# Open problems session 1

## Gleb Koshevoy

### Good Exercise:

For any set of vectors $A$ in $\mathbb{Z}^n$, let $H(A)$ be the family of integral polytopes whose facet normals are in $A$ directions and let $Ed(A)$ be the family of integral polytopes whose edge directions are in $A$ directions  (H for Hoffman or Hyperplanes, and Ed for Edmond or Edges).

Let $A_n=\{e_i - e_j : i \neq j \text{ in } \{1,\dots,n\}\} \cup \{\pm e_i : i \in \{1,\dots,n\}\}$, which are edge directions of the simplex with vertices $0,e_1,\dots,e_n$.  It is a maximal unimodular system.  It is easy to see that $H(A_n)$ (also called $L^\#$ convex polytopes or alcoved polytopes) is closed under intersection.  The family $Ed(A_n)$ (also called $M^\#$ convex polytopes or generalized polymatroids or generalized permutohedra) is not closed under intersection (e.g. take two pyramid in an octahedron, which is the matroid polytope of $U_{4,2}$ that do not share a square face).

But $Ed(A_n)$ has the property that the intersection of any two (but not three!) polytopes in the the family is an integral polytope.  We will call this property (*).

Problem (proven by Isedora Heller): Suppose $A$ is a maximal cardinality unimodular system with property (*).  Is $A = A_n$ after a change of lattice basis?

Remark: Let $G$ be a graph and $A$ the matrix whose columns are $e_i - e_j$ for all edges $(i,j)$ in the graph.  We can then construct a graphical matrix $[I | A]$and cographical matrix $[I | A^T]$ of $G$Seymour’s decomposition theorem says that every unimodular system is obtained from graphical matrices, cographical matrices, and an exceptional system $R_{10}$ via two operation $\oplus$, which is putting the matrices (whose columns are the vectors) in a block diagonal form, and $\oplus'$ is like $\oplus'$ with an extra row containing a one and the rest zero.

### Question:

Let $\mathcal{L}$ be a finite collection of some linear subspaces of $\mathbb{Q}^n$.  We say that $\mathcal{L}$ is unimodular if for any subcollection $\{L_1,\dots,L_r\} \subset \mathcal{L}$, the group $\mathbb{Z}^n / (\mathbb{Z}^n \cap L_1+\cdots L_r)$ is torsion-free.

Conjecture: Suppose $A$ is a unimodular set of vectors such that the lines spanned by vectors in $A$ form a maximal unimodular system of subspaces, i.e. we cannot add any other subspaces to the collection while keeping the unimodularity property.  Then $A = A_n$ for some $n$ after a change of lattice basis.

## John Weymark

Intersect the boundary of the sets $Q_a$‘s (see the slides) with the set of valuations, which can be assumed to be convex. Call this intersection $S$

Question: Write $late S$ as the intersection of a tropical hyperplane with the set of valuations.  Find conditions under which this is possible.  How do you find the tropical hyperplane?