Mather-Yau Theorem in Positive Characteristic

TL;DR

This paper extends the Mather-Yau theorem to positive characteristic fields by demonstrating that the isomorphism type of isolated hypersurface singularities can be determined by higher Tjurina and Milnor algebras, with effective bounds.

Contribution

It introduces a positive characteristic analogue of the Mather-Yau theorem using higher Tjurina and Milnor algebras with explicit bounds.

Findings

Isomorphism type determined by higher Tjurina algebra in positive characteristic

Effective bounds provided for the higher Tjurina algebra

Similar results established for higher Milnor algebra

Abstract

The well-known Mather-Yau theorem says that the isomorphism type of the local ring of an isolated complex hypersurface singularity is determined by its Tjurina algebra. It is also well known that this result is wrong as stated for power series f in K[[x]] over fields K of positive characteristic. In this note we show that, however, also in positive characteristic the isomorphism type of an isolated hypersurface singularity f is determined by an Artinian algebra, namely by a "higher Tjurina algebra" for sufficiently high index, for which we give an effective bound. We prove also a similar version for the "higher Milnor algebra" considered as K[[f]]-algebra.