Here, to begin with, F and G are plane curves, i.e., polynomials in X and Y over an algebraically closed ground field k of characteristic zero. More generally, we let F and G be meromorphic curves, i.e., polynomials in Y over the (formal) meromorphic series field k((X)). In terms of the the contact structure of F and G, we shall produce a factorization of J(F,G). Note that if G = −X then J(F,G) = FY ; in this special case, our results generalize some results of Merle [Me], Delgado [De], and Kuo-Lu [KL] who studied the situation when F has one (Merle) or two (Delgado) or more (Kuo-Lu) branches. These authors restricted their attention to the analytic case, i.e., when F is a polynomial in Y over the (formal) power series ring k[[X ]]. With an eye on the Jacobian Conjecture, we are particularly interested in the meromorphic case. The main technique we use is the method of Newton Polygon, i.e., the method of deformations, characteristic sequences, truncations, and contact sets given in Abhyankar’s 1977 Kyoto paper [Ab]. In Sections 2 to 5 we shall review the relevant material from [Ab]. In Section 6 we shall introduce the tree of contacts and in Sections 7 to 9 we shall show how this gives rise to the factorizations. The said Jacobian Conjecture predicts that if the Jacobian of two bivariate polynomials F (X,Y ) and G(X,Y ) is a nonzero constant then the variables X and Y can be expressed as polynomials in F and G, i.e., if 0 6= J(F,G) ∈ k for F and G in k[X,Y ] then k[F,G] = k[X,Y ]. We hope that the results of this paper may contribute towards a better understanding of this Bivariate Conjecture, and hence also of its obvious Multivariate Incarnation. Section 2: Deformations We are interested in studying polynomials in indeterminates X and Y over an algebraically closed ground field k of characteristic zero. To have more elbow room to maneuver, we consider the larger ring R = k((X))[Y ] of polynomials in Y over k((X)), i.e., with coefficients in k((X)), where k((X)) is the meromorphic series field in X over k. Given any g = g(X,Y ) = ∑