Georgia Institute of Technology College of Sciences

The theory of stable polynomials features a key notion called proper position, which generalizes interlacing of real-rooted polynomials to higher dimensions. In a recent paper, I introduced a Lorentzian analog of proper position and used it to give a new characterization of elementary quotients of valuated matroids. This connects the local structure of spaces of Lorentzian polynomials with the incidence geometry of tropical linear spaces. A central object in this connection is the moduli space of codimension-1 tropical linear subspaces of a given tropical linear space. In this talk, I will show some new structural results on this moduli space and their implications for Lorentzian polynomials.