Lojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

TL;DR

This paper provides a new formula for the Łojasiewicz exponent of ideal tuples in complex space using Newton filtrations and Rees mixed multiplicities, identifying cases where the gradient map's exponent is maximized.

Contribution

It introduces a novel expression for Łojasiewicz exponents based on Newton filtrations and Rees mixed multiplicities, expanding understanding of semi-weighted homogeneous functions.

Findings

Derived a formula for Łojasiewicz exponents using Newton filtrations.

Identified classes of semi-weighted homogeneous functions with maximal gradient Łojasiewicz exponent.

Abstract

We give an expression for the {\L}ojasiewicz exponent of a wide class of n-tuples of ideals $(I_1,..., I_n)$ in $\O_n$ using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation of {\L}ojasiewicz exponents in terms of Rees mixed multiplicities. As a consequence, we obtain a wide class of semi-weighted homogeneous functions $(\mathbb{C}^n,0)\to (\mathbb{C},0)$ for which the {\L}ojasiewicz of its gradient map $\nabla f$ attains the maximum possible value.