Łojasiewicz exponents and Farey sequences (Q321253)

Let \(I\) be an ideal of the ring of formal power series \(\mathbb{K}[[x,y]]\) where \(\mathbb{K}\) is an algebraically closed field of arbitrary characteristic. Let \(\dim _{\mathbb{K}}\mathbb{K}[[x,y]]/I<+\infty .\) The authors prove that the Łojasiewicz exponent \(\mathcal{L}_{0}(I)\) defined by \[ \mathcal{L}_{0}(I):=\sup_{\phi \in \Phi }\left( \inf_{f\in I}\frac{\text{ ord}f\circ \phi }{\text{ord}\phi }\right) \] where \(\Phi \) is the set of all parametrizations \(\phi =(\phi _{1},\phi _{2})\neq (0,0)\) such that \(\phi _{1}(0)=0,\) \(\phi _{2}(0)=0,\) is a Farey number i.e. \(\mathcal{L}_{0}(I)=N+\frac{b}{a},\) where \(N,a,b\in \mathbb{N}\) and \(0<b<a<N.\) The result is a generalization of the same result from complex case \(\mathbb{K=C}\) to arbitrary field [\textit{A. Płoski}, Banach Cent. Publ. 20, 353--364 (1988; Zbl 0661.32018)].