# An explicit incidence theorem in F_p

Research paper by **Harald Andres Helfgott, Misha Rudnev**

Indexed on: **19 Jan '10**Published on: **19 Jan '10**Published in: **Mathematics - Combinatorics**

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#### Abstract

Let $P = A\times A \subset \mathbb{F}_p \times \mathbb{F}_p$, $p$ a prime.
Assume that $P= A\times A$ has $n$ elements, $n<p$. See $P$ as a set of points
in the plane over $\mathbb{F}_p$. We show that the pairs of points in $P$
determine $\geq c n^{1 + {1/267}}$ lines, where $c$ is an absolute constant.
We derive from this an incidence theorem: the number of incidences between a
set of $n$ points and a set of $n$ lines in the projective plane over $\F_p$
($n<\sqrt{p}$) is bounded by $C n^{{3/2}-{1/10678}}$, where $C$ is an absolute
constant.