Bounds on the number of cells and the dimension of the Dressian

TL;DR

This paper establishes upper bounds on the number of cells and the dimension of the Dressian of a matroid, revealing how these quantities grow with the size and rank of the matroid.

Contribution

It provides the first explicit upper bounds on the number of cells and the dimension of the Dressian for general matroids, including uniform matroids.

Findings

Upper bounds on the number of cells in the Dressian grow roughly as binomial coefficients times a logarithmic factor.

The dimension of the Dressian is bounded above by a ratio involving binomial coefficients and the matroid's parameters.

Bounds are comparable to lower bounds from valuations derived from sparse paving matroids.

Abstract

The {\em Dressian} of a matroid $M$ is the set of all valuations of $M$. This Dressian is the support of a polyhedral complex $\mathcal{Dr}(M)$ whose open cells correspond 1-1 with matroid subdivisions of the matroid polytope of $M$. We present upper bounds on the number of cells and the dimension of $\mathcal{Dr}(M)$. For matroids $M$ of rank $r\geq 3$ on $n$ elements we show that $$ \ln\#\mathcal{Dr}(M)\leq {\binom{n}{r}} O\left(\frac{\ln(n)^2}{n}\right)\text{ as }n\rightarrow\infty,\qquad\text{and}\qquad\dim \mathcal{Dr}(M)\leq {\binom{n}{r}}\frac{3}{n-r+3},$$ as well as some more detailed bounds that incorporate structural properties of such $M$. For uniform matroids $M=U(r,n)$, these upper bounds are comparable to lower bounds derived from valuations that are constructed from sparse paving matroids.