Book contents
- Frontmatter
- Contents
- Introduction
- List of Talks
- Participants
- Some Recent Combinatorial Applications of Borsuk-Type Theorems
- On Extremal Finite Sets in the Sphere and Other Metric Spaces
- Metric and Geometric Properties of Sets of Permutations
- Infinite Geometric Groups and Sets
- Intersection and Containment Problems Without Size Restrictions
- Distance-Transitive Graphs of Valency k, 8 ≤ k ≤ 13
- Latin Square Determinants
- A Computer Search for a Projective Plane of Order 10
- Matroids, Algebraic and Non Algebraic
- Algebraic Properties of a General Convolution
- Quasi Groups, Association Schemes, and Laplace Operators on Almost Periodic Functions
- Geometric Methods in Group Theory
- Problem Section
Matroids, Algebraic and Non Algebraic
Published online by Cambridge University Press: 05 April 2013
- Frontmatter
- Contents
- Introduction
- List of Talks
- Participants
- Some Recent Combinatorial Applications of Borsuk-Type Theorems
- On Extremal Finite Sets in the Sphere and Other Metric Spaces
- Metric and Geometric Properties of Sets of Permutations
- Infinite Geometric Groups and Sets
- Intersection and Containment Problems Without Size Restrictions
- Distance-Transitive Graphs of Valency k, 8 ≤ k ≤ 13
- Latin Square Determinants
- A Computer Search for a Projective Plane of Order 10
- Matroids, Algebraic and Non Algebraic
- Algebraic Properties of a General Convolution
- Quasi Groups, Association Schemes, and Laplace Operators on Almost Periodic Functions
- Geometric Methods in Group Theory
- Problem Section
Summary
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 e1,…,en in K are algebraically dependent over F when there is a non-zero polynomial p(X1,…,Xn) ϵ F[X] such that p(e1…,en) = 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.
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- Algebraic, Extremal and Metric Combinatorics 1986 , pp. 166 - 174Publisher: Cambridge University PressPrint publication year: 1988
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