Related papers: Algebraic Aspects in Tropical Mathematics

The tropical semifield, i.e., the real numbers enhanced by the operations of addition and maximum, serves as a base of tropical mathematics. Addition is an abelian group operation, whereas the maximum defines an idempotent semigroup…

This paper lays out a foundation for a theory of supertropical algebraic geometry, relying on commutative $\nu$-algebra. To this end, the paper introduces $\mathfrak{q}$-congruences, carried over $\nu$-semirings, whose distinguished ghost…

In this paper we give an elementary proof of the Fundamental Theorem of Algebra for polynomials over the rational tropical semi-ring. We prove that, tropically, the rational numbers are algebraically closed. We provide a simple algorithm…

This paper concerns the notion of a symmetric algebra and its generalization to a quasi-symmetric algebra. We study the structure of these algebras in respect to their hull-kernel regularity and existence of some ideals, especially the…

The main aim of this paper is to show the interconnections between {\L}ukasiewicz logic and algebraic geometry using algebraic, geometric and logical instruments. We continue our investigation into a new algebraic geometry based on…

This survey consists of two parts. Part 1 is devoted to amoebas. These are images of algebraic subvarieties in the complex torus under the logarithmic moment map. The amoebas have essentially piecewise-linear shape if viewed at large.…

We develop the algebraic polynomial theory for "supertropical algebra," as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of "ghost elements," which also play the key role in our…

In the last few years there has been a growing interest towards methods for statistical inference and learning based on computational geometry and, notably, tropical geometry, that is, the study of algebraic varieties over the min-plus…

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…

For $l,n \in \mathbb{N}$ we define tonal partition algebra $P^l_n$ over $\mathbb{Z}[\delta]$. We construct modules $\{ \Delta_{\underline{\mu}} \}_{\underline{\mu}}$ for $P^l_n$ over $\mathbb{Z}[\delta]$, and hence over any integral domain…

We develop basic notions and methods of algebraic geometry over the algebraic objects called hyperrings. Roughly speaking, hyperrings generalize rings in such a way that an addition is `multi-valued'. This paper largely consisits of two…

The main objective of this paper is to show that the notion of type which was developed within the frames of logic and model theory has deep ties with geometric properties of algebras. These ties go back and forth from universal algebraic…

We introduce algebraic structures on the polyvector fields of an algebraic torus that serve to compute multiplicities in tropical and log Gromov-Witten theory while also connecting to the mirror symmetry dual deformation theory of complex…

We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define…

A new tropical plactic algebra is introduced in which the Knuth relations are inferred from the underlying semiring arithmetics, encapsulating the ubiquitous plactic monoid $\mathcal{P}_n$. This algebra manifests a natural framework for…

Real algebraic geometry is the study of semi-algebraic sets, subsets of $\R^k$ defined by Boolean combinations of polynomial equalities and inequalities. The focus of this thesis is to study quantitative results in real algebraic geometry,…

Let $I$ be an ideal of the ring of Laurent polynomials $K[x_1^{\pm1},\ldots,x_n^{\pm1}]$ with coefficients in a real-valued field $(K,v)$. The fundamental theorem of tropical algebraic geometry states the equality…

In tropical geometry, one studies algebraic curves using combinatorial techniques via the tropicalization procedure. The tropicalization depends on a map to an algebraic torus and the combinatorial methods are most useful when the…

In a series of work [Wor22], [Wor21] and [CW20], algebraic capacities were introduced in an algebraic manner for polarized algebraic surfaces and applied to the symplectic embedding problems. In this paper, we give a reformulation of…

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…