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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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,*Representations of matroids*, Combinatorial Mathematics and its Applications (D. J. A. Welsh, ed.), Academic Press, London and New York, 1971, pp. 149-169. MR**0278974 (43:4700)****[2]**J. Kahn,*Characteristic sets of matroids*, J. London Math. Soc.**26**(1982), 207-217. MR**675165 (84j:05044)****[3]**S. Lang,*Algebra*, Addison-Wesley, Reading, Mass., 1971. MR**0197234 (33:5416)****[4]**T. Lazarson,*The representaiton problem for independence functions*, J. London Math. Soc.**33**(1958), 21-25. MR**0098701 (20:5156)****[5]**D. J. A. Welsh,*Matroid theory*, Academic Press, London, 1976. MR**0427112 (55:148)****[6]**D. J. Winter,*The structure of fields*, Springer-Verlag, New York-Heidelberg-Berlin, 1974. MR**0389873 (52:10703)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
05B35

Retrieve articles in all journals with MSC: 05B35

Additional Information

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

Article copyright:
© Copyright 1985
American Mathematical Society