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

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.