For a plane curve singularity \(f\colon (\mathbb C^2,0)\to (\mathbb C,0)\) and a nonsingular \(l\colon (\mathbb{C}^2,0)\to (\mathbb{C},0)\) (whose zero set is not a component of \(f=0\)), one considers the polar curve of the mapping \((l,f)\) and its image. Any equation for this image is called the discriminant \(\mathcal D(u,v)\). Its Newton diagram is the Jacobian Newton diagram, introduced by B. Teissier [Invent. Math. 40, 267–292 (1977; Zbl 0446.32002)]. It depends only on the topological type of \((l,f)\).
This paper studies the factorisation of the discriminant. It specifies formulas for the weighted initial forms of each factor. In particular, it is determined for which curves the discriminant is non-degenerate for its Newton diagram. For irreducible \(f\) this is the case if and only if there are no lattice points inside the compact edges. Therefore non-degeneracy depends only on the topological type of \((l,f)\). This is no longer true for reducible curves. The Authors obtain a criterion for non-degeneracy. They give an example of curves with arbitrary many singular branches and non-degenerate discriminants.
Finally they show that non-degeneracy of the discriminant of \((l,f)\) depends only on the curve \(f=0\) (with \(l\) fixed). Fot \(f\) unitangent one can also change \(l\), provided it is transverse. An example shows that the condition unitangent cannot be omitted.