Related papers: A Brief Survey on Lattice Zonotopes
It was observed by Bump et al. that Ehrhart polynomials in a special family exhibit properties similar to the Riemann {\zeta} function. The construction was generalized by Matsui et al. to a larger family of reflexive polytopes coming from…
This is an expository introduction to simplicial sets and simplicial homotopy theory with particular focus on relating the combinatorial aspects of the theory to their geometric/topological origins. It is intended to be accessible to…
This chapter combines an introduction and research survey about Schubert varieties. The theme is to combinatorially classify their singularities using a family of polynomial ideals generated by determinants.
The purpose of this note is to give an exposition of some interesting combinatorics and convex geometry concepts that appear in algebraic geometry in relation to counting the number of solutions of a system of polynomial equations in…
In this paper we focus on two new families of polynomials which are connected with exponential polynomials and geometric polynomials. We discuss their generalizations and show that these new families of polynomials and their generalizations…
Many (if not most) of convex polytopes, important for combinatorial and algebraic geometry, are closely related to secondary polytopes of point configurations, or base polytopes of submodular functions, or their numerous variations and…
A lattice polytope is called spanning if its lattice points affinely span the ambient lattice. We show as a corollary to a general result in the Ehrhart theory of lattice polytopes that the $h^*$-vector of a spanning lattice polytope has no…
The h^*-polynomial of a lattice polytope is the numerator of the generating function of the Ehrhart polynomial. Let P be a lattice polytope with h^*-polynomial of degree d and with linear coefficient h^*_1. We show that P has to be a…
A mixed lattice is a lattice-type structure consisting of a set with two partial orderings, and generalizing the notion of a lattice. Mixed lattice theory has previously been studied in various algebraic structures, such as groups and…
The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful;…
We classify general systems of polynomial equations with a single solution, or, equivalently, collections of lattice polytopes of minimal positive mixed volume. As a byproduct, this classification provides an algorithm to evaluate the…
Ellipsoids are a common representation for reachability analysis, because they can be transformed efficiently under affine maps, and allow conservative approximation of Minkowski sums, which let one incorporate uncertainty and linearization…
Neighborly polytopes are those that maximize the number of faces in each dimension among all polytopes with the same number of vertices. Despite their extremal properties they form a surprisingly rich class of polytopes, which has been…
We introduce permutrees, a unified model for permutations, binary trees, Cambrian trees and binary sequences. On the combinatorial side, we study the rotation lattices on permutrees and their lattice homomorphisms, unifying the weak order,…
Hypertoric varieties are quaternionic analogues of toric varieties, important for their interaction with the combinatorics of matroids as well as for their prominent place in the rapidly expanding field of algebraic symplectic and…
There is a simple formula for the Ehrhart polynomial of a cyclic polytope. The purpose of this paper is to show that the same formula holds for a more general class of polytopes, lattice-face polytopes. We develop a way of decomposing any…
A hypertoric variety is a quaternionic analogue of a toric variety. Just as the topology of toric varieties is closely related to the combinatorics of polytopes, the topology of hypertoric varieties interacts richly with the combinatorics…
The purpose of this thesis is to study classical combinatorial objects, such as polytopes, polytopal complexes, and subspace arrangements, using tools that have been developed in combinatorial topology, especially those tools developed in…
A polynomial homotopy is a family of polynomial systems in one parameter, which defines solution paths starting from known solutions and ending at solutions of a system that has to be solved. We consider paths leading to isolated singular…
Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this paper we explore this correspondence to classify smooth lattice polytopes…