Related papers: Exploring tropical differential equations
We construct polyhedral families of valuations on the branching algebra of a morphism of reductive groups. This establishes a connection between the combinatorial rules for studying a branching problem and the tropical geometry of the…
We explore several facets of tropical subrepresentations of a linear representation of a group over the tropical semifield $\mathbb{T}$. A key role in the study of tropical subrepresentations is played by two types of modules over a…
An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on…
This paper is about the combinatorics of finite point configurations in the tropical projective space or, dually, of arrangements of finitely many tropical hyperplanes. Moreover, arrangements of finitely many tropical halfspaces can be…
The paper describes different approaches to generalize the trapezoidal method to fractional differential equations. We analyze the main theoretical properties and we discuss computational aspects to implement efficient algorithms. Numerical…
We propose a generalization of tropical curves by dropping the rationality and integrality requirements while preserving the balancing condition. An interpretation of such curves as critical points of a certain quadratic functional allows…
We develop the basic theory of projective modules and splitting in the more general setting of systems. Systems provide a common language for most tropical algebraic approaches including supertropical algebra, hyperrings (specifically…
In this note, we propose a novel technique to reduce the algorithmic complexity of neural network training by using matrices of tropical real numbers instead of matrices of real numbers. Since the tropical arithmetics replaces…
This paper surveys {\it tropical modifications}, which have already become a folklore in tropical geometry. Tropical modifications are used in tropical intersection theory, tropical Hodge theory, and in the study of singularities. They…
The objective of this paper is to lay out the algebraic theory of supertropical vector spaces and linear algebra, utilizing the key antisymmetric relation of ``ghost surpasses.''Special attention is paid to the various notions of ``base,''…
We introduce tropical singular intersection homologies (non-GM and GM) with the tropical coefficients on rational polyhedral spaces using their filtrations. We investigate their fundamental properties. In the non-GM case, we give a…
This is the first paper in a series of eight where in the first three we develop a systematic approach to the geometric algebras of multivectors and extensors, followed by five papers where those algebraic concepts are used in a novel…
This document is a slightly expanded version of a series of talks given by J. Giansiracusa at the workshop `Geometry over semirings' at Universitat Aut\`{o}noma de Barcelona in July 2025. In the first lecture we introduce tropical…
We introduce a notion of elliptic differential graded Lie algebra. The class of elliptic algebras contains such examples as the algebra of differential forms with values in endomorphisms of a flat vector bundle over a compact manifold, etc.…
A "tropical ideal" is an ideal in the idempotent semiring of tropical polynomials that is also, degree by degree, a tropical linear space. We introduce a construction based on transversal matroids that canonically extends any principal…
In this paper, the tropical differential Gr\"obner basis is studied, which is a natural generalization of the tropical Gr\"obner basis to the recently introduced tropical differential algebra. Like the differential Gr\"obner basis, the…
This paper proposes a tropical geometry-based edge detection framework that reformulates convolution and gradient computations using min-plus and max-plus algebra. The tropical formulation emphasizes dominant intensity variations,…
Let $K$ be a real closed field with a nontrivial non-archimedean absolute value. We study a refined version of the tropicalization map, which we call real tropicalization map, that takes into account the signs on $K$. We study images of…
In this paper we will present an ongoing project which aims to use model theory as a suitable mathematical setting for studying the formalism of quantum mechanics. We will argue that this approach provides a geometric semantics for such…
The tropical semiring is a semiring of extended real numbers, where the operations of `max' and `+' replace the usual addition and multiplication, respectively. Difference equations obtained from the ultradiscrete limit of discrete…