On the number of common bases of two matroids

M1=(E,F1) and M2=(E,F2) are two simple matroids on the same set E. We assume that M1 and M2 have a common basis and we want to know a least bound for the number of common bases of M1 and M2. To solve this problem, we study the dimension of the intersection of the polytopes K(M1) and K(M2); where K(Mi) is the polytope having as extreme points the representative vectors of the bases of M>i,i=1,2. Applications of the least number of common bases of two matroids are given for some combinatorial optimisation problems. © 1983.