A reduction of algebraic representations of matroids
Proc. Amer. Math. Soc. 100 (1987), 388-389
Primary 05B35; Secondary 12F20
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Abstract: We prove the following result conjectured by M. J. Piff in his thesis (1972).
Theorem. Let be a matroid with an algebraic representation over a field , where is transcendental over . Then has an algebraic representation over .
The proof depends on Noether's normalization theorem and the place extension theorem. We obtain the following corollary.
Corollary. If a matroid is algebraic over a field , then any minor of is algebraic over .
Lang, Introduction to algebraic geometry, Interscience
Publishers, Inc., New York-London, 1958. MR 0100591
M. J. Piff, Some problems in combinatorial theory, Ph.D. thesis, Oxford, 1972.
J. A. Welsh, Matroid theory, Academic Press [Harcourt Brace
Jovanovich, Publishers], London-New York, 1976. L. M. S. Monographs, No. 8.
0427112 (55 #148)
- S. Lang, Introduction to algebraic geometry, Interscience, New York, 1958. MR 0100591 (20:7021)
- M. J. Piff, Some problems in combinatorial theory, Ph.D. thesis, Oxford, 1972.
- D. J. A. Welsh, Matroid theory, Academic Press, London, 1976. MR 0427112 (55:148)
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