Invariance of Hironaka's characteristic polyhedron

TL;DR

This paper proves that certain features of Hironaka's characteristic polyhedron are invariant under changes in embedding and depend solely on the singularity and a specific flag, establishing their role as intrinsic invariants.

Contribution

It demonstrates that faces of Hironaka's characteristic polyhedron depend only on the singularity and a linear form, leading to embedding-independent numerical invariants.

Findings

Faces of the polyhedron depend only on the singularity and a flag.

Certain numerical data from the polyhedron are invariants of the singularity.

These invariants do not depend on the embedding of the singularity.

Abstract

We show that given a face of Hironaka's characteristic polyhedron, it does only depend on the singularity and a flag defined by the linear form determining the face. As a consequence we get that certain numerical data obtained from the characteristic polyhedron are invariants of the singularity. In particular, they do not depend on an embedding.