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Our work is motivated by the connection established between Lorentzian polynomials and the Dressian in the seminal work of Br\"and\'en and Huh on Lorentzian polynomials. We analyze this relation for the class of positroids, and are able to show that in this case, we can relate a multiaffine homogenous stable polynomial to it. Additionally, we also highlight that a conjecture for matroids posed by Br\"and\'en and Huh is true when considered over the class of Rayleigh matroids which strictly contain the class of positroids. We collect these findings along with other results for further exploration., Comment: Comments are welcome!

We introduce the notion of lattice path matroidal subdivisions, or LPM subdivisions for short, and show that these subdivisions are regular and hence the weight vectors for them lie in the Dressian. This leads us to explore the structure of the set of these weights inside the Dressian and owing to the fact that Lattice path matroids are positroids, we move to the positive Dressian which in turn is equal to the positive tropical Grassmannian, an object of immense interest currently in Physics. This is related to the amplituhedron and positive configuration space, which we describe here and wish to explore these connections further., Comment: Comments are welcome

Metric trees and metric tree arrangements index cones in the polyhedral fan structure in the Dressian $Dr(2,n)$ and $Dr(3,n)$ respectively. We introduce the notion of generalized metric tree arrangements which parameterize points in $Dr(k,n)$ and extend previously known results to $Dr(k,n)$ along with providing explicit examples of these generalized metric tree arrangements. We study the adjacency of cones in the positive Dressian $Dr_{>0}(3, n)$ and introduce generalized Whitehead moves which provide a condition for adjacency of maximal cones in $Dr_{>0}(3,n)$ in terms of the associated metric tree arrangements., Comment: Comments are welcome!

We state the fundamental theorem of projective geometry for semimodules over semirings, which is facilitated by recent work in the study of bases in semimodules defined over semirings. In the process we explore in detail the linear algebra setup over semirings. We also provide more explicit results to understand the implications of our main theorem on maps between tropical lines in the tropical plane. Along with this we also look at geometrical connections to the rich theory of tropical geometry, Comment: Comments welcome!

Given a lattice polygon $P$ with $g$ interior lattice points, we associate to it the moduli space of tropical curves of genus $g$ with Newton polygon $P$. We completely classify the possible dimensions such a moduli space can have. For non-hyperelliptic polygons the dimension must be between $g$ and $2g+1$, and can take on any integer value in this range, with exceptions only in the cases of genus $3$, $4$, and $7$. We provide a similar result for hyperelliptic polygons, for which the range of dimensions is from $g$ to $2g-1$. In the case of non-hyperelliptic polygons, our results also hold for the moduli space of algebraic curves that are non-degenerate with respect to $P$., Comment: 17 pages, 15 figures

We study the classical result by Bruijn and Erd\H os regarding the bound on the number of lines determined by a $n$-point configuration in the plane, and in the light of the recently proven Tropical Sylvester-Gallai theorem, come up with a tropical version of the above-mentioned result. In this work, we introduce stable tropical lines, which help in answering questions pertaining to incidence geometry in the tropical plane. Projective duality in the tropical plane helps in translating the question for stable lines to stable intersections that have been previously studied in depth. Invoking duality between Newton subdivisions and line arrangements, we are able to classify stable intersections with shapes of cells in subdivisions, and this ultimately helps us in coming up with a bound. In this process, we also encounter various unique properties of linear Newton subdivisions which are dual to tropical line arrangements., Comment: Comments welcome!

Two lattice points are visible to one another if there exist no other lattice points on the line segment connecting them. In this paper we study convex lattice polygons that contain a lattice point such that all other lattice points in the polygon are visible from it. We completely classify such polygons, show that there are finitely many of lattice width greater than $2$, and computationally enumerate them. As an application of this classification, we prove new obstructions to graphs arising as skeleta of tropical plane curves., Comment: 24 pages, 20 figures