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 \(J(F,G)= F_X G_Y- F_Y G_X\) 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 contact structure of \(F\) and \(G\), we produce a factorization of \(J(F,G)\). Note that if \(G= -X\) then \(J(F,G)= F_Y\); in this special case, our results generalize some results of M. Merle [Invent. Math. 41, 103-111 (1977; Zbl 0371.14003)], F. Delgado de la Mata [Compos. Math. 92, No. 3, 327-375 (1994; Zbl 0816.14012)] and T.-C. Kuo and Y.-C. Lu [Topology 16, 299-310 (1977; Zbl 0378.32001)] who studied the situation when \(F\) has one (Merle) or two (Delgado) ore 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 the author’s 1977 Kyoto paper [S. S. Abhyankar in: Algebraic Geometry, Proc. int. Symp., Kyoto 1977, 249-414 (1977; Zbl 0408.14010)]. In §§2-5 we review the relevant material from this article. In §6 we introduce a tree of contacts and in §§7-9 we 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 \(0\neq J(F,G)\in k\) 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.