Generic lemniscates of algebraic functions
An algebraic function is a holomorphic map of a compact Riemann surface \(C\) of genus \(g\) to the Riemann sphere \(\mathbb{P}^1\). The associated real-valued function \(|f|^2\) has as critical points the zeros and poles of \(f\) plus the critical points of \(f\). The union of the singular level sets of \(|f|^2\) is called the lemniscate configuration \(D\) of \(f\), and consists of the zeros and poles of \(f\) plus certain singular curves. One can associate a graph whose vertices \(v,v',\dots\) are the connected components of the lemniscate configuration \(D\), and an edge connects \(v\) and \(v'\) if following the gradient of an integral curve stemming from a point of \(v\) hits \(D\) in a point of \(v'\) (one can also give integral multiplicity, called weight, to these edges). One has an open set \(L\), in the ``Hurwitz'' space of algebraic functions, where the singularities of the level sets are at most one double point, and where all the zeros and poles are simple. To each connected component of \(L\) is associated a unique graph, but does conversely a graph determine the connected component? For polynomials the converse problem was solved by the second author and \textit{M. Paluszny} [Topology 30, 623-640 (1991; Zbl 0755.57002)]. Here a description of the connected components of \(L\) is provided, and it is shown that for rational functions the graph determines the connected component, but that this no longer holds for genus at least 1 and degree at least 3. The paper also contains an introductory part on Hurwitz spaces and their topology, and also some application to the topology of ODE's on the Riemann sphere.
scientific article; zbMATH DE number 1031540