Comparison of the upper bounds for the extreme points of the polytopes of line-stochastic tensors

Abstract

We call a real multi-dimensional array a {\em tensor} for short. In enumerating vertices of the polytopes of stochastic tensors, different approaches have been used: {(1)} Combinatorial method via Latin squares; {(2)} Analytic (topological) approach by using hyperplanes; {(3)} Computational geometry (polytope theory) approach; and (4) Optimization (linear programming) approach. As all these approaches are worthy of consideration and investigation in the enumeration problem, various bounds have been obtained. This note is to compare the existing upper bounds arose from different approaches.