A set of axioms for defining a matroid in terms of its bases is given by the Steinitz exchange lemma. In this paper, we show these axioms are not independent, and find a subcollection defining the same structure.
A special motivation is given by the GraBmann variety and by oriented matroids, where we present improved versions of known results.
A reduction in Steinitz exchange lemma
The set ~' of bases of a linear space, as to the combinatorial viewpoint, is characterized by the Steinitz exchange lemma: Given any two bases B, B'~,~, it holds:
This structure forms a matroid; more precisely, if E= {1,2 ..... n} and 04:9~c2 E verifies the lemma, we say ~' is a matroid on E. This means that, for a given field K, if ff..,cK(,") is the Graflmann variety over K (of rank r on E) and -7~ff .... then ~(E):={{21,22 ..... 2,}=EI3(21,22 ..... 2,}4:0} is a matroid. We recall that Se(~,,, means exactly -7(21,22 ..... 2,}=det(Xzl,Xz2 ..... Xzr ) for some element X = (XI, X2 ..... X,)eK'", which, on the other hand, occurs if and only if p(-7)= 0~for every GraBmann-Plficker polynomial p . n Unsurprisingly, it may happen, for 0 4: ~ e K(r), that ~'(~) is a matroid but ,7 ¢ ft,,,; however, in this case we must have p(-7)4:0 even for some three-termed