Characteristic polyhedra of singularities without completion

Let $(R,M,k)$ be a regular local G-ring with regular system of parameters $(u_1,\dots ,u_d,y)$. We prove that the Hironaka characteristic polyhedron $\mathrm{\Delta }(f;u_1,\dots ,u_d)$, $f\in ̸(u_1,\dots ,u_d)$ of a hypersurface singularity $X=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}R\mathrm{/}(f)$ can be computed in some system of coordinates belonging to $R$. No assumption on the residue characteristic is required.