12 research outputs found
Matrices commuting with a given normal tropical matrix
Consider the space of square normal matrices over
, i.e., and .
Endow with the tropical sum and multiplication .
Fix a real matrix and consider the set of matrices
in which commute with . We prove that is a finite
union of alcoved polytopes; in particular, is a finite union of
convex sets. The set of such that is
also a finite union of alcoved polytopes. The same is true for the set
of such that .
A topology is given to . Then, the set is a
neighborhood of the identity matrix . If is strictly normal, then
is a neighborhood of the zero matrix. In one case, is
a neighborhood of . We give an upper bound for the dimension of
. We explore the relationship between the polyhedral complexes
, and , when and commute. Two matrices,
denoted and , arise from , in connection with
. The geometric meaning of them is given in detail, for one example.
We produce examples of matrices which commute, in any dimension.Comment: Journal versio
Convexity of tropical polytopes
We study the relationship between min-plus, max-plus and Euclidean convexity
for subsets of . We introduce a construction which associates to
any max-plus convex set with compact projectivisation a canonical matrix called
its dominator. The dominator is a Kleene star whose max-plus column space is
the min-plus convex hull of the original set. We apply this to show that a set
which is any two of (i) a max-plus polytope, (ii) a min-plus polytope and (iii)
a Euclidean polytope must also be the third. In particular, these results
answer a question of Sergeev, Schneider and Butkovic and show that row spaces
of tropical Kleene star matrices are exactly the "polytropes" studied by Joswig
and Kulas.Comment: 12 page
Enumerating Polytropes
Polytropes are both ordinary and tropical polytopes. We show that tropical
types of polytropes in are in bijection with cones of a
certain Gr\"{o}bner fan in restricted
to a small cone called the polytrope region. These in turn are indexed by
compatible sets of bipartite and triangle binomials. Geometrically, on the
polytrope region, is the refinement of two fans: the fan of
linearity of the polytrope map appeared in \cite{tran.combi}, and the bipartite
binomial fan. This gives two algorithms for enumerating tropical types of
polytropes: one via a general Gr\"obner fan software such as \textsf{gfan}, and
another via checking compatibility of systems of bipartite and triangle
binomials. We use these algorithms to compute types of full-dimensional
polytropes for , and maximal polytropes for .Comment: Improved exposition, fixed error in reporting the number maximal
polytropes for , fixed error in definition of bipartite binomial
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 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 , 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 Ehrhart theory and tropical volume
We introduce a novel intrinsic volume concept in tropical geometry. This is achieved by developing the foundations of a tropical analog of lattice point counting in polytopes. We exhibit the basic properties and compare it to existing measures. Our exposition is complemented by a brief study of arising complexity questions
Distances on the tropical line determined by two points
Let . Write if is a multiple of
. Two different points and in uniquely
determine a tropical line , passing through them, and stable under
small perturbations. This line is a balanced unrooted semi--labeled tree on
leaves. It is also a metric graph.
If some representatives and of and are the first and second
columns of some real normal idempotent order matrix , we prove that the
tree is described by a matrix , easily obtained from . We also
prove that is caterpillar. We prove that every vertex in
belongs to the tropical linear segment joining and . A vertex, denoted
, closest (w.r.t tropical distance) to exists in . Same for
. The distances between pairs of adjacent vertices in and the
distances \dd(p,pq), \dd(qp,q) and \dd(p,q) are certain entries of the
matrix . In addition, if and are generic, then the tree
is trivalent. The entries of are differences (i.e., sum of principal
diagonal minus sum of secondary diagonal) of order 2 minors of the first two
columns of .Comment: New corrected version. 31 pages and 9 figures. The main result is
theorem 13. This is a generalization of theorem 7 to arbitrary n. Theorem 7
was obtained with A. Jim\'enez; see Arxiv 1205.416