On the topology of rational functions in two complex variables (Q445786)

Let \(F=f/g:\mathbb{C}^{2}\setminus \{g=0\}\rightarrow \mathbb{C}\) be a rational function, where \(f,g\in \mathbb{C[}x,y\mathbb{]}\) and \(f,g\) have no common factor. Outside a finite set \(B(F)\subset \mathbb{C}\) (called the bifurcation set of \(F\)) the mapping \(F\) over \(\mathbb{C}\setminus B(F) \) is a locally trivial fibration of class \(C^{\infty }.\) The author gives various characterizations of \(B(F),\) almost analogous to characterizations in the case of polynomial functions. If \(\deg f>\deg g\) then \[ B(F)=K_{0}(F)\cup K_{1}(F)\cup B_{\infty }(F), \] where: \(K_{0}(F)\) -- the set of critical values of \(F,\) \(K_{1}(F):=\{t_{0} \in \mathbb{C}:\) there exists \(P\in \mathbb{C}^{2}\) such that \(f(P)=g(P)=0\) and the Milnor numbers of \(f-t_{0}g\) and \(f-tg\) at \(P\) are different for \(t\) generic\(\}\) and \(B_{\infty }(F)\) is the set of critical values of \(F\) at infinity (by definition \(t_{0}\notin B_{\infty }(F)\) if \(F\) is a trivial fibration over a neighbourhood of \(t_{0}\) outside a compact set in \(\mathbb{C }^{2}\)). The most important and interesting are characterizations of elements of \( B_{\infty }(F).\) The author gives such characterizations (under some assumptions on degrees of \(f\) and \(g\)) in terms of the known conditions:\ Malgrange, \(M\)-tameness and Euler characteristics of fibers of \(F.\)