L ojasiewicz exponents and Farey sequences

\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