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arXiv · 1301.1359

The distribution of rational points and polynomial maps on an affine variety over a finite field on average

Kit-Ho Mak·Alexandru Zaharescu
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Abstract

Let $p$ be a prime, let $V/\mathbb{F}_p$ be an absolutely irreducible affine variety inside the affine $r$-space. In this paper, we consider the problem of how often a box $B$ will contain the expected number of points. In particular, we give a lower bound on the volume of $B$ that guarantees almost all translations of $B$ in the $r$-space contain the expected number of points. This shows that the Weil estimate holds in smaller regions in an "almost all" sense.

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Kit-Ho Mak, Alexandru Zaharescu. 2013-09-22. The distribution of rational points and polynomial maps on an affine variety over a finite field on average. https://arxiv.org/abs/1301.1359

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