Solutions to two problems on permanents

TL;DR

This paper resolves two open problems in the theory of permanents, proving a conjecture about the concavity of certain quotients of permanents and characterizing the nonnegativity of alpha-permanents for positive semidefinite matrices.

Contribution

It proves Bapat's conjecture on the concavity of quotients of permanents and completely characterizes the values of alpha for which alpha-permanents are nonnegative.

Findings

Proved Bapat's conjecture using hyperbolic polynomial properties.

Provided a complete solution to the nonnegativity problem of alpha-permanents.

Disproved the previously conjectured answer for the nonnegativity of alpha-permanents.

Abstract

In this note we settle two open problems in the theory of permanents by using recent results from other areas of mathematics. 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 $\alpha$ 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.