6 problems

A $d$-dimensional polystochastic matrix of order $n$ is a nonnegative $d$-dimensional array whose entries on every line sum to $1$. Its permanent is the sum of the products selecte…

Let $\Omega_1(d,n)$ denote the set of $1$-polystochastic $d$-dimensional matrices of order $n$, let $\operatorname{per_1}(A)$ denote the permanent, and let $J_n^d$ be the $d$-dimen…

A $d$-dimensional permutation of order $n$ is a $d$-dimensional $0$-$1$ array in which every line contains exactly one $1$. Its permanent counts the transversals of the correspondi…

Let $\Omega_1(d,n)$ be the set of $1$-polystochastic $d$-dimensional matrices of order $n$, and let a $(d-1)$-permutation matrix be an element of the corresponding set of $(d-1)$-p…

Let $\Omega_1(d,n)$ be the polytope of $1$-polystochastic $d$-dimensional matrices of order $n$, and let a local extremum mean a local maximum or local minimum of the permanent on…

Let $A$ be a $d$-dimensional matrix of order $n$, let $\Lambda_{d-1}(d,n)$ denote the set of $(d-1)$-permutation matrices, let $\operatorname{Hull}(\Lambda_{d-1}(d,n))$ be their co…