Related papers: Second main theorem with tropical hypersurfaces an…

Given an algebraic variety defined over a discrete valuation field and a skeleton of its Berkovich analytification, the tropicalization process transforms function field of the variety to a semifield of tropical functions on the skeleton.…

We consider the question of when points in tropical affine space uniquely determine a tropical hypersurface. We introduce a notion of multiplicity of points so that this question may be meaningful even if some of the points coincide. We…

This is a foundational paper in tropical linear algebra, which is linear algebra over the min-plus semiring. We introduce and compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the…

We consider the classical Picard's problem for non-parabolic complete K\"ahler manifolds with non-negative Ricci curvature. Based on the global Green function approach, we give a positive answer to Picard's problem under certain condition…

We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and…

We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as $\mathbb{T} = (\mathbb{R}\cup…

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 the first part of a series of three papers. In this paper, we establish a Big Picard type theorem for holomorphic maps $f:Y \to X$, where $Y$ is a ramified covering of the punctured disc $\mathbb{D}^*$ with small ramification and…

Let $V$ be a projective subvariety of $\mathbb P^n(\mathbb C)$. A family of hypersurfaces $\{Q_i\}_{i=1}^q$ in $\mathbb P^n(\mathbb C)$ is said to be in $N$-subgeneral position with respect to $V$ if for any $1\le i_1<\cdots <i_{N+1}$, $…

Tropical roots of tropical polynomials have been previously studied and used to localize roots of classical polynomials and eigenvalues of matrix polynomials. We extend the theory of tropical roots from tropical polynomials to tropical…

The concepts of tropical-semiring and tropical hypersurface, are extended for an arbitrary ordered group. Then, we define the tropicalization of a polynomial with coefficients in a Krull-valued field. After a close study of the properties…

A version of the second main theorem of Nevanlinna theory is proved, where the ramification term is replaced by a term depending on a certain composition operator of a meromorphic function of small hyper-order. As a corollary of this result…

We prove that the rational function semifield of a tropical curve is finitely generated as a semifield over the tropical semifield $\boldsymbol{T} := ( \boldsymbol{R} \cup \{ - \infty \}, \operatorname{max}, +)$ by giving a specific finite…

Let $X = X_\Sigma$ be a toric surface and $(\check{X}, W)$ be its Landau-Ginzburg (LG) mirror where $W$ is the Hori-Vafa potential. We apply asymptotic analysis to study the extended deformation theory of the LG model $(\check{X}, W)$, and…

We consider a version of the symmetric Anderson impurity model (compactified) which has a non-Fermi liquid weak coupling regime. We find that in the Majorana fermion representation, perturbation theory can be conveniently developed in terms…

We investigate Krull dimensions of semirings and semifields dealt in tropical geometry. For a congruence $C$ on a tropical Laurent polynomial semiring $\boldsymbol{T}[X_1^{\pm}, \ldots, X_n^{\pm}]$, a finite subset $T$ of $C$ is called a…

In this paper, we prove some difference analogue of second main theorems of meromorphic mapping from Cm into an algebraic variety V intersecting a finite set of fixed hypersurfaces in subgeneral position. As an application, we prove a…

In this paper, we study the growth, in terms of the Nevanlinna characteristic function, of meromorphic solutions of three types of second order nonlinear algebraic ordinary differential equations. We give all their meromorphic solutions…

In this paper, we establish a second main theorem for holomorphic curve intersecting hypersurfaces in general position in projective space with level of truncation. As an application, we reduce the number hypersurfaces in uniqueness problem…

Let $f$ be a meromorphic function on the complex plane $\mathbb C$ with the maximum function of its modulus $M(r,f)$ on circles centered at zero of radius $r$. A number of classical, well-known and widely used results allow us to estimate…