Characteristic polyhedra of singularities without completion

TL;DR

This paper demonstrates that the Hironaka characteristic polyhedron of a hypersurface singularity in a regular local G-ring can be computed within the ring itself without needing to complete the ring or assume residue characteristic conditions.

Contribution

It proves that the characteristic polyhedron can be determined in the original ring without completion or characteristic restrictions, extending previous results.

Findings

Characteristic polyhedron computable in original ring

No residue characteristic restrictions needed

Applicable to hypersurface singularities in G-rings

Abstract

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.