Characteristic Sets of Matroids

Matroids are combinatorial structures that generalize the properties of linear independence. But not all matroids have linear representations. Furthermore, the existence of linear representations depends on the characteristic of the fields, and the linear characteristic set is the set of characteristics of fields over which a matroid has a linear representation. The algebraic independence in a field extension also defines a matroid, and also depends on the characteristic of the fields. The algebraic characteristic set is defined in the similar way as the linear characteristic set. The linear representations and characteristic sets are well studied. But the algebraic representations and characteristic sets received much less attention, and the possible algebraic characteristic sets are still not completely known. This dissertation is a study of possible pairs of linear-algebraic characteristic sets of matroids. Furthermore, if a matroid has an algebraic representation over a positive characteristic field, then the matroid can be represented by a particular set of linear matroids in a field of the same characteristic, called Frobenius flock. In this dissertation, we also have studied Frobenius flock representations, and possible flock characteristic sets