On the Algebraic Characteristic Set for a Class of Matroids1

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 Mp (p a prime) the vector representations which were determined by T. Lazarson long ago. One member of this class, Mi, is the important Fano matroid which plays a crucial role in many parts of matroid theory. We prove that Mp has algebraic representations only over fields of characteristic p. The proof depends on derivations in fields. Using derivations we transform an algebraic representation of Mp into a vector representation. We will assume that the reader is familiar with the elements of matroid theory and, in particular, with the notion of an algebraic matroid [5, Chapters 1 and 11]. The matroid Mp can be defined by its vector representation over the prime field GF(p), the column vectors of the matrix /l 0 ••• 0 0 1 0 1 ••• 1 1\ 0 1 ■•• 0 0 110 •■• 1 1 1 0 0 ••• 0 0 111 ■■■ 11 0 0 ••• 10 111 ••• 0 1 V0 0 ••• 0 1 1 1 1 •■• 1 0/ with p + 1 rows and 2p + 3 columns. The rank of the matroid Mp is p + 1. Using the vector representation of Mp it is easy to derive an algebraic representation over GF(p) according to [5, Chapter 11.2]. Let Xi,X2,...,Xp+i be algebraically independent transcendentals over GF(p). Let yo = x\ + ■ ■ ■ + xp+i and y i — yo — x¿ for i = 1,... ,p + 1. The elements of Mp are then represented by ii,...,xp+i,yo, j/i,...,yP+i in this order corresponding to the columns of the matrix above. Of course, there are many other algebraic representations of Mp. Let me just mention one when p = 2: replace "+" everywhere in the p¿'s by the composition "*" defined by a*b = (a + b)(l + ab)-1. But are there representations of Mp over Received by the editors September 6, 1984 and, in revised form, November 11, 1984. 1980 Mathematics Subject Classification. Primary 05B35, 12F99. 1 Research supported by the Swedish Natural Science Research Council. ©1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page