Blurred combinatorics in resolution of singularities: (A little) Beyond the characteristic polytope

Abstract

We introduce a variation of the well-known Newton–Hironaka polytope for algebroid hypersurfaces. This combinatorial object is a perturbed version of the original one, parameterized by a real number $\epsilon \in {\mathbb{R}}_{\ge 0}$. For well-chosen values of the parameter, the objects obtained are very close to the original, while at the same time presenting more (hopefully interesting) information in a way that does not depend on the choice of parameter.