Bounds on the number of connected components for tropical prevarieties

For a tropical prevariety in Rn given by a system of k tropical polynomials in n variables with degrees at most d, we prove that its number of connected components is less than k+7n−

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

Upper bounds on Betti numbers of tropical prevarieties

We prove upper bounds on the sum of Betti numbers of tropical prevarieties in dense and sparse settings. In the dense setting the bound is in terms of the volume of Minkowski sum of Newton polytopes of defining tropical polynomials, or, alternatively, via the maximal degree of these polynomials. In sparse setting, the bound involves the number of the monomials.Comment: 9 pages. Upper bounds are slightly improve

Computing Tropical Varieties

The tropical variety of a $d$-dimensional prime ideal in a polynomial ring with complex coefficients is a pure $d$-dimensional polyhedral fan. This fan is shown to be connected in codimension one. We present algorithmic tools for computing the tropical variety, and we discuss our implementation of these tools in the Gr\"obner fan software \texttt{Gfan}. Every ideal is shown to have a finite tropical basis, and a sharp lower bound is given for the size of a tropical basis for an ideal of linear forms.Comment: 22 pages, 2 figure

Tropical Positivity and Semialgebraic Sets from Polytopes

This dissertation presents recent contributions in tropical geometry with a view towards positivity, and on certain semialgebraic sets which are constructed from polytopes. Tropical geometry is an emerging field in mathematics, combining elements of algebraic geometry and polyhedral geometry. A key in establishing this bridge is the concept of tropicalization, which is often described as mapping an algebraic variety to its 'combinatorial shadow'. This shadow is a polyhedral complex and thus allows to study the algebraic variety by combinatorial means. Recently, the positive part, i.e. the intersection of the variety with the positive orthant, has enjoyed rising attention. A driving question in recent years is: Can we characterize the tropicalization of the positive part? In this thesis we introduce the novel notion of positive-tropical generators, a concept which may serve as a tool for studying positive parts in tropical geometry in a combinatorial fashion. We initiate the study of these as positive analogues of tropical bases, and extend our theory to the notion of signed-tropical generators for more general signed tropicalizations. Applying this to the tropicalization of determinantal varieties, we develop criteria for characterizing their positive part. Motivated by questions from optimization, we focus on the study of low-rank matrices, in particular matrices of rank 2 and 3. We show that in rank 2 the minors form a set of positive-tropical generators, which fully classifies the positive part. In rank 3 we develop the starship criterion, a geometric criterion which certifies non-positivity. Moreover, in the case of square-matrices of corank 1, we fully classify the signed tropicalization of the determinantal variety, even beyond the positive part. Afterwards, we turn to the study of polytropes, which are those polytopes that are both tropically and classically convex. In the literature they are also established as alcoved polytopes of type A. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and h^*-polynomials of lattice polytropes. These algorithms are applied to all polytropes of dimensions 2,3 and 4, yielding a large class of integer polynomials. We give a complete combinatorial description of the coefficients of volume polynomials of 3-dimensional polytropes in terms of regular central subdivisions of the fundamental polytope, which is the root polytope of type A. Finally, we provide a partial characterization of the analogous coefficients in dimension 4. In the second half of the thesis, we shift the focus to study semialgebraic sets by combinatorial means. Intersection bodies are objects arising in geometric tomography and are known not to be semialgebraic in general. We study intersection bodies of polytopes and show that such an intersection body is always a semialgebraic set. Computing the irreducible components of the algebraic boundary, we provide an upper bound for the degree of these components. Furthermore, we give a full classification for the convexity of intersection bodies of polytopes in the plane. Towards the end of this thesis, we move to the study of a problem from game theory, considering the correlated equilibrium polytope $P_G$ of a game G from a combinatorial point of view. We introduce the region of full-dimensionality for this class of polytopes, and prove that it is a semialgebraic set for any game. Through the use of oriented matroid strata, we propose a structured method for classifying the possible combinatorial types of $P_G$, and show that for (2 x n)-games, the algebraic boundary of each stratum is a union of coordinate hyperplanes and binomial hypersurfaces. Finally, we provide a computational proof that there exists a unique combinatorial type of maximal dimension for (2 x 3)-games.:Introduction 1. Background 2. Tropical Positivity and Determinantal Varieties 3. Multivariate Volume, Ehrhart, and h^*-Polynomials of Polytropes 4. Combinatorics of Correlated Equilibri

Tropical varieties, maps and gossip

Tropical geometry is a relatively new field of mathematics that studies the tropicalization map: a map that assigns a certain type of polyhedral complex, called a tropical variety, to an embedded algebraic variety. In a sense, it translates algebraic geometric statements into combinatorial ones. An interesting feature of tropical geometry is that there does not exist a good notion of morphism, or map, between tropical varieties that makes the tropicalization map functorial. The main part of this thesis studies maps between different classes of tropical varieties: tropical linear spaces and tropicalizations of embedded unirational varieties. The first chapter is a concise introduction to tropical geometry. It collects and proves the main theorems. None of these results are new. The second chapter deals with tropicalizations of embedded unirational varieties. We give sufficient conditions on such varieties for there to exist a (not necessarily injective) parametrization whose naive tropicalization is surjective onto the associated tropical variety. The third chapter gives an overview of the algebra related to tropical linear spaces. Where fields and vector spaces are the central objects in linear algebra, so are semifields and modules over semifields central to tropical linear algebra and the study of tropical linear spaces. Most results in this chapter are known in some form, but scattered among the available literature. The main purpose of this chapter is to collect these results and to determine the algebraic conditions that suffice to give linear algebra over the semifield a familiar feel. For example, under which conditions are varieties cut out by linear polynomials closed under addition and scalar multiplication? The fourth chapter comprises the biggest part of the thesis. The techniques used are a combination of tropical linear algebra and matroid theory. Central objects are the valuated matroids introduced by Andreas Dress and Walter Wenzl. Among other things the chapter contains a classification of functions on a tropical linear space whose cycles are tropical linear subspaces, extending an old result on elementary extensions of matroids by Henry Crapo. It uses Mikhalkin’s concept of a tropical modification to define the morphisms in a category whose objects are all tropical linear spaces. Finally, we determine the structure of an open submonoid of the morphisms from affine 2-space to itself as a polyhedral complex. Finally, the fifth and last chapter is only indirectly related to maps. It studies a certain monoid contained in the tropicalization of the orthogonal group: the monoid that is generated by the distance matrices under tropical matrix multiplication (i.e. where addition is replaced by minimum, and multiplication by addition). This monoid generalizes a monoid that underlies the well-known gossip problem, to a setting where information is transmitted only with a certain degree accuracy. We determine this so-called gossip monoid for matrices up to size 4, and prove that in general it is a polyhedral monoid of dimension equal to that of the orthogonal group