Tropical Algebraic Sets, Ideals and An Algebraic Nullstellensatz

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.Comment: 21 pages, 2 figure

Kernels in tropical geometry and a Jordan-H\"older Theorem

A correspondence exists between affine tropical varieties and algebraic objects, following the classical Zariski correspondence between irreducible affine varieties and the prime spectrum of the coordinate algebra in affine algebraic geometry. Although in this context the natural analog of the polynomial ring over a field is the polynomial semiring over a semifield (without a zero element), one obtains homomorphic images of coordinate algebras via congruences rather than ideals, which complicates the algebraic theory considerably. In this paper, we pass to the semifield $F(\lambda_1, \dots, \lambda_n)$ of fractions of the polynomial semiring, for which there already exists a well developed theory of kernels, which are normal convex subgroups; this approach enables us to switch the structural roles of addition and multiplication and makes available much of the extensive theory of chains of homomorphisms of groups, including the Jordan-Holder theory. The parallel of the zero set now is the 1-set. These notions are refined in the language of supertropical algebra to $\nu$-kernels and $1^\nu$-sets, lending more precision to the theory. In analogy to Hilbert's celebrated Nullstellensatz which provides a correspondence between radical ideals and zero sets, we develop a correspondence between $1^\nu$-sets and a well-studied class of $\nu$-kernels of the rational semifield called polars, originating from the theory of lattice-ordered groups. This correspondence becomes simpler and more applicable when restricted to a special kind of kernel, called principal, intersected with the kernel generated by $F$. We utilize this theory to study tropical roots in tropical geometry. As an application, we develop composition series and convexity degree, leading to a tropical version of the Jordan-H\"{o}lder theorem.Comment: 58 pages. The Jordan-H\"older Theorem paper is merged into this one, a few more misprints are corrected, and the abstract is improve

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.Comment: 22 page

Algebraic Matroids and Set-theoretic Realizability of Tropical Varieties

To each prime ideal in a polynomial ring over a field we associate an algebraic matroid and show that it is preserved under tropicalization. This gives a necessary condition for a tropical variety to be set-theoretically realizable from a prime ideal. We also show that there are infinitely many Bergman fans that are not set-theoretically realizable as the tropicalization of any ideal.Comment: 4 pages. Minor revisions and changes to the title and abstract. To appear in JCT

Tropical Effective Primary and Dual Nullstellens\"atze

Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows to study properties of mathematical objects such as algebraic varieties and algebraic curves from the computational point of view. This makes it important to study both mathematical and computational aspects of tropical polynomials. In this paper we prove a tropical Nullstellensatz and moreover we show an effective formulation of this theorem. Nullstellensatz is a natural step in building algebraic theory of tropical polynomials and its effective version is relevant for computational aspects of this field. On our way we establish a simple formulation of min-plus and tropical linear dualities. We also observe a close connection between tropical and min-plus polynomial systems

Puiseux power series solutions for systems of equations

We give an algorithm to compute term by term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton's method for plane algebraic curves replacing the Newton polygon by the tropical variety of the ideal generated by the system. As a corollary we deduce a property of tropical varieties of quasi-ordinary singularities.Comment: 19 pages. To appear in International Journal of Mathematics. Major changes: Several sections added explaining the geometrical meaning of the series solutions and a corollary about quasi-ordinary singularitie

Cech Cohomology of Semiring Schemes

A semiring scheme generalizes a scheme in such a way that the underlying algebra is that of semirings. We generalize \v{C}ech cohomology theory and invertible sheaves to semiring schemes. In particular, when $X=\mathbb{P}^n_M$, a projective space over a totally ordered idempotent semifield $M$, we show that \v{C}ech cohomology theory is in agreement with the classical computation. Finally, we classify all invertible sheaves on $X=\mathbb{P}^n_M$ by computing the Picard group of $X$ explicitly.Comment: 14 pages, Section 4 is rewritten with more general result

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.Comment: 18 page

An informal overview of triples and systems

We describe triples and systems, expounded as an axiomatic algebraic umbrella theory for classical algebra, tropical algebra, hyperfields, and fuzzy rings.Comment: 12 pages, improved introduction and reference

The Boundary of Amoebas

The computation of amoebas has been a challenging open problem for the last dozen years. The most natural approach, namely to compute an amoeba via its boundary, has not been practical so far since only a superset of the boundary, the contour, is understood in theory and computable in practice. We define and characterize the extended boundary of an amoeba, which is sensitive to some degenerations that the topological boundary does not detect. Our description of the extended boundary also allows us to distinguish between the contour and the boundary. This gives rise not only to new structural results in amoeba theory, but in particular allows us to compute hypersurface amoebas via their boundary in any dimension. In dimension two this can be done using Gr\"obner bases alone. We introduce the concept of amoeba bases, which are sufficient for understanding the amoeba of an ideal. We show that our characterization of the boundary is essential for the computation of these amoeba bases and we illustrate the potential of this concept by constructing amoeba bases for linear systems of equations.Comment: Minor corrections and improved readability; 26 pages, 5 figure