A logical implication between two conjectures on matrix permanents

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2025-07-10 DOI:10.1016/j.laa.2025.07.011

Léo Pioge , Kamil K. Pietrasz , Benoit Seron , Leonardo Novo , Nicolas J. Cerf

引用次数: 0

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 nontrivial 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. Interestingly, such a relationship between these two mathematical conjectures appears from considerations on their quantum physics implications.

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关于矩阵恒等式的两个猜想之间的逻辑蕴涵

我们证明了巴帕特和桑德关于正半定矩阵恒性的两个老猜想之间的逻辑蕴涵。尽管德鲁里最近已经否定了这两个猜想，但这个逻辑暗示在两个看似无关的条件之间产生了一个重要的联系，一个正半定矩阵可能满足这两个条件。作为一个推论，已知服从第一个猜想的矩阵类随后被立即证明服从第二个猜想。相反，我们通过展示第二个猜想的先前未知类型的反例来揭示第一个猜想的新反例。有趣的是，这两个数学猜想之间的这种关系出现在对它们的量子物理含义的考虑中。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

期刊介绍： Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.