Related papers: Working with Tropical Meromorphic Functions of One…
This paper introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e. summation and maximum. Although our framework is combinatorial,…
The theory of tropical series, that we develop here, firstly appeared in the study of the growth of pluriharmonic functions. Motivated by waves in sandpile models we introduce a dynamic on the set of tropical series, and it is…
Tropical Geometry and Mathematical Morphology share the same max-plus and min-plus semiring arithmetic and matrix algebra. In this chapter we summarize some of their main ideas and common (geometric and algebraic) structure, generalize and…
In this paper, we prove normality criteria for families of meromorphic functions involving sharing of a holomorphic function by a certain class of differential polynomials. Results in this paper extends the works of different authors…
We introduce and study a special class of ideals, called tropical ideals, in the semiring of tropical polynomials, with the goal of developing a useful and solid algebraic foundation for tropical geometry. The class of tropical ideals…
A very brief introduction to tropical and idempotent mathematics is presented. Applications to classical mechanics and geometry are especially examined.
In this paper, the tropical Nevanlinna theory is extended for piecewise polynomial continuous functions. By constructing the $n$-th Poisson-Jensen formula, the $n$-th tropical counting, proximity, and characteristic functions are…
We define scrollar invariants of tropical curves with a fixed divisor of rank 1. We examine the behavior of scrollar invariants under specialization, and provide an algorithm for computing these invariants for a much-studied family of…
In this paper, we give an explicit description of tropical cohomology of smooth algebraic varieties over trivially valued fields. We also construct ``monodromy weight'' spectral sequences for tropical cohomology of geometric strictly…
Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces…
This thesis delves into the geometry of abstract tropical curves, exploring their complete linear system and associated tropical submodules. We establish a lower bound on the dimension of tropical submodules in terms of the Baker-Norine…
In the existing literature, many researchers consider the uniqueness of the power of a meromorphic function with its derivative counterpart share certain values or small functions. Here we consider the same problem under the aegis of a more…
In this paper, it is shown that every polynomial function is mixed monotone globally with a polynomial decomposition function. For univariate polynomials, the decomposition functions can be constructed from the Gram matrix representation of…
In the previous works, the rational function semifields of abstract tropical curves were characterized. In this paper, we give a contravariant categorical equivalence between the category of abstract tropical curves with morphisms and the…
We study how geometric properties of tropical convex sets and polytopes, which are of interest in many application areas, manifest themselves in their algebraic structure as modules over the tropical semiring. Our main results establish a…
This paper attempts to describe some of the results obtained in the iteration theory of transcendental meromorphic functions, not excluding the case of entire functions. The reader is not expected to be familiar with the iteration theory of…
We investigate the spectral properties of rooted trees with the intention of improving the currently existing results that deal with this matter. The concept of an assigned rational function is recursively defined for each vertex of a…
Tropical varieties capture combinatorial information about how coordinates of points in a classical variety approach zero or infinity. We present algorithms for computing the rays of a complex and real tropical curve defined by polynomials…
In an isomorphic copy of the ring of symmetric polynomials we study some families of polynomials which are indexed by rational weight vectors. These families include well known symmetric polynomials, such as the elementary, homogeneous, and…
We obtain a measure theoretical characterization of polynomials among rational functions on $\mathbb{P}^1$, which generalizes a theorem of Lopes. Our proof applies both classical and dynamically weighted potential theory.