  The multivariate integer Chebyshev problem is to find polynomials with
integer coefficients that minimize the supremum norm over a compact set in
$\C^d.$ We study this problem on general sets, but devote special attention to
product sets such as cube and polydisk. We also establish a multivariate analog
of the Hilbert-Fekete upper bound for the integer Chebyshev constant, which
depends on the dimension of space. In the case of single variable polynomials
in the complex plane, our estimate coincides with the Hilbert-Fekete result.
