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

The { Dressian} of a matroid M is the set of all valuations of M. This Dressian is the support of a polyhedral complex 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 Dr(M). For matroids M of rank r≥ 3 on n elements we show that ln#Dr(M)≤ nr O(ln(n)²/n) as n→∞, Dr(M)≤ nr3/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.