A partition result for algebraic varieties

Shalev, Aner

doi:10.1090/S0002-9939-1991-1042273-5

A partition result for algebraic varieties
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Proc. Amer. Math. Soc. 111 (1991), 619-624 Request permission

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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