# The sequence of mixed \L ojasiewicz exponents associated to pairs of ideals

## TL;DR

This paper studies the sequence of mixed Łojasiewicz exponents for pairs of monomial ideals, providing combinatorial formulas and relations with other invariants in complex analytic geometry.

## Contribution

It introduces a combinatorial expression for the sequence when one ideal is diagonal and explores its relations with other numerical invariants.

## Key findings

- Derived a combinatorial formula for the sequence when J is diagonal
- Established relations between Łojasiewicz exponents and other invariants
- Analyzed the sequence for monomial ideals of finite colength

## Abstract

We analyze the sequence $\mathcal L^*_J(I)$ of mixed \L ojasiewicz exponents attached to any pair $I,J$ of monomial ideals of finite colength of the ring of analytic function germs $(\mathbb C^n,0)\to \mathbb C$. In particular, we obtain a combinatorial expression for this sequence when $J$ is diagonal. We also show several relations of $\mathcal L^*_J(I)$ with other numerical invariants associated to $I$ and $J$.