Chains of the Eggers tree and polar curves (Q1884057)

Let us consider a curve \(C \subset {\mathbb C}^{2}\) defined by an holomorphic function \(f(x,y)=0\). We may assume that \(f(0,0)=0\). Its polar curve has equation \(P \frac{\partial f}{\partial x} +Q \frac{\partial f}{\partial y} + Rf=0\), where \(P\) and \(Q\) are functions, at least one of them non-vanishing at \((0,0)\). Up to a correct choice of the function \(f\), we have that its polar curve is defined by \(\frac{\partial f}{\partial y}=0\). Let us assume we have this correct choice for \(f\). We say that the polar curve is transverse to \(C\) if the line \(x=0\) is transverse to \(C\). In the transversal situation, the point \((0,0)\) is a singular point of the polar curve with multiplicity one less than for \(C\). One would like to say something more about the type of singularity of the polar curve if one knows the type of singularity of \(C\). A general statement is given by \textit{M. Merle} [Invent. Math. 41, 103--111 (1977; Zbl 0371.14003)] in the case that \(C\) consists of a single branch. In the case of several branches a general theorem is given by \textit{E. García Barroso} [Invariants des singularités de courbes planes et courbure des fibres de Milnor. Doctoral thesis, La Laguna (1996)], but the known proofs involve messy combinatorial arguments. In this paper the author reduces these to easy calculations by introducing additional structure on the Eggers tree.