A tropical Nullstellensatz

Symmetric Polynomials in Tropical Algebra Semirings

The growth of tropical geometry has generated significant interest in the tropical semiring in the past decade. However, there are other semirings in tropical algebra that provide more information, such as the symmetrized (max, +), Izhakian-Rowen's extended and supertropical semirings. In this paper we identify in which of these upper-bound semirings we can express symmetric polynomials in terms of elementary ones. This allows us to determine the tropical algebra semirings where an analogue of the Fundamental Theorem of Symmetric Polynomials holds and to what extent.

On the difference between `tropical functions' and real-valued functions

I introduce the concept of integral closure for elements and ideals in idempotent semirings, and establish how it corresponds to its namesake in commutative algebra. In the case of free semirings, integral closure can be understood in terms of a certain monoid of convex bodies under Minkowski sum. I argue that integral closure therefore accounts for the difference between `tropical functions' and real functions.

热带 adic 空间 I：拓扑半环的连续谱

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

本文为构建 adic 空间的热带类比，提出以拓扑半环上的素同余空间替代连续赋值空间，建立拓扑幂等半环理论，并以收敛幂级数半环为例验证其几何信息与维数行为。

Solving Linear Systems over Tropical Semirings through Normalization Method and its Applications

In this paper, we introduce and analyze a normalization method for solving a system of linear equations over tropical semirings. We use a normalization method to construct an associated normalized matrix, which gives a technique for solving the system. If solutions exist, the method can also determine the degrees of freedom of the system. Moreover, we present a procedure to determine the column rank and the row rank of a matrix. Flowcharts for this normalization method and its applications are included as well.

Ideals of polynomial semirings in tropical mathematics

We describe the ideals, especially the prime ideals, of semirings of polynomials over layered domains, and in particular over supertropical domains. Since there are so many of them, special attention is paid to the ideals arising from layered varieties, for which we prove that every prime ideal is a consequence of finitely many binomials. We also obtain layered tropical versions of the classical Principal Ideal Theorem and Hilbert Basis Theorem.

On the Maximal Solution of A Linear System over Tropical Semirings

In this paper, we present methods for solving a system of linear equations, $ AX=b $, over tropical semirings. To this end, if possible, we first reduce the order of the system through some row-column analysis, and obtain a new system with fewer equations and variables. We then use the pseudo-inverse of the system matrix to solve the system if solutions exist. Moreover, we propose a new version of Cramer's rule to determine the maximal solution of the system. Maple procedures for computing the pseudo-inverse are included as well.

Linear independence over tropical semirings and beyond

We investigate different notions of linear independence and of matrix rank that are relevant for max-plus or tropical semirings. The factor rank and tropical rank have already received attention, we compare them with the ranks defined in terms of signed tropical determinants or arising from a notion of linear independence introduced by Gondran and Minoux. To do this, we revisit the symmetrization of the max-plus algebra, establishing properties of linear spaces, linear systems, and matrices over the symmetrized max-plus algebra. In parallel we develop some general technique to prove combinator…

Krull-tropical hypersurfaces

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 of the operator "tropicalization" we conclude with an extension of Kapranov's theorem to algebraically closed fields together with a valuation over an ordered group.

置换群作用的热带不变量

Tropical Invariants for Permutation Group Actions

本文研究置换群作用下的热带不变量，证明当群由2-循环生成时，不变多项式子半环有限生成，并展示分离不变量在构造轨道空间嵌入中的应用。

Layered Tropical Mathematics

Generalizing supertropical algebras, we present a "layered" structure, "sorted" by a semiring which permits varying ghost layers, and indicate how it is more amenable than the "standard" supertropical construction in factorizations of polynomials, description of varieties, properties of the resultant, and for mathematical analysis and calculus, in particular with respect to multiple roots of polynomials. Explicit examples and comparisons are given for various sorting semirings such as the natural numbers and the positive rational numbers, and we see how this theory relates to some recent devel…

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 s…

The Tropical Rank of a Tropical Matrix

In this paper we further develop the theory of matrices over the extended tropical semiring. Introducing a notion of tropical linear dependence allows for a natural definition of matrix rank in a sense that coincides with the notions of tropical regularity and invertibility.

上界半环中的对称多项式

Symmetric Polynomials in Upper-bound Semirings

本文研究了上界半环中对称多项式的性质，提出并完善了完全初等上界半环的特征描述。

Subordinate quadratic forms and isometric maps over semirings

The paper expands the theory of quadratic forms on modules over a semiring R, introduced in [12]-[14], especially in the setup of tropical and supertropical algebra. Isometric linear maps induce subordination on quadratic forms, and provide a main tool in our current study. These maps allow lifts and pushdowns of quadratic forms on different modules, preserving basic characteristic properties.

Quadratic and Symmetric Bilinear Forms on Modules with Unique Base Over a Semiring

We study quadratic forms on free modules with unique base, the situation that arises in tropical algebra, and prove the analog of Witt's Cancellation Theorem. Also, the tensor product of an indecomposable bilinear module $(U, γ)$ with an indecomposable quadratic module $(V,q) $ is indecomposable, with the exception of one case, where two indecomposable components arise.

Syzygies over the Polytope Semiring

Tropical geometry and its applications indicate a "theory of syzygies" over polytope semirings. Taking cue from this indication, we study a notion of syzygies over the polytope semiring. We begin our exploration with the concept of Newton basis, an analogue of Gröbner basis that captures the image of an ideal under the Newton polytope map. The image ${\rm New}(I)$ of a graded ideal $I$ under the Newton polytope is a graded sub-semimodule of the polytope semiring. Analogous to the Hilbert series, we define the notion of Newton-Hilbert series that encodes the rank of each graded piece of ${\rm N…

热带微分方程的一般框架

A general framework for tropical differential equations

本文基于幂等半环与幂等微分代数构建热带微分方程的一般框架，定义微分方程及其解集的热带化，显著细化Grigoriev框架，并证明Payne逆极限定理的微分类比，将热带化极限与Berkovich解析化的微分变体联系起来。

Algebraic structures of tropical mathematics

Tropical mathematics often is defined over an ordered cancellative monoid $\tM$, usually taken to be $(\RR, +)$ or $(\QQ, +)$. Although a rich theory has arisen from this viewpoint, cf. [L1], idempotent semirings possess a restricted algebraic structure theory, and also do not reflect certain valuation-theoretic properties, thereby forcing researchers to rely often on combinatoric techniques. In this paper we describe an alternative structure, more compatible with valuation theory, studied by the authors over the past few years, that permits fuller use of algebraic theory especially in underst…

Modules and Lie Semialgebras over Semirings with a Negation Map

In this article, we present the basic definitions of modules and Lie semialgebras over semirings with a negation map. Our main example of a semiring with a negation map is ELT algebras, and some of the results in this article are formulated and proved only in the ELT theory. When dealing with modules, we focus on linearly independent sets and spanning sets. We define a notion of lifting a module with a negation map, similarly to the tropicalization process, and use it to prove several theorems about semirings with a negation map which possess a lift. In the context of Lie semialgebras over sem…

Tropical compound matrix identities

We prove identities on compound matrices in extended tropical semirings. Such identities include analogues to properties of conjugate matrices, powers of matrices and~$\adj(A)\det(A)^{ -1}$, all of which have implications on the eigenvalues of the corresponding matrices. A tropical Sylvester-Franke identity is provided as well.

关于预序半域和半环的比较定理 II

Abstract Vergleichsstellensätze for preordered semifields and semirings II

本文研究预序半域和半环的比较定理，扩展了第一部分的结果，不再假设$1 \ge 0$，并探讨了预序半域中不等式的关系及类型分类。

预序半域和半环的抽象比较定理 I

Abstract Vergleichsstellensätze for preordered semifields and semirings I

本文针对实代数中研究预序半环和半域的新兴子领域，发展了若干基础性结果，提出了两个抽象比较定理，并展示了其如何恢复Strassen的比较定理及Krivine-Kadison-Dubois的经典正性定理。

Supertropical Quadratic Forms I

We initiate the theory of a quadratic form $q$ over a semiring $R$. As customary, one can write $$q(x+y) = q(x) + q(y)+ b(x,y),$$ where $b$ is a companion bilinear form. But in contrast to the ring-theoretic case, the companion bilinear form need not be uniquely defined. Nevertheless, $q$ can always be written as a sum of quadratic forms $q = κ+ ρ,$ where $κ$ is quasilinear in the sense that $κ(x+y) = κ(x) + κ(y),$ and $ρ$ is rigid in the sense that it has a unique companion. In case that $R$ is a supersemifield (cf. Definition 4.1 below) and $q$ is defined on a free $R$-module, we obtain an e…

Proto-exact categories of modules over semirings and hyperrings

\emph{Proto-exact categories}, introduced by Dyckerhoff and Kapranov, are a generalization of Quillen exact categories which provide a framework for defining algebraic K-theory and Hall algebras in a \emph{non-additive} setting. This formalism is well-suited to the study of categories whose objects have strong combinatorial flavor. In this paper, we show that the categories of modules over semirings and hyperrings - algebraic structures which have gained prominence in tropical geometry - carry proto-exact structures. In the first part, we prove that the category of modules over a semiring is e…

加法幺半群中的阿基米德类

Archimedean classes in additive monoids

本文针对加法幂等半环上模中常见的被和吸收子模，推进了对该类子模及更广泛加法幺半群的研究，重点探讨其阿基米德类与商结构。

T-对上的线性代数

Linear algebra over T-pairs

本文研究了T-对上的线性代数，探讨了矩阵秩的相等性问题，并在热带代数及相关领域如超环和模糊环中应用。