  A $d$-dimensional matrix is called \emph{$1$-polystochastic} if it is
non-negative and the sum over each line equals~$1$. Such a matrix that has a
single $1$ in each line and zeros elsewhere is called a \emph{$1$-permutation}
matrix. A \emph{diagonal} of a $d$-dimensional matrix of order $n$ is a choice
of $n$ elements, no two in the same hyperplane. The \emph{permanent} of a
$d$-dimensional matrix is the sum over the diagonals of the product of the
elements within the diagonal.
  For a given order $n$ and dimension $d$, the set of $1$-polystochastic
matrices forms a convex polytope that includes the $1$-permutation matrices
within its set of vertices. For even $n$ and odd $d$, we give a construction
for a class of $1$-permutation matrices with zero permanent. Consequently, we
show that the set of $1$-polystochastic matrices with zero permanent contains
at least $n^{n^{3/2}(1/2-o(1))}$ $1$-permutation matrices and contains a
polytope of dimension at least $cn^{3/2}$ for fixed $c,d$ and even
$n\to\infty$. We also provide counterexamples to a conjecture by Taranenko
about the location of local extrema of the permanent.
  For odd $d$, we give a construction of $1$-permutation matrices that
decompose into a convex linear sum of positive diagonals. These combine with a
theorem of Taranenko to provide counterexamples to a conjecture by Dow and
Gibson generalising van der Waerden's conjecture to higher dimensions.
