Notes on Foregger's conjecture

Abstract

This thesis is devoted to investigation of some properties of the permanent function over the set Omega_n of n-by-n doubly stochastic matrices. It contains some basic properties as well as some partial progress on Foregger's conjecture. CONJECTURE[Foregger] For every n\in N, there exists k=k(n)>1 such that, for every matrix A\in Omega_n, per(A^k)<=per(A).

In this thesis the author proves the following result.

THEOREM For every c>0, n\in N, for all sufficiently large k=k(n,c), for all A\in\Omega_n which minimum nonzero entry exceeds c, per(A^k)<=per(A).

This theorem implies that for every A\in\Omega_n, there exists k=k(n,A)>1 such that per(A^k)<=per(A).