The log canonical threshold of products of ideals and mixed \L ojasiewicz exponents

TL;DR

This paper establishes a precise lower bound for the log canonical threshold of the product of two ideals in terms of their mixed Łojasiewicz exponents, with equality conditions related to integral closure and powers of the maximal ideal.

Contribution

It provides a sharp lower bound for the log canonical threshold of product ideals using mixed Łojasiewicz exponents, advancing understanding of singularity invariants.

Findings

Derived a sharp lower bound for the log canonical threshold of products of ideals.

Identified conditions for equality involving integral closure and powers of the maximal ideal.

Connected mixed Łojasiewicz exponents with algebraic properties of ideals.

Abstract

Given two ideals $I$ and $J$ of the ring $\mathcal O_n$ of analytic function germs $f:(\mathbb C^n,0)\to \mathbb C$, we show a sharp lower bound for the log canonical threshold of $IJ$ in terms of the sequences of mixed {\L}ojasiewicz exponents of them. In particular, in the case where $J$ is the maximal ideal, the corresponding equality holds if and only if the integral closure of $I$ equals some power of the maximal ideal.