Related papers: Point-Line Geometry in the Tropical Plane
A special case of a combinatorial theorem of De Bruijn and Erdos asserts that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvatal suggested a possible generalization of this assertion in…
This paper is about the combinatorics of finite point configurations in the tropical projective space or, dually, of arrangements of finitely many tropical hyperplanes. Moreover, arrangements of finitely many tropical halfspaces can be…
We establish new bounds on the number of tangencies and orthogonal intersections determined by an arrangement of curves. First, given a set of $n$ algebraic plane curves, we show that there are $O(n^{3/2})$ points where two or more curves…
Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on…
In this paper, we prove that a set of $N$ points in ${\bf R}^2$ has at least $c{N \over \log N}$ distinct distances, thus obtaining the sharp exponent in a problem of Erd\"os. We follow the set-up of Elekes and Sharir which, in the spirit…
We describe recent work connecting combinatorics and tropical/non-Archimedean geometry to Diophantine geometry, particularly the uniformity conjectures for rational points on curves and for torsion packets of curves. The method of…
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…
In this paper we generalize correspondence theorems of Mikhalkin and Nishinou-Siebert providing a correspondence between algebraic and parameterized tropical curves. We also give a description of a canonical tropicalization procedure for…
Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows to study properties of mathematical objects such as algebraic varieties…
We introduce tropical dual numbers as an extension of tropical semiring. By this innovation, one can work with honest ideals, instead of congruences, and recover the Euclidean topology on affine tropical spaces similar to Zariski's approach…
We study questions in incidence geometry where the precise position of points is `blurry' (e.g. due to noise, inaccuracy or error). Thus lines are replaced by narrow tubes, and more generally affine subspaces are replaced by their small…
In this paper, we study tropicalisations of families of curves with a singularity in a fixed point. The tropicalisation of such a family is a linear tropical variety. We describe its maximal dimensional cones using results about linear…
The tropical Grassmannian parameterizes tropicalizations of linear spaces, while the Dressian parameterizes all planes in $\TP^{n-1}$. We study these parameter spaces and we compute them explicitly for $n \leq 7$. Planes are identified with…
The main result of this paper is a formula for the limit cycle of a 1-parameter family of subvarieties of a tropical compactification, expressed in terms of tropical intersections. Our theorem generalizes results of…
Tropical geometry gives a bound on the ranks of divisors on curves in terms of the combinatorics of the dual graph of a degeneration. We show that for a family of examples, curves realizing this bound might only exist over certain…
Via correspondence theorems, rational log Gromov--Witten invariants of the plane can be computed in terms of tropical geometry. For many cases, there exists a range of algorithms to compute tropically: for instance, there are (generalized)…
Splice type surface singularities were introduced by Neumann and Wahl as a generalization of the class of Pham-Brieskorn-Hamm complete intersections of dimension two. Their construction depends on a weighted tree called a splice diagram. In…
We consider the tropicalization of tangent lines to a complete intersection curve $X$ in $\mathbb{P}^n$. Under mild hypotheses, we describe a procedure for computing the tropicalization of the image of the Gauss map of $X$ in terms of the…
We introduce a notion of Lorentzian proper position in close analogy to proper position of stable polynomials. Using this notion, we give a new characterization of elementary quotients of M-convex function that parallels the Lorentzian…
This paper has been written to illustrate the power of techniques from tropical geometry and mirror symmetry for studying the KSBA moduli space of surfaces on or near the Noether line. We focus on the moduli space of octic double planes…