On the algebraic characteristic set for a class of matroids
Proc. Amer. Math. Soc. 95 (1985), 147-151
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Abstract: The independent sets of an algebraic matroid are sets of algebraically independent transcendentals over a field . If a matroid is isomorphic to an algebraic matroid the latter is called an algebraic representation of . 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 ). The corresponding problem for algebraic representations is open.
We consider a class of matroids ( a prime) the vector representations which were determined by T. Lazarson long ago. One member of this class, , is the important Fano matroid which plays a crucial role in many parts of matroid theory. We prove that has algebraic representations only over fields of characteristic .
The proof depends on derivations in fields. Using derivations we transform an algebraic representation of into a vector representation.
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- S. Lang, Algebra, Addison-Wesley, Reading, Mass., 1971. MR 0197234 (33:5416)
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- D. J. Winter, The structure of fields, Springer-Verlag, New York-Heidelberg-Berlin, 1974. MR 0389873 (52:10703)
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