Related papers: A specialization inequality for tropical complexes
In this article we define a natural tropicalization procedure for closed subsets of log-regular varieties in the case of constant coefficients and study its basic properties. This framework allows us to generalize some of Tevelev's results…
We consider optimization problems that are formulated and solved in the framework of tropical mathematics. The problems consist in minimizing or maximizing functionals defined on vectors of finite-dimensional semimodules over idempotent…
We study the perimeter inequality under circular symmetrisation, and we provide a full geometric characterisation of equality cases. A careful inspection of the proof shows that a similar characterisation holds true also for the perimeter…
We introduce adic tropicalizations for subschemes of toric varieties as limits of Gubler models associated to polyhedral covers of the ordinary tropicalization. Our main result shows that Huber's adic analytification of a subscheme of a…
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…
Sparse polynomial systems with vertical coefficient dependencies arise naturally when describing the critical points of optimization problems and, when augmented with linear forms, the steady states of chemical reaction networks. Moreover,…
We present a simple and elementary procedure to sketch the tropical conic given by a degree--two homogeneous tropical polynomial. These conics are trees of a very particular kind. Given such a tree, we explain how to compute a defining…
Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of…
We construct qualitatively new examples of superabundant tropical curves which are non-realizable in genus $3$ and $4$. These curves are in $\mathbb{R}^3$ and $\mathbb{R}^4$ respectively, and have properties resembling canonical embeddings…
We show that the tropicalization of an irreducible variety over a complete or algebraically closed valued field is connected through codimension 1, giving an affirmative answer in all characteristics to a question posed by Einsiedler, Lind,…
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…
We compare strongly recursive tropical linear series as defined by Farkas, Jensen, and Payne with combinatorial limit linear series as defined by Amini and Gierczak. We show that strongly recursive tropical linear series of rank $r$ are…
Each Gr\"obner stratum of a tropical variety is a connected set of points, all of which induce the same initial subscheme. The Gr\"obner stratification is a coarsening of the decomposition into Gr\"obner polyhedra, and has the advantage…
Tropical polyhedra seem to play a central role in static analysis of softwares. These tropical geometrical objects play also a central role in parity games especially mean payoff games and energy games. And determining if an initial state…
For suitable metrics on the locally symmetric space associated to a maximal representation, we prove inequalities between the length of the boundary and the lengths of orthogeodesics that generalize the classical Basmajian's identity from…
We study a tropical linear regression problem consisting in finding the best approximation of a set of points by a tropical hyperplane. We establish a strong duality theorem, showing that the value of this problem coincides with the maximal…
We introduce and study minimal (with respect to inclusion) solutions of systems of tropical linear differential equations. We describe the set of all minimal solutions for a single equation. It is shown that any tropical linear differential…
In this paper, we study the deformation theory of degenerate algebraic curves on singular varieties which appear as the degenerate limit of families of varieties. For this purpose, we systematically develop a new method to calculate the…
For a linear difference equation with the coefficients being computable sequences, we establish algorithmic undecidability of the problem of determining the dimension of the solution space including the case when some additional prior…
In a recent paper, Matthew Baker and Oliver Lorscheid showed that Descartes's Rule of Signs and Newton's Polygon Rule can both be interpreted as multiplicities of polynomials over hyperfields. Hyperfields are a generalization of fields…