Singular foliations of toric type (Q1576395)

A germ \(\mathcal F\) of a singular foliation at \(0\in{\mathbb C}^2\) is said to be of toric type if there is a toric modification of \({\mathbb C}^2\) that desingularizes \(\mathcal F\). The authors offer an algorithm that achieves this, if it is possible at all, in an explicit way, by choosing coordinates in \({\mathbb C}^2\) suitably. To do this, they choose a normal crossings divisor \(D\) which is invariant under \(\mathcal F\), and write the form \(\omega\) that determines \(\mathcal F\) in terms of local coordinates in which \(D\subseteq \{xy=0\}\). This determines a Newton polygon, and the coordinates are then to be modified so as to simplify the polygon.