Study of the permanent conjecture and some of its generalizations. II

Abstract:

In this paper we investigate in a more systematic manner some of the topics initiated in part I of the paper with same title [5]. More specifically, we study in greater detail the properties of the function $E(y)$ defined in [5] attached to convex polytopes, whose properties in the special case of the space of doubly stochastic matrices are connected with the permanent conjecture. Some close links with Perron-Frobenius theory are developed, and we obtain as a by-product of our study what is, I believe, a new expression for the maximum eigenvalue of a nonnegative matrix, which leads to some new estimates of the same. A final section of the paper investigates some purely algebraic properties of $E(y)$, and we obtain some very interesting information connecting a doubly stochastic matrix and its transversals. In order to keep this paper as self-contained as possible, facts used here drawn from part I are stated with as much explicit detail as possible.