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
Full-text PDF Free Access
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.
-  A. W. Ingleton, Representation of matroids, Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), Academic Press, London, 1971, pp. 149–167. MR 0278974
-  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
-  Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
-  T. Lazarson, The representaiton problem for independence functions, J. London Math. Soc. 33 (1958), 21-25. MR 0098701 (20:5156)
-  D. J. A. Welsh, Matroid theory, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. L. M. S. Monographs, No. 8. MR 0427112
-  David Winter, The structure of fields, Springer-Verlag, New York-Heidelberg, 1974. Graduate Texts in Mathematics, No. 16. MR 0389873
- 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)
- J. Kahn, Characteristic sets of matroids, J. London Math. Soc. 26 (1982), 207-217. MR 675165 (84j:05044)
- S. Lang, Algebra, Addison-Wesley, Reading, Mass., 1971. MR 0197234 (33:5416)
- T. Lazarson, The representaiton problem for independence functions, J. London Math. Soc. 33 (1958), 21-25. MR 0098701 (20:5156)
- D. J. A. Welsh, Matroid theory, Academic Press, London, 1976. MR 0427112 (55:148)
- 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