Ehrhart Series of Polytopes Related to Symmetric Doubly-Stochastic Matrices

Robert Davis

doi:10.37236/4692

Abstract

In Ehrhart theory, the $h^*$-vector of a rational polytope often provides insights into properties of the polytope that may be otherwise obscured. As an example, the Birkhoff polytope, also known as the polytope of real doubly-stochastic matrices, has a unimodal $h^*$-vector, but when even small modifications are made to the polytope, the same property can be very difficult to prove. In this paper, we examine the $h^*$-vectors of a class of polytopes containing real doubly-stochastic symmetric matrices.

Highlights

Denote by Sn the polytope containing all real n × n symmetric matrices with nonnegative entries such that every row and column sum is 2
For a rational polytope P ⊆ Rn of dimension d, consider the counting function LP(m) = |mP ∩ Zn|, where mP is the m-th dilate of P
Let den P denote the least common multiple of the denominators appearing in the coordinates of the vertices of P

Summary

Introduction

Denote by Sn the polytope containing all real n × n symmetric matrices with nonnegative entries such that every row and column sum is 2. 2. For any term ordering, every variable in ISn appears in a degree-two binomial in G. A lattice polytope P ⊆ Rn is called integrally closed if, for every v ∈ mP ∩ Zn, there are m points v1, .

Objectives

Results

Conclusion