Miruna-Stefana Sorea, MiS Leipzig: The shapes of level curves of real polynomials near strict local minima

Referentin: Miruna-Stefana Sorea: The shapes of level curves of real polynomials near strict local minima

Dienstag,  19.11.2019, 15:10 h

Ort:Raum 48-436

We consider a real bivariate polynomial function vanishing at the origin and exhibiting a strict local minimum at this point. We work in a neighbourhood of the origin in which the non-zero level sets of this polynomial are smooth Jordan curves. Whenever the origin is a Morse critical point, the sufficiently small levels become boundaries of convex topological disks. Otherwise, these level curves may fail to be convex.

The aim of this talk is two-fold. Firstly, we introduce a combinatorial object, called the Poincaré-Reeb tree, to encode the non-convexity of a smooth, compact, connected component of a real algebraic plane curve in a neighbourhood of a strict local minimum. And secondly, we want to characterise all possible topological types of such trees. To this end, we construct a family of bivariate polynomial functions with non-Morse strict local minima, realising a large class of these trees. In our construction we first reduce the problem to the univariate polynomials, using a tool inspired by Ghys’s work and the real polar curve. Then we give a new and effective construction of Morse polynomials whose associated permutation (the so-called “Arnold snake”) is separable. Finally, we show how this shape can be transferred back to the bivariate setting.