The Tjurina number for Sebastiani-Thom type isolated hypersurface singularities

TL;DR

This paper derives a formula for the Tjurina number of joins of isolated hypersurface singularities, characterizes singularities with small Milnor-Tjurina difference, and establishes new bounds relating these invariants.

Contribution

It introduces a formula for the Tjurina number of joins of hypersurface singularities and provides new bounds and characterizations related to Milnor and Tjurina numbers.

Findings

Formula for Tjurina number of joins of singularities

Characterization of singularities with small Milnor-Tjurina difference

New upper bounds for the quotient of Milnor and Tjurina numbers

Abstract

In this note we provide a formula for the Tjurina number of a join of isolated hypersurface singularities in separated variables. From this we are able to provide a characterization of isolated hypersurface singularities whose difference between the Milnor and Tjurina numbers is less or equal than two arising as the join of isolated hypersurface singularities in separated variables. Also, we are able to provide new upper bounds for the quotient of Milnor and Tjurina numbers of certain join of isolated hypersurface singularities. Finally, we deduce an upper bound for the quotient of Milnor and Tjurina numbers in terms of the singularity index of any isolated hypersurface singularity, not necessarily a join of singularities.