Related papers: Graphs and obstruction theory for algebraic curves
We continue, in this second article, the study of the the algebraic tools which play a role in tropical algebra. We especially examine here the polynomial algebras over idempotent semi-fields. this work is motivated by the development of…
We discuss, following Mikhalkin, Brugall\'e, and many others, the counting of curves on toric surfaces with prescribed genus, Newton polygon, and intersection pattern with the toric boundary divisor, both at assigned and unassigned points.…
In algebraic geometry, trigonal curves can always be embedded into Hirzebruch surfaces. In tropical geometry, the notion of trigonality does not have a unique translation. We focus on the characterization in terms of the existence of a…
This paper will contribute to a practical problem, Urban Traffic. We will investigate those features, try to simplify the complexity and formulize this dynamic system. These contents mainly contain how to analyze a decision problem with…
Every graph $\Gamma$ can be embedded in the plane with a minimal number of edge intersections, called its classical crossing number $\text{cross}\left(\Gamma\right)$. In this paper, we prove that if $\Gamma$ is a metric graph it can be…
Intersection graphs are very important in both theoretical as well as application point of view. Depending on the geometrical representation, different type of intersection graphs are defined. Among them interval, circular-arc, permutation,…
This paper is the first part in a series of three papers devoted to the study of enumerative invariants of abelian surfaces through the tropical approach. In this paper, we consider the enumeration of genus $g$ curves of fixed degree…
Associated to any hypergraph is a toric ideal encoding the algebraic relations among its edges. We study these ideals and the combinatorics of their minimal generators, and derive general degree bounds for both uniform and non-uniform…
This note is supposed to answer some questions on deformation theory in derived algebraic geometry. We show that derived algebraic geometry allows for a geometrical interpretation of the full cotangent complex and gives a natural setting…
Tropical Geometry and Mathematical Morphology share the same max-plus and min-plus semiring arithmetic and matrix algebra. In this chapter we summarize some of their main ideas and common (geometric and algebraic) structure, generalize and…
Inspired by notorious combinatorial optimization problems on graphs, in this paper we consider a series of related problems defined using a metric space and topology determined by a graph. Particularly, we present the Independent Set,…
It is known that isomorphisms of graph Jacobians induce cyclic bijections on the associated graphs. We characterize when such cyclic bijections can be strengthened to graph isomorphisms, in terms of an easily computed divisor. The result…
We consider the enumeration of tropical curves in M\"obius strips for two different lattice structures and relate them to the enumeration of curves in two rational ruled surfaces over a complex elliptic curve. Using this correspondence, we…
In the 1990's, Itenberg and Haas studied the relations between combinatorial data in Viro's patchworking and the topology of the resulting non-singular real algebraic curves in the projective plane. Using recent results from Renaudineau and…
We calculate the intersection ring of three-dimensional graph manifolds with rational coefficients and give an algebraic characterization of these rings when the manifold's underlying graph is a tree. We are able to use this…
The paper gives an overview of recent advances in structural equation modeling. A structural equation model is a multivariate statistical model that is determined by a mixed graph, also known as a path diagram. Our focus is on the…
Tropical algebraic geometry offers new tools for elimination theory and implicitization. We determine the tropicalization of the image of a subvariety of an algebraic torus under any homomorphism from that torus to another torus.
The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry. In this survey, we briefly describe some open problems related to algebraic curves which can be…
We introduce the notion of tropical area of a tropical curve defined in an open subset of $\mathbb R^n$. We prove that the number of vertices of a tropical curve is bounded by the area of the curve. The approach is totally elementary yet…
We derive various inequalities involving the intersection number of the curves contained in geodesics and tight geodesics in the curve graph. While there already exist such inequalities on tight geodesics, our method applies in the setting…