Linear independence over tropical semirings and beyond
Marianne Akian, Stephane Gaubert, Alexander Guterman
发表信息：
Proceedings of the International Conference on Tropical and Idempotent Mathematics, G.L. Litvinov and S.N. Sergeev Eds, volume 495 in Contemporary Mathematics, pp 1--38, 2009, AMS
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 combinatorial and polynomial identities for matrices over semirings that we illustrate by a number of examples.
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.
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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…
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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.
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.
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…
This paper introduces the foundations of the polynomial algebra and basic structures for algebraic geometry over the extended tropical semiring. Our development, which includes the tropical version for the fundamental theorem of algebra, leads to the reduced polynomial semiring -- a structure that provides a basis for developing a tropical analogue to the classical theory of commutative algebra. The use of the new notion of tropical algebraic com-sets, built upon the complements of tropical algebraic sets, eventually yields the tropical algebraic Nullstellensatz.
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.
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.
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.
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.
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.
We develop notions of valuations on a semiring, with a view toward extending the classical theory of abstract nonsingular curves and discrete valuation rings to this general algebraic setting; the novelty of our approach lies in the implementation of hyperrings to yield a new definition (\emph{hyperfield valuation}). In particular, we classify valuations on the semifield $\mathbb{Q}_{max}$ (the max-plus semifield of rational numbers) and also valuations on the `function field' $\mathbb{Q}_{max}(T)$ (the semifield of rational functions over $\mathbb{Q}_{max}$) which are trivial on $\mathbb{Q}_{…
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…
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…
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…
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…
We interpret a valuation $v$ on a ring $R$ as a map $v: R \to M$ into a so called bipotent semiring $M$ (the usual max-plus setting), and then define a \textbf{supervaluation} $ϕ$ as a suitable map into a supertropical semiring $U$ with ghost ideal $M$ (cf. [IR1], [IR2]) covering $v$ via the ghost map $U \to M$. The set $\Cov(v)$ of all supervaluations covering $v$ has a natural ordering which makes it a complete lattice. In the case that $R$ is a field, hence for $v$ a Krull valuation, we give a complete explicit description of $\Cov(v)$. The theory of supertropical semirings and supervaluati…
\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…
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…
