On the algebraic characteristic set for a class of matroids

Abstract:

The independent sets of an algebraic matroid are sets of algebraically independent transcendentals over a field $k$. If a matroid $M$ is isomorphic to an algebraic matroid the latter is called an algebraic representation of $M$. 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 ${M_p}$ ($p$ a prime) the vector representations which were determined by T. Lazarson long ago. One member of this class, ${M_2}$, is the important Fano matroid which plays a crucial role in many parts of matroid theory. We prove that ${M_p}$ has algebraic representations only over fields of characteristic $p$. The proof depends on derivations in fields. Using derivations we transform an algebraic representation of ${M_p}$ into a vector representation.