Iterated inverse images of plane curve singularities (Q1942179)

The authors consider analytic morphisms \(\varphi: S \rightarrow S\), \(\varphi(O)=O\), where \(O \) is a fixed point of a smooth analytic surface \(S\). They use concepts as multiplicity \( e(\varphi)\), fundamental point and trunk of \(\varphi\) introduced in [\textit{E. Casas-Alvero}, Asian J. Math. 11, No. 3, 373--426 (2007; Zbl 1137.14006)]. The morphism \(\varphi\) is called dicritical whenever \(O\) is the unique point of its trunk, with multiplicity \(e(\varphi)\), and is called to be tranverse if and only if either is dicritical or the first neighboring point of its trunk is not fundamental. The main result of the paper has to do with iterations of \(\varphi \) and assert on the one hand that for morphisms certain intersection numbers and multiplicities are exponential in the number of iterations with basis the multiplicity of \( \varphi\); on the other hand, it is proved that transverse morphisms are characterized by the property that \( e(\varphi^i) = e(\varphi)^i\) for \(i \geq 0\). In this case, the Poincaré series associated with the elements \( e(\varphi^i) \) is rational and equal to \((1- e(\varphi))^{-1}\). Some comments and an example concerning non-transverse morphisms are also included in the paper.