Degree of \(C^0\)-sufficiency of an analytic germ with respect to a principal ideal
Let \(f\in {\mathbb C}\{ x,y\} \) be a germ of an analytic function of two complex va\-ria\-bles. If \(I\) is an ideal of \( {\mathbb C}\{ x,y\} \) then \(f\) is said to be \(C^0\)-sufficient of order \(r\) with respect to \(I\) if for any \(h\in {\mathbb C}\{ x,y\}\) such that \(f-h\in I^{r+1}\), the germs \(f\) and \(h\) are topologically equivalent. The author proves that if \(f\) has at most isolated singularity at zero and \(I\) is an ideal generated by a germ \(g\in {\mathbb C}\{ x,y\} \) then the degree of \(C^0\)-sufficiency of \(f\) with respect to \(I\), which is a minimum of integers \(r\) such that \(f\) is \(C^0\)-sufficient of order \(r\) with respect to \(I\) is equal to \(max\{{m_{\delta }\over n_{\delta }} : \delta \in \Delta \}\). In this formulae \(\delta \) is an irreducible component of the critical curve and \(u=av^{{m_{\delta }\over n_{\delta }}}+\dots \) is its Poiseaux expansion.
scientific article; zbMATH DE number 2133555
scientific article