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

DOI:
https://doi.org/10.1090/S0002-9939-1985-0796464-6

MathSciNet review:
796464

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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 [**2**]). 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.

**[1]**A. W. Ingleton,*Representation of matroids*, Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), Academic Press, London, 1971, pp. 149–167. MR**0278974****[2]**Jeff Kahn,*Characteristic sets of matroids*, J. London Math. Soc. (2)**26**(1982), no. 2, 207–217. MR**675165**, https://doi.org/10.1112/jlms/s2-26.2.207**[3]**Serge Lang,*Algebra*, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR**0197234****[4]**T. Lazarson,*The representation problem for independence functions*, J. London Math. Soc.**33**(1958), 21–25. MR**0098701**, https://doi.org/10.1112/jlms/s1-33.1.21**[5]**D. J. A. Welsh,*Matroid theory*, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. L. M. S. Monographs, No. 8. MR**0427112****[6]**David Winter,*The structure of fields*, Springer-Verlag, New York-Heidelberg, 1974. Graduate Texts in Mathematics, No. 16. MR**0389873**

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DOI:
https://doi.org/10.1090/S0002-9939-1985-0796464-6

Article copyright:
© Copyright 1985
American Mathematical Society