Characteristic polyhedra of singularities without completion -- Part II

TL;DR

This paper demonstrates that Hironaka's characteristic polyhedron, a key tool in understanding singularities, can be determined without completion under certain conditions related to the local ring's properties.

Contribution

It extends the understanding of characteristic polyhedra by showing they can be computed without completion for G-rings satisfying specific conditions.

Findings

Polyhedron determination without completion under G-ring conditions

Conditions include Henselian property, polynomiality, or mild singularity conditions

Applicable when residue field is perfect

Abstract

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.