On the algebraic characteristic set for a class of matroids
Bernt Lindström · Proceedings of the American Mathematical Society · 1985
The independent sets of an algebraic matroid are sets of algebraically independent transcendentals over a field k k . If a matroid M M is isomorphic to an algebraic matroid the latter is called an algebraic representation of M M . Vector representations of matroids are defined similarly. A matroid may have algebraic (resp. vector) representations over fields of different characteristics. The problem in which characteristic sets are possible for vector representations was recently answered (see [ 2 ]). The corresponding problem for algebraic representations is open. We consider a class of matroids M p {M_p} ( p p a prime) the vector representations which were determined by T. Lazarson long ago. One member of this class, M 2 {M_2} , is the important Fano matroid which plays a crucial role in many parts of matroid theory. We prove that M p {M_p} has algebraic representations only over fields of characteristic p p . The proof depends on derivations in fields. Using derivations we transform an algebraic representation of M p {M_p} into a vector representation.