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A reduction of algebraic representations of matroids

Author: Bernt Lindström
Journal: Proc. Amer. Math. Soc. 100 (1987), 388-389
MSC: Primary 05B35; Secondary 12F20
MathSciNet review: 884485
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Abstract: We prove the following result conjectured by M. J. Piff in his thesis (1972).

Theorem. Let $ M$ be a matroid with an algebraic representation over a field $ F(t)$, where $ t$ is transcendental over $ F$. Then $ M$ has an algebraic representation over $ F$.

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 $ F$, then any minor of $ M$ is algebraic over $ F$.

References [Enhancements On Off] (What's this?)

  • [1] S. Lang, Introduction to algebraic geometry, Interscience, New York, 1958. MR 0100591 (20:7021)
  • [2] M. J. Piff, Some problems in combinatorial theory, Ph.D. thesis, Oxford, 1972.
  • [3] D. J. A. Welsh, Matroid theory, Academic Press, London, 1976. MR 0427112 (55:148)

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Article copyright: © Copyright 1987 American Mathematical Society

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