Discriminant of a germ \((g,f): (\mathbb{C}^2,0)\to (\mathbb{C}^2,0)\) and contact quotients in the resolution of \(f\cdot g\) (Q1284683)

Let \(f: \mathbb C^2\to\mathbb C\) and \(g: \mathbb C^2\to\mathbb C\) be two germs of plane curves. The Jacobian of the map \((g,f)\) defines the Jacobian curve whose image is the discriminant. The first Puiseux pairs of all branch give a set of rational numbers, which are called Jacobian quotients. In the case where \(g\) is a transverse linear form one obtains the polar quotients. The author describes how to compute these quotients in terms of a simultaneous embedded resolution of \(f\) and \(g\). As application the case that the reduced Jacobian curve is smooth is studied.