Tropical points of multiplicity m

We recall basic notions of tropical geometry and describe a new necessary condition for a point to be the tropicalization of a point of multiplicity $m$. For plane tropical curves the result is easily visible on a subdivision of the Newton polygon of the curve: the sum of areas of some faces should be at least $3m^2/8$. If time permits, we will present ideas of the proof (the key for the story is the lattice width of some planar sets) and reformulate the result in terms of the Bergman fan of the matroid determined by linear conditions which a point of multiplicity $m$ imposes on the curve's coefficients.