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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the algebraic characteristic set for a class of matroids


Author: Bernt Lindström
Journal: Proc. Amer. Math. Soc. 95 (1985), 147-151
MSC: Primary 05B35
MathSciNet review: 796464
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0796464-6
Article copyright: © Copyright 1985 American Mathematical Society