A logical implication between two conjectures on matrix permanents
Rings and Algebras
2025-08-04 v1 Combinatorics
Quantum Physics
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.
Cite
@article{arxiv.2508.00111,
title = {A logical implication between two conjectures on matrix permanents},
author = {Léo Pioge and Kamil K. Pietrasz and Benoit Seron and Leonardo Novo and Nicolas J. Cerf},
journal= {arXiv preprint arXiv:2508.00111},
year = {2025}
}