arXiv subjects

Kalina Mincheva

Publications and source records attributed to Kalina Mincheva.

18 recordsLinked to original sources

arXiv2026

Integral closure for (additively idempotent) semirings

In commutative ring theory there are multiple equivalent definitions of integrality. These notions diverge when working with idempotent semirings. In this paper we present these different definitions of integrality for semirings and explore the relations between them. As a tool, we prove a Cayley-Hamilton theorem over additively idempotent semirings, which may be of broader interest. In examples, we compute integral closures of coordinate semirings in their total semiring of fractions and integral closures of sub-semirings of coordinate semirings. Such computation gives avenues to defining and understanding the normalization of tropical varieties as well as computing normalization of varieties tropically.

math.AC↗

arXiv2026

Geometric classification of primes modulo a (bend) congruence

In this paper we continue the program to develop the algebraic foundations of tropical (algebraic) geometry. We give strong characterizations of prime congruences containing a given congruence on a toric semiring. We give four applications of this result. (1) We prove an analogue of the strong Nullstellensatz for congruences with finite tropical basis. This extends the existing result of Joó-Mincheva to cases, such as the bend congruence of a tropical(ized) ideal, where the congruence is not finitely generated. (2) We show that, if $I$ is the ideal of an affine variety not contained in the coordinate hyperplanes, then $\mathbb{T}[x_1, \dots, x_n]/\sqrt{\operatorname{Bend}(\operatorname{trop} I)}$ is cancellative. This result has applications to the integral closure (as per Tolliver) of $\mathbb{T}[x_1, \dots, x_n]/\operatorname{Bend}(\operatorname{trop} I)$ which we explore in a forthcoming paper. (3) We show that $\mathbb{T}[x_1, \dots, x_n]/\sqrt{\operatorname{Bend}(\operatorname{trop} I)}$ is the tropical function semiring on $\operatorname{trop} V(I)$, which creates a bridge between the algebraic approach to non-embedded tropicalization in the work of J. Song and the bend congruence approach of Giansiracusa-Giansiracusa and Maclagan-Rincón. (4) As a consequence of one of our lemmas, we describe the closure of a polyhedron in a tropical toric variety even when the polyhedron is not compatible with the fan defining the tropical toric variety.

math.AG↗

arXiv2026

Prime ideals in the Boolean polynomial semiring

In this article, we disprove a conjecture of F. Alarcón and D. Anderson and give a complete classification of the prime ideals in the one variable polynomial semiring with coefficients in Boolean semifield. We group the prime ideals of $\mathbb{B}[x]$ into three classes, indexed by integers.

math.AC↗

arXiv2025

Varieties of prime tropical ideals and the dimension of the coordinate semiring

In this note we study the relationship between ideals and congruences of the tropical polynomial and Laurent polynomial semirings. We show that the variety of a non-zero prime ideal of the tropical (Laurent) polynomial semiring consists of at most one point. We also prove a result relating the dimension of an affine tropical variety and the dimension of its coordinate semiring.

math.AG↗

arXiv2025

Geometric interpretation of valuated term (pre)orders

Valuated term orders are studied for the purposes of Gröbner theory over fields with valuation. The points of a usual tropical variety correspond to certain valuated terms preorders. Generalizing both of these, the set of all ``well-behaved'' valuated term preorders is canonically in bijection with the points of a space introduced in our previous work on tropical adic geometry. In this paper we interpret these points geometrically by explicitly characterizing them in terms of classical polyhedral geometry. This characterization gives a bijection with equivalence classes of flags of polyhedra as well as a bijection with a class of prime filters on a lattice of polyhedral sets. The first of these also classifies valuated term orders. The second bijection is of the same flavor as the bijections from [van der Put and Schneider, 1995] in non-archimedean analytic geometry and indicates that the results of that paper may have analogues in tropical adic geometry.

math.AG↗

arXiv2025

Representation theory over semifields

We study and classify representations of a torsion group $G$ over an idempotent semifield with special attention on the case over the Boolean semifield $\mathbb{B}$. In subsequent work we extend this theory to studying representations of matroids of low rank.

math.RT↗

arXiv2025

Matroidal representations of low rank

We study tropical subrepresentations of the Boolean regular representation $\mathbb{B}[G]$ of a finite group $G$. These are equivalent to the matroids on ground set $G$ for which left-multiplication by each element of $G$ is a matroid automorphism. We completely classify the tropical subrepresentations of $\mathbb{B}[G]$ for rank 3. When $G$ is an abelian group, our approach can be seen as a generalization of Golomb rulers. In doing so, we also introduce an interesting class of matroids obtained from equivalence relations on finite sets.

math.CO↗

arXiv2024

Tropical representations and valuated matroids

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 semiring: weakly free and quasi-free modules. We also investigate subgroups of $\text{GL}_n(K)$ for $K=\mathbb{T}$, $ \mathbb{R}_{\geq 0}$, and automorphisms of weakly free modules and tropical prevarieties defined by tropical linear equations. As an application of our results, we provide an intrinsic description of tropical subrepresentation via certain quasi-free modules, and prove that a tropical subrepresentation is equivalent to a valuated matroidal representation.

math.RT↗

arXiv2024

Tropical adic spaces I: The continuous spectrum of a topological semiring

Towards building tropical analogues of adic spaces, we study certain spaces of prime congruences as a topological semiring replacement for the space of continuous valuations on a topological ring. This requires building the theory of topological idempotent semirings, and we consider semirings of convergent power series as a primary example. We consider the semiring of convergent power series as a topological space by defining a metric on it. We check that, in tropical toric cases, the proposed objects carry meaningful geometric information. In particular, we show that the dimension behaves as expected. We give an explicit characterization of the points in terms of classical polyhedral geometry in a follow up paper.

math.AG↗

arXiv2024

Equivariant vector bundles on toric schemes over semirings

We introduce a notion of equivariant vector bundles on schemes over semirings. We do this by considering the functor of points of a locally free sheaf. We prove that every toric vector bundle on a toric scheme $X$ over an idempotent semifield equivariantly splits as a sum of toric line bundles. We then study the equivariant Picard group $\text{Pic}_G(X)$. Finally, we prove a version of Klyachko's classification theorem for toric vector bundles over an idempotent semifield.

math.AG↗

arXiv2023

Vector bundles on tropical schemes

We define vector bundles for tropical schemes, and explore their properties. The paper largely consists of three parts; (1) we study free modules over zero-sum free semirings, which provide the necessary algebraic background for the theory (2) we relate vector bundles on tropical schemes to topological vector bundles and vector bundles on monoid schemes, and finally (3) we show that all line bundles on a tropical scheme can be lifted to line bundles on a usual scheme in the affine case.

math.AG↗

arXiv2022

$\mathcal{T}$-semiring pairs

We develop a general axiomatic theory of algebraic pairs, which simultaneously generalizes several algebraic structures, in order to bypass negation as much as feasible. We investigate several classical theorems and notions in this setting including fractions, integral extensions, and Hilbert's Nullstellensatz. Finally, we study a notion of growth in this context.

math.RA↗

arXiv2020

Homology of systemic modules

We develop the rudiments of a tropical homology theory, based on "triples" and "systems." Results include a version of Schanuel's lemma, projective dimension, the homology semi-module, and a weak Snake lemma.

math.RA↗

arXiv2019

Projective systemic modules

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 hyperfields), and fuzzy rings. This enables us to prove analogues of classical theorems for tropical and hyperring theory in a unified way. In this context we prove a Dual Basis Lemma and versions of Schanuel's Lemma.

math.AC↗

arXiv2018

Picard groups for tropical toric schemes

From any monoid scheme $X$ (also known as an $\mathbb{F}_1$-scheme) one can pass to a semiring scheme (a generalization of a tropical scheme) $X_S$ by scalar extension to an idempotent semifield $S$. We prove that for a given irreducible monoid scheme $X$ (satisfying some mild conditions) and an idempotent semifield $S$, the Picard group $Pic(X)$ of $X$ is stable under scalar extension to $S$ (and in fact to any field $K$). In other words, we show that the groups $Pic(X)$ and $Pic(X_S)$ (and $Pic(X_K)$) are isomorphic. In particular, if $X_\mathbb{C}$ is a toric variety, then $Pic(X)$ is the same as the Picard group of the associated tropical scheme. The Picard groups can be computed by considering the correct sheaf cohomology groups. We also define the group $CaCl(X_S)$ of Cartier divisors modulo principal Cartier divisors for a cancellative semiring scheme $X_S$ and prove that $CaCl(X_S)$ is isomorphic to $Pic(X_S)$.

math.AG↗

arXiv2018

Computing toric degenerations of flag varieties

We compute toric degenerations arising from the tropicalization of the full flag varieties $\mathcal{F}\ell_4$ and $\mathcal{F}\ell_5$ embedded in a product of Grassmannians. For $\mathcal{F}\ell_4$ and $\mathcal{F}\ell_5$ we compare toric degenerations arising from string polytopes and the FFLV polytope with those obtained from the tropicalization of the flag varieties. We also present a general procedure to find toric degenerations in the cases where the initial ideal arising from a cone of the tropicalization of a variety is not prime.

math.AG↗

arXiv2017

Prime congruences of idempotent semirings and a Nullstellensatz for tropical polynomials

A new definition of prime congruences in additively idempotent semirings is given using twisted products. This class turns out to exhibit some analogous properties to the prime ideals of commutative rings. In order to establish a good notion of radical congruences it is shown that the intersection of all primes of a semiring can be characterized by certain twisted power formulas. A complete description of prime congruences is given in the polynomial and Laurent polynomial semirings over the tropical semifield ${\pmb T}$, the semifield $\mathbb{Z}_{max}$ and the two element semifield $\mathbb{B}$. The minimal primes of these semirings correspond to monomial orderings, and their intersection is the congruence that identifies polynomials that have the same Newton polytope. It is then shown that every finitely generated congruence in each of these cases is an intersection of prime congruences with quotients of Krull dimension $1$. An improvement of a result of A. Bertram and R. Easton from 2013 is proven which can be regarded as a Nullstellensatz for tropical polynomials.

math.AC↗

arXiv2015

On the dimension of polynomial semirings

In our previous work, motivated by the study of tropical polynomials, a definition for prime congruences was given for an arbitrary commutative semiring. It was shown that for additively idempotent semirings this class exhibits some analogous properties to prime ideals in ring theory. The current paper focuses on the resulting notion of Krull dimension, which is defined as the length of the longest chain of prime congruences. Our main result states that for any additively idempotent semiring $A$, the semiring of polynomials $A[x]$ and the semiring of Laurent polynomials $A(x)$, we have $\dim A[x] = \dim A(x) = \dim A + 1$.

math.AC↗