The shapes of level curves of real polynomials near strict local minima

  • Miruna-Ştefana Sorea (MPI MiS, Leipzig)
G3 10 (Lecture hall)


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 curves of this function are smooth Jordan curves. Whenever the origin is a Morse critical point, the sufficiently small levels become boundaries of convex disks. Otherwise, these level curves may fail to be convex.

The aim of this talk is two-fold. Firstly, to study a combinatorial object measuring this nonconvexity; it is a planar rooted tree. And secondly, we want to characterise all possible topological types of these objects. To this end, we construct a family of polynomial functions with non-Morse strict local minima realising a large class of such trees.

Mirke Olschewski

MPI for Mathematics in the Sciences Contact via Mail

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