Related papers: Shrinking dynamic on multidimensional tropical ser…
We prove that if a multiple trigonometric series is spherically Abel summable everywhere to an everywhere finite function $f(x)$ which is bounded below by an integrable function, then the series is the Fourier series of $f(x)$ if the…
A new multidimensional optimization problem is considered in the tropical mathematics setting. The problem is to minimize a nonlinear function defined on a finite-dimensional semimodule over an idempotent semifield and given by a conjugate…
To quantify the geometric expressivity of transformers, we introduce a tropical geometry framework to characterize their exact spatial partitioning capabilities. By modeling self-attention as a vector-valued tropical rational map, we prove…
We consider toric maximum likelihood estimation over the field of Puiseux series and study critical points of the likelihood function using tropical methods. This problem translates to finding the intersection points of a tropical affine…
After endowing the space of diagrams of probability spaces with an entropy distance, we study its large-scale geometry by identifying the asymptotic cone as a closed convex cone in a Banach space. We call this cone the tropical cone, and…
In this paper we further develop the theory of matrices over the extended tropical semiring. Introducing a notion of tropical linear dependence allows for a natural definition of matrix rank in a sense that coincides with the notions of…
We investigate a canonical extension of a conventional combinatorial notion of reduced divisors to a notion of tropical projections, which can be defined as the unique minimizers of the so-called $B$-pseudonorms with respect to compact…
Complex algebraic varieties become easy piecewise-linear objects after passing to the so-called tropical limit. Geometry of these limiting objects is known as tropical geometry. In this short survey we take a look at motivation and…
Generalizing supertropical algebras, we present a "layered" structure, "sorted" by a semiring which permits varying ghost layers, and indicate how it is more amenable than the "standard" supertropical construction in factorizations of…
A tropical (or min-plus) semiring is a set $\mathbb{Z}$ (or $\mathbb{Z \cup \{\infty\}}$) endowed with two operations: $\oplus$, which is just usual minimum, and $\odot$, which is usual addition. In tropical algebra the vector $x$ is a…
This is a survey on an analogue of tropical convexity developed over the max-min semiring, starting with the descriptions of max-min segments, semispaces, hyperplanes and an account of separation and non-separation results based on…
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…
Tropical convective clouds evolve over a wide range of temporal and spatial scales, and this makes them difficult to simulate numerically. Here, we propose that their statistical properties can be derived within a simplified…
Systems biology uses large networks of biochemical reactions to model the functioning of biological cells from the molecular to the cellular scale. The dynamics of dissipative reaction networks with many well separated time scales can be…
The motivic nearby fiber is an invariant obtained from degenerating a complex variety over a disc. It specializes to the Euler characteristic of the original variety but also contains information on the variation of Hodge structure…
We give an overview of recently implemented polymake features for computations in tropical geometry. The main focus is on explicit examples rather than technical explanations. Our computations employ tropical hypersurfaces, moduli of…
The tropical Nevanlinna theory is Nevanlinna theory for tropical functions or maps over the max-plux semiring by using the approach of complex analysis. The main purpose of this paper is to study the second main theorem with tropical…
It has been proposed that the number of tropical cyclones as a function of the energy they release is a decreasing power-law function, up to a characteristic energy cutoff determined by the spatial size of the ocean basin in which the storm…
We show that the asymptotic behavior of the two main competing notions of rank of a linear series on a tropical curve is governed by asymptotic invariants, closely paralleling the theory of volumes in algebraic geometry. We introduce and…
The growth of tropical geometry has generated significant interest in the tropical semiring in the past decade. However, there are other semirings in tropical algebra that provide more information, such as the symmetrized (max, +),…