On the quotient of Milnor and Tjurina numbers for two‐dimensional isolated hypersurface singularities

Abstract

In this paper we give a complete answer to a question posed by Dimca and Greuel about the quotient of the Milnor and Tjurina numbers of a plane curve singularity. We put this question into a general framework of the study of the difference of Milnor and Tjurina numbers for isolated complete intersection singularities showing its connection with other problems in singularity theory.

On the Approximate Polar Curves of Foliations

“…In [6, Question 4.2], Dimca and Greuel present a family of curves {C m } m≥1 such that the quotient µ (Cm) τ (Cm) is strictly increasing with limit 4/3 and they ask if this property is true for any plane curve singularity. The question was affirmatively answered in [1], [10], and [15] for the irreducible case and in [2] for the reduced case.…”