Related papers: Real Tropical Quartics and their Bitangents: Count…
In this paper we consider plane quartics with to involutions. We compute the Dixmier invariants, the bitangents and the Matrix representation problem of these curves, showing that they have symbolic solutions for the last two questions.
This paper studies the curvatures of amoebas and real amoebas (i.e. essentially logarithmic curvatures of the complex and real parts of a real algebraic hypersurface) and of tropical and real tropical hypersurfaces. If V is a tropical…
We study rational double Hurwitz cycles, i.e. loci of marked rational stable curves admitting a map to the projective line with assigned ramification profiles over two fixed branch points. Generalizing the phenomenon observed for double…
The purpose of this note is to give an exposition of some interesting combinatorics and convex geometry concepts that appear in algebraic geometry in relation to counting the number of solutions of a system of polynomial equations in…
We study representations of tropical linear spaces as intersections of tropical hyperplanes of circuits. For several classes of matroids, we describe minimal tropical bases. We also show that every realizable tropical linear space has a…
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…
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…
The eigenvalues of a matrix polynomial can be determined classically by solving a generalized eigenproblem for a linearized matrix pencil, for instance by writing the matrix polynomial in companion form. We introduce a general scaling…
In this paper, we characterize the polynomiality of surfaces of revolution by means of the polynomiality of an associated plane curve. In addition, if the surface of revolution is polynomial, we provide formulas for computing a polynomial…
In the present article, we provide examples of fake quadrics, that is, minimal complex surfaces of general type with the same numerical invariants as the smooth quadric in $\PP ^3$ which are quotients of the bidisc by an irreducible lattice…
We reconcile the discrepancy between the complex and tropical counts of some enumerative problems reducing to positive characteristic. Each problem that we consider suggests a prime with special behaviour. Modulo this prime, the solutions…
The counting of alternating tangles in terms of their crossing number, number of external legs and connected components is presented here in a unified framework using quantum field-theoretic methods applied to a matrix model of colored…
A graph profile records all possible densities of a fixed finite set of graphs. Profiles can be extremely complicated; for instance the full profile of any triple of connected graphs is not known, and little is known about hypergraph…
Numerical equivalence of algebraic cycles is defined abstractly by intersection numbers. Classically, for smooth complex proper toric varieties, the quotients by numerical equivalence with rational coefficients can be described…
We introduce the first graph kernels for metric graphs via tropical algebraic geometry. In contrast to conventional graph kernels based on graph combinatorics such as nodes, edges, and subgraphs, our metric graph kernels are purely based on…
A tropical complete intersection curve C in R^(n+1) is a transversal intersection of n smooth tropical hypersurfaces. We give a formula for the number of vertices of C given by the degrees of the tropical hypersurfaces. We also compute the…
Lattice polytope representation of natural numbers is introduced based on the fundamental theorem of arithmetic. The combinatorial and geometric properties of the polytopes are studied using Polymake and Qhull software. The volume of the…
In this paper, we investigate the computational complexity of the knapsack problem and subset sum problem for the following tropical algebraic structures. We consider the semigroup of square matrices of size $k \times k$ with non-negative…
The Catalan triangle, as well as a Fuss-Catalan triangle, enter a problem of counting particular tied arc diagrams. This setting allows us to prove some combinatorial properties of these triangles.
In a previous work of the authors, a result to algorithmically compute the topology types of the level curves of an algebraic surface, is given. From this result, here we derive applications based on level curves to determine some…