Related papers: Tropical Plane Geometric Constructions: a Transfer…
We show that points in the intersection of the tropicalizations of subvarieties of a torus lift to algebraic intersection points with expected multiplicities, provided that the tropicalizations intersect in the expected dimension. We also…
We introduce a new method to evaluate algebraic integrals over the simplex numerically. This new approach employs techniques from tropical geometry and exceeds the capabilities of existing numerical methods by an order of magnitude. The…
Geometrization of physical theories have always played an important role in their analysis and development. In this contribution we discuss various aspects concerning the geometrization of physical theories: from classical mechanics to…
This paper proposes the use of combinatorial techniques from tropical geometry to build the 120 tritangent planes to a given smooth algebraic space sextic. Although the tropical count is infinite, tropical tritangents come in 15 equivalence…
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…
We introduce new invariants of the projective plane (and, more generally, of certain toric surfaces) that arise from the appropriate enumeration of real elliptic curves. These invariants admit a refinement (according to the quantum index)…
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…
A "tropical ideal" is an ideal in the idempotent semiring of tropical polynomials that is also, degree by degree, a tropical linear space. We introduce a construction based on transversal matroids that canonically extends any principal…
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…
We study the stationary descendant Gromov-Witten theory of toric surfaces by combining and extending a range of techniques - tropical curves, floor diagrams, and Fock spaces. A correspondence theorem is established between tropical curves…
Cylindrical algebraic decomposition is a classical construction in real algebraic geometry. Although there are many algorithms to compute a cylindrical algebraic decomposition, their practical performance is still very limited. In this…
This is a follow-up paper of arXiv:1805.00115, where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in arXiv:1509.07453 allowed us to…
We define tropical analogues of the notions of linear space and Plucker coordinate and study their combinatorics. We introduce tropical analogues of intersection and dualization and define a tropical linear space built by repeated…
In this contribution, a mathematical framework is constructed to relate and compare non-linear partial differential equations (PDEs) in the category of smooth manifolds. In particular, it can be used to compare those aspects of field…
In this survey, we discuss linear series on tropical curves and their relation to classical algebraic geometry, describe the main techniques of the subject, and survey some of the recent major developments in the field, with an emphasis on…
We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as…
This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial…
We use the tropical geometry approach to compute absolute and relative Gromov-Witten invariants of complex surfaces which are $\CC P^1$-bundles over an elliptic curve. We also show that the tropical multiplicity used to count curves can be…
This basic introduction to tropical geometry is hopefully accessible to a first years student in mathematics. The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro's patchworking.…
With the use of mathematical techniques of tropical geometry, it was shown by Mikhalkin some twenty years ago that certain Gromov-Witten invariants associated with topological quantum field theories of pseudoholomorphic maps can be computed…