Positiveness of the permanent of 4-dimensional polystochastic matrices of order 4

TL;DR

This paper reviews known results and proves that the permanent of any 4-dimensional polystochastic matrix of order 4 is always positive, confirming a key property in multidimensional matrix theory.

Contribution

It establishes the positiveness of the permanent for all 4-dimensional polystochastic matrices of order 4, a previously unproven property.

Findings

Permanent of every 4D polystochastic matrix of order 4 is > 0

Overview of known results on polystochastic matrices

New proof of positiveness property

Abstract

A nonnegative multidimensional matrix is called polystochastic if the sum of its entries over each line is equal to $1$. In this paper we overview known results on positiveness of the permanent of polystochastic matrices and prove that the permanent of every $4$-dimensional polystochastic matrix of order $4$ is greater than zero.