Nuij type pencils of hyperbolic polynomials

Nuij’s theorem states that if a polynomial p ∈ ℝ[z] is hyperbolic (i.e., has only real roots), then p+sp'' is also hyperbolic for any s ∈ ℝ. We study other perturbations of hyperbolic polynomials of the form pa(z, s) := . We give a full characterization of those a = (a1 , . . . , ad ) ∈ ℝd for which pa(z, s) is a pencil of hyperbolic polynomials. We also give a full characterization of those a = (a1 , . . . , ad ) ∈ ℝd for which the associated families pa(z, s) admit universal determinantal representations. In fact, we show that all these sequences come fromspecial symmetric Toeplitz matrices.