Dicritical singularities of holomorphic vector fields (Q2725681)

A differential equation of the form \(w=a(x,y)dy-b(x,y)dx=0\), where \(a(x,y)\)and \(b(x,y)\) are holomorphic defined in an open subset \(U\subset \mathbb{C}^2\), induces a foliation \(F\) on \(U\) whose leaves are Riemann surfaces tangent at each point of \(U\) to the kernel of \(w\). NEWLINENEWLINENEWLINEThe author of this paper studied the local properties of \(F\) near an isolated singular point. By using the resolution theorem of Seidenberg, the author gives necessary and sufficient condition that the foliation \(F\) admits a meromorphic first integral near \(0\in \mathbb{C}^2 \).NEWLINENEWLINEFor the entire collection see [Zbl 0959.00033].