On the polar quotients of an analytic plane curve (Q1609928)

Let \(f \in \mathbb{C}\{x,y\}\) be a reduced power series and \(t\) a power series of order 1 not dividing \(f\). The polar quotients of \(f\) with respect to \(t\) are defined by \[ Q(f,t) = \left\{\frac{(f,\varphi)_0}{(t,\varphi)_0} \Bigg|\varphi \text{ an irreducible factor of } \frac{\partial(t,f)}{\partial(x,y)}\right\}, \] here \((f,\varphi)_0\) is the intersection multiplicity of \(f\) and \(\varphi\) at 0. An explicit formula in terms of characteristic exponents and intersection multiplicities of the branches for the polar quotients is given.