Solutions to two problems on permanents
2012 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 436, no 1, 53-58 p.Article in journal (Refereed) Published
In this note we settle two open problems in the theory of permanents by using recent results from other areas of mathematics. Both problems were recently discussed in Bapat's survey . Bapat conjectured that certain quotients of permanents, which generalize symmetric function means, are concave. We prove this conjecture by using concavity properties of hyperbolic polynomials. Motivated by problems on random point processes, Shirai and Takahashi raised the problem: Determine all real numbers a for which the alpha-permanent (or alpha-determinant) is nonnegative for all positive semidefinite matrices. We give a complete solution to this problem by using recent results of Scott and Sokal on completely monotone functions. It turns out that the conjectured answer to the problem is false.
Place, publisher, year, edition, pages
2012. Vol. 436, no 1, 53-58 p.
Permanent, alpha-Permanent, alpha-Determinant, Positivity, Hyperbolic polynomial, Complete monotonicity, Symmetric function mean
IdentifiersURN: urn:nbn:se:kth:diva-53388DOI: 10.1016/j.laa.2011.06.022ISI: 000297431200005ScopusID: 2-s2.0-80055063559OAI: oai:DiVA.org:kth-53388DiVA: diva2:470325
QC 201112282011-12-282011-12-282011-12-28Bibliographically approved