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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The distribution of solutions to equations over finite fields

Author: Todd Cochrane
Journal: Trans. Amer. Math. Soc. 293 (1986), 819-826
MSC: Primary 11T41; Secondary 11D72
MathSciNet review: 816328
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Abstract: Let $ {\mathbb{F}_q}$ be the finite field in $ q = {p^f}$ elements, $ \underline F (\underline x )$ be a $ k$-tuple of polynomials in $ {\mathbb{F}_q}[{x_1}, \ldots ,{x_n}]$, $ V$ be the set of points in $ \mathbb{F}_q^n$ satisfying $ \underline F (\underline x ) = \underline 0 $ and $ S$, $ T$ be any subsets of $ \mathbb{F}_q^n$. Set $ \phi (V,\underline 0 ) = \vert V\vert - {q^{n - k}}$,

$\displaystyle \phi (V,\underline y ) = \sum\limits_{\underline x \in V} {e\left... ...ot \underline y )} \right)\quad {\text{for}}\;\underline y \ne \underline 0 ,} $

and $ \Phi (V) = {\max _{\underline y }}\vert\phi (V,\underline y )\vert$. We use finite Fourier series to show that $ (S + T) \cap V$ is nonempty if $ \vert S\vert\vert T\vert > {\Phi ^2}(V){q^{2k}}$. In case $ q = p$ we deduce from this, for example, that if $ C$ is a convex subset of $ {\mathbb{R}^n}$ symmetric about a point in $ {\mathbb{Z}^n}$, of diameter $ < 2p$ (with respect to the sup norm), and $ \operatorname{Vol} (C) > {2^{2n}}\Phi (V){p^k}$, then $ C$ contains a solution of $ \underline F (\underline x ) \equiv \underline 0 (\bmod p)$.

We also show that if $ B$ is a box of points in $ \mathbb{F}_q^n$ not contained in any $ (n - 1)$-dimensional subspace and $ \vert B\vert > 4 \cdot {2^{nf}}\Phi (V){q^k}$, then $ B \cap V$ contains $ n$ linearly independent points.

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Article copyright: © Copyright 1986 American Mathematical Society