\L ojasiewicz exponents and Farey sequences

TL;DR

This paper studies the Łojasiewicz exponent of ideals in formal power series rings and proves that for finite codimension ideals, this exponent is always a Farey number, a specific type of rational number.

Contribution

It establishes that the Łojasiewicz exponent for finite codimension ideals is always a Farey number, linking algebraic invariants to Farey sequences.

Findings

Łojasiewicz exponent is a Farey number for finite codimension ideals.

The exponent can be an integer or a rational number of the form N + b/a.

The result holds over algebraically closed fields of arbitrary characteristic.

Abstract

\noindent Let $I$ be an ideal of the ring of formal power series $\bK[[x,y]]$ with coefficients in an algebraically closed field $\bK$ of arbitrary characteristic. Let $\Phi$ denote the set of all parametrizations $\varphi=(\varphi_1,\varphi_2)\in \bK[[t]]^2$, where $\varphi \neq (0,0)$ and $\varphi (0,0)=(0,0)$. The purpose of this paper is to investigate the invariant \[ \Lo(I)=\sup_{\varphi \in \Phi}\left(\inf_{f\in I} \frac{\ord f \circ \varphi}{\ord \varphi}\right) \] \noindent called the {\it \L ojasiewicz exponent} of $I$. Our main result states that for the ideals $I$ of finite codimension the \L ojasiewicz exponent $\Lo(I)$ is a Farey number i.e. an integer or a rational number of the form $N+\frac{b}{a}$, where $a,b,N$ are integers such that $0<b<a<N$.