# The {\L}ojasiewicz Exponent via The Valuative Hamburger-Noether Process

## TL;DR

This paper demonstrates the equivalence of two definitions of the Łojasiewicz exponent for ideals in power series rings over algebraically closed fields using the Valuative Hamburger-Noether process, advancing understanding in algebraic geometry.

## Contribution

It introduces a novel application of the Hamburger-Noether process to establish the equivalence of definitions of the Łojasiewicz exponent in a general characteristic setting.

## Key findings

- Proves the equivalence of two definitions of Łojasiewicz exponent
- Applies the Hamburger-Noether process in a new context
- Extends results to fields of any characteristic

## Abstract

Let $k$ be an algebraically closed field of any characteristic. We apply the Hamburger-Noether process of successive quadratic transformations to show the equivalence of two definitions of the {\L}ojasiewicz exponent $\mathfrak{L}(\mathfrak{a})$ of an ideal $\mathfrak{a}\subset k[[x,y]]$.