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Abstract: 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 [[2]R.B. Bapat, Recent developments and open problems in the theory of permanents, Math. Student 76 (2007) 55–69.]. 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 for which the -permanent (or -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. [Copyright &y& Elsevier]

The Heine-Stieltjes theorem describes the polynomial solutions, ( v, f) such that T( f)= vf, to specific second-order differential operators, T, with polynomial coefficients. We extend the theorem to concern all (nondegenerate) differential operators preserving the property of having only real zeros, thus solving a conjecture of B. Shapiro. The new methods developed are used to describe intricate interlacing relations between the zeros of different pairs of solutions. This extends recent results of Bourget, McMillen and Vargas for the Heun equation and answers their question of how to generalize their results to higher degrees. Many of the results are new even for the classical case. [ABSTRACT FROM AUTHOR]

Abstract: The majorization order on induces a natural partial ordering on the space of univariate hyperbolic polynomials of degree n. We characterize all linear operators on polynomials that preserve majorization, and show that it is sufficient (modulo obvious degree constraints) to preserve hyperbolicity. [ABSTRACT FROM AUTHOR]