  Let $(R,M,k)$ be a regular local G-ring with regular system of parameters
$(u_1, \ldots ,u_d,y)$. We prove that the Hironaka characteristic polyhedron
$\Delta (f;u_1, \ldots ,u_d)$, $f \not \in (u_1, \ldots ,u_d)$ of a
hypersurface singularity $X={\rm Spec}R/(f)$ can be computed in some system of
coordinates belonging to $R$. No assumption on the residue characteristic is
required.
