Mixed \L ojasiewicz exponents, log canonical thresholds of ideals and bi-Lipschitz equivalence

TL;DR

This paper explores the relationships between jasiewicz exponents and log canonical thresholds of ideals in complex spaces, providing formulas, inequalities, and invariance results under bi-Lipschitz transformations.

Contribution

It introduces the notion of bi-Lipschitz equivalence of ideals and proves invariance of key invariants under this equivalence, along with effective formulas for monomial ideals.

Findings

Effective formulas for ideals with monomial integral closure

An inequality relating jasiewicz exponents and log canonical thresholds

Bi-Lipschitz invariance of these invariants

Abstract

We study the \L ojasiewicz exponent and the log canonical threshold of ideals of $\mathcal O_n$ when restricted to generic subspaces of $\mathbb C^n$ of different dimensions. We obtain effective formulas of the resulting numbers for ideals with monomial integral closure. An inequality relating these numbers is also proven. We also introduce the notion of bi-Lipschitz equivalence of ideals and we prove the bi-Lipschitz invariance of \L ojasiewicz exponents and log canonical thresholds of ideals.