Blurred combinatorics in resolution of singularities: (a little) beyond the characteristic polytope

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, parametrized by a real number. 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 which does not depend on the choice of parameter.