Characteristic polyhedra of singularities without completion

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 $Δ(f;u_1, \ldots ,u_d)$, $f \not \in (u_1, \ldots ,u_d)$ 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.

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