Characteristic polyhedra of singularities without completion -- Part II

Hironaka’s characteristic polyhedron is an important combinatorial object reflecting the local nature of a singularity. We prove that it can be determined without passing to the completion if the local ring is a G-ring and if additionally either it is Henselian, or a certain polynomiality condition $(\mathrm{P}\mathrm{o}\mathrm{l})$ holds, or a mild condition $(\ast )$ on the singularity holds. For example, the latter is fulfilled if the residue field is perfect.