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* *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* .

**[1]**Serge Lang,*Introduction to algebraic geometry*, Interscience Publishers, Inc., New York-London, 1958. MR**0100591****[2]**M. J. Piff,*Some problems in combinatorial theory*, Ph.D. thesis, Oxford, 1972.**[3]**D. J. A. Welsh,*Matroid theory*, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. L. M. S. Monographs, No. 8. MR**0427112**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1987-0884485-6

Article copyright:
© Copyright 1987
American Mathematical Society