相关论文: A tropical version of Hilbert polynomial (in dimen…

The tropical arithmetic operations on $\mathbb{R}$ are defined by $a\oplus b=\min\{a,b\}$ and $a\otimes b=a+b$. Let $A$ be a tropical matrix and $k$ a positive integer, the problem of Tropical Matrix Factorization (TMF) asks whether there…

In this paper we study general tropical linear spaces locally: For any basis B of the matroid underlying a tropical linear space L, we define the local tropical linear space L_B to be the subcomplex of L consisting of all vectors v that…

In previous works, an approach to the study of cyclic functions in reproducing kernel Hilbert spaces has been presented, based on the study of so called \emph{optimal polynomial approximants}. In the present article, we extend such approach…

An unconstrained optimization problem is formulated in terms of tropical mathematics to minimize a functional that is defined on a vector set by a matrix and calculated through multiplicative conjugate transposition. For some particular…

Tropical implicitization means computing the tropicalization of a unirational variety from its parametrization. In the case of a hypersurface, this amounts to finding the Newton polytope of the implicit equation, without computing its…

We present tools and definitions to study abstract tropical manifolds in dimension 2, which we call simply tropical surfaces. This includes explicit descriptions of intersection numbers of 1-cycles, normal bundles to some curves and…

Some results on singularities of plurisubharmonic functions are put into the context of tropical mathematics.

Each metric graph has canonically associated to it a polarized real torus called its tropical Jacobian. A fundamental real-valued invariant associated to each polarized real torus is its tropical moment. We give an explicit and efficiently…

It is well known that for two univariate polynomials over complex number field the number of their common roots is equal to the order of their resultant. In this paper, we show that this fundamental relationship still holds for the tropical…

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…

The algebraic foundation of tropical polynomial algebra provides the framework for the geometric construction of the supplement and the reversal of tropical varieties, thereby inducing a duality of reduced tropical varieties; for classes of…

We consider Hilbert-type functions associated with difference (not necessarily inversive) field extensions and systems of algebraic difference equations in the case when the translations are assigned some integer weights. We will show that…

We explain how to tropicalize scalar quantum field theory and show that tropicalized massive scalar quantum field theory is exactly solvable. This exact solution manifests as a non-linear recursion equation fulfilled by the expansion…

We prove an analogue of Clifford's inequality for tropical curves. Next we focus on the hyperelliptic case and we characterize divisors attaining equality. Finally we speculate whether inequality in tropical Clifford's Theorem does imply…

Let $I$ be a homogeneous ideal in the polynomial ring $R = k[z_1, \cdots, z_n]$ , where $k$ is an algebraically closed field of characteristic zero. Macaulay's Theorem provides constraints on the Hilbert function of $I$ or $R/I$ from one…

We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of…

We consider a binary classifier defined as the sign of a tropical rational function, that is, as the difference of two convex piecewise linear functions. The parameter space of ReLU neural networks is contained as a semialgebraic set inside…

We define multidimensional tropical series, i.e. piecewise linear functions which are tropical polynomials locally but may contain an infinite number of monomials. Tropical series appeared in the study of the growth of pluriharmonic…

Deep neural networks show great success when input vectors are in an Euclidean space. However, those classical neural networks show a poor performance when inputs are phylogenetic trees, which can be written as vectors in the tropical…

We introduce tropical holonomic sequences of a given order and calculate their entropy in case of the second order.