  Let us fix a prime $p$. The Erd\H{o}s-Ginzburg-Ziv problem asks for the
minimum integer $s$ such that any collection of $s$ points in the lattice
$\mathbb{Z}^n$ contains $p$ points whose centroid is also a lattice point in
$\mathbb{Z}^n$. For large $n$, this is essentially equivalent to asking for the
maximum size of a subset of $\mathbb{F}_p^n$ without $p$ distinct elements
summing to zero.
  In this paper, we give a new upper bound for this problem for any fixed prime
$p\geq 5$ and large $n$. In particular, we prove that any subset of
$\mathbb{F}_p^n$ without $p$ distinct elements summing to zero has size at most
$C_p\cdot \left(2\sqrt{p}\right)^n$, where $C_p$ is a constant only depending
on $p$. For $p$ and $n$ going to infinity, our bound is of the form
$p^{(1/2)\cdot (1+o(1))n}$, whereas all previously known upper bounds were of
the form $p^{(1-o(1))n}$ (with $p^n$ being a trivial bound).
  Our proof uses the so-called multi-colored sum-free theorem which is a
consequence of the Croot-Lev-Pach polynomial method. This method and its
consequences were already applied by Naslund as well as by Fox and the author
to prove bounds for the problem studied in this paper. However, using some key
new ideas, we significantly improve their bounds.
