Matroids, Algebraic and Non Algebraic
Bernt Lindström · Cambridge University Press eBooks · 1988
INTRODUCTION After the definition of matroids in the 30's a lot of work has been done on vector representations of matroids. Much less work has been devoted to algebraic representations of matroids in the proper sense: after the pioneer work of S. MacLane fifty years ago comes the work of A.W. Ingleton et al. fifteen years ago. Not more than five years ago I decided to try to solve some of the open problems in this neglected field. Recently more people have become interested in it. I intend to give a survey of contributions I know about. First I recall some definitions. For a better introduction I recommend Chapter 11 of the book of Welsh (1976). Let F be a fixed field and K an extension of F. Elements e 1 ,…,e n in K are algebraically dependent over F when there is a non-zero polynomial p(X 1 ,…,X n ) ϵ F[X] such that p(e 1 …,e n ) = 0. If E is a finite subset of K then the algebraically independent subsets of E over F give the independent sets of a matroid M(E). Such a matroid is called algebraic. The rank r(A) of a subset A ⊆ E in this matroid is the transcendence degree tr.d. F F(A) of the field F(A) over F.