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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A partition result for algebraic varieties


Author: Aner Shalev
Journal: Proc. Amer. Math. Soc. 111 (1991), 619-624
MSC: Primary 11G25; Secondary 14G15
MathSciNet review: 1042273
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Abstract: Let $ K$ be a finite field. It is shown that, given positive integers $ d$ and $ r$, there exists $ M = M(d,r)$, such that any variety $ V = V(f) \subseteq {K^n}$, defined by a polynomial $ f$ of degree $ d$ in $ n \geq M$ variables over $ K$, can be partitioned into affine subspaces, each of dimension $ r$. This result, relying on a theorem of R. Brauer, holds in fact for many other fields, including algebraically closed fields. It may provide a partial structural explanation to a divisibility phenomenon discovered by J. Ax.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1042273-5
PII: S 0002-9939(1991)1042273-5
Article copyright: © Copyright 1991 American Mathematical Society