Faye Pasley Simon | Searcharxiv

arXiv2021

Invariant hyperbolic curves: determinantal representations and applications to the numerical range

Here we study the space of real hyperbolic plane curves that are invariant under actions of the cyclic and dihedral groups and show they have determinantal representations that certify this invariance. We show an analogue of Nuij's theorem for the set of invariant hyperbolic polynomials of a given degree. The main theorem is that every invariant hyperbolic plane curve has a determinantal representation using a block cyclic weighted shift matrix. This generalizes previous work by Lentzos and the first author, as well as by Chien and Nakazato. One consequence is that if the numerical range of a matrix is invariant under rotation, then it is the numerical range of a block cyclic weighted shift matrix.

math.AG↗

arXiv2020

Tropical convex hulls of polyhedral sets

In this paper we focus on the tropical convex hull of convex sets and polyhedral complexes. We give a vertex description of the tropical convex hull of a line segment and a ray. %in \RR^{n+1}/\RR\mathbf{1}. Next we show that tropical convex hull and ordinary convex hull commute in two dimensions and characterize tropically convex polyhedra in any dimension. %$\mathbb{R}^3/\mathbb {R}\mathbf{1}$. Finally we show that the dimension of a tropically convex fan depends on the coordinates of its rays and give a lower bound on the degree of a fan tropical curve using only tropical techniques.

math.CO↗