Introduction and Basic Definitions

The purpose of this chapter is to provide background and general results concerning coordinatizations, while the more specialized subtopics of binary and unimodular matroids are covered in later chapters. The first section of this chapter is devoted to definitions and notational conventions. The second section concerns linear and projective equivalence of coordinatizations. Although they are not usually explicitly considered in other expositions of matroid coordinatization, these equivalence relations are very useful in working with examples of coordinatizations, as well as theoretically useful as in Proposition 1.2.5. Section 1.3 involves the preservation of coordinatizability under certain standard matroid operations, including duality and minors. The next section presents some well-known counterexamples, and Section 1.5 considers characterizations of coordinatizability, especially characterizations by excluded minors. The final five sections are somewhat more technical in nature, and may be omitted by the reader who desires only an introductory survey. Section 1.6 concerns the bracket conditions, another general characterization of coordinatizability. Section 1.7 presents techniques for construction of a matroid requiring a root of any prescribed polynomial in a field over which we wish to coordinatize it. These techniques are extremely useful in the construction of examples and counterexamples, yet are not readily available in other works, except Greene (1971). The last three sections concern characteristic sets, the use of transcendentals in coordinatizations, and algebraic representation (i.e., modeling matroid dependence by algebraic dependence). Some additional topics which could have been considered here, such as chain groups, are omitted because they are well-covered in other readily available sources, such as Welsh (1976).