  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{Pol}) $ holds, or a mild condition $(*) $ on the singularity holds.
For example, the latter is fulfilled if the residue field is perfect.
