Measuring the local non-convexity of real algebraic plane curves

We study the real Milnor fibre of real bivariate polynomial functions vanishing at the origin, with an isolated local minimum at this point. We work in a neighbourhood of the origin in which its non-zero level sets are smooth Jordan curves. Whenever the origin is a Morse critical point, the sufficiently small levels become boundaries of convex disks. Otherwise, they may fail to be convex, as was shown by Coste.

In order to measure the non-convexity of the level curves, we introduce a new combinatorial object, called the Poincaré-Reeb tree, and show that locally the shape stabilises and that no spiralling phenomena occur near the origin. Our main objective is to characterise all topological types of asymptotic Poincaré-Reeb trees. To this end, we construct a family of polynomials with non-Morse strict local minimum at the origin, realising a large class of such trees.

As a preliminary step, we reduce the problem to the univariate case, via the interplay between the polar curve and its discriminant. Here we give a new and constructive proof of the existence of Morse polynomials whose associated permutation (the so-called “Arnold snake”) is separable, using tools inspired from Ghys’s work.