A logical implication between two conjectures on matrix permanents

TL;DR

This paper establishes a logical link between two longstanding conjectures on matrix permanents, showing how the validity of one implies the other, and providing new counterexamples for both.

Contribution

It proves a logical implication between two old conjectures on positive semidefinite matrices' permanents, connecting their validity and counterexamples.

Findings

A logical implication between the two conjectures is established.

Classes of matrices satisfying the first conjecture are shown to satisfy the second.

New counterexamples to the first conjecture are identified.

Abstract

We prove a logical implication between two old conjectures stated by Bapat and Sunder about the permanent of positive semidefinite matrices. Although Drury has recently disproved both conjectures, this logical implication yields a non-trivial link between two seemingly unrelated conditions that a positive semidefinite matrix may fulfill. As a corollary, the classes of matrices that are known to obey the first conjecture are then immediately proven to obey the second one. Conversely, we uncover new counterexamples to the first conjecture by exhibiting a previously unknown type of counterexamples to the second conjecture.