Lionel Lang: On the topology of amoebas of curves

In order to understand the topology of the logarithmic projection from a planar curve V onto its amoeba, we study its critical locus \(S(V) \subset C\). When V is constraint in a fixed linear system |L|, we show that the study of topological pairs (V,S(V)) fits into a framework similar to Hilbert's 16-th problem: out of some real codimension 1 walls in |L|, S(V) is a collection of pairwise disjoint ovals in the smooth curve V. We provide bounds for the number of ovals and discuss their sharpness. Along the way, we use extensively the Lyashko–Looijenga mapping (LL) associated to the Log-Gauss map on V. We show in particular that LL is algebraic and extends to nodal curves.