The permanent and diagonal products on the set of nonnegative matrices with bounded rank

TL;DR

This paper proposes conjectures on the maximum values of the permanent and diagonal products for nonnegative matrices with bounded rank, relating these bounds to row and column sums.

Contribution

It introduces new conjectures linking the permanent and diagonal products to matrix rank, row sums, and column sums, extending understanding of these functions on nonnegative matrices.

Findings

Conjecture that the permanent of a singular nonnegative matrix is bounded by half the minimum of row and column sum products.

Conjecture that the diagonal product of a singular nonnegative matrix is bounded by a quarter of the minimum of row and column sum products.

Provides equivalent formulations of these bounds based on matrix rank, row sums, and column sums.

Abstract

We formulate conjectures regarding the maximum value and maximizing matrices of the permanent and of diagonal products on the set of stochastic matrices with bounded rank. We formulate equivalent conjectures on upper bounds for these functions for nonnegative matrices based on their rank, row sums and column sums. In particular we conjecture that the permanent of a singular nonnegative matrix is bounded by 1/2 times the minimum of the product of its row sums and the product of its column sums, and that the product of the elements of any diagonal of a singular nonnegative matrix is bounded by 1/4 times the minimum of the product of its row sums and the product of its column sums.