The limit empirical spectral distribution of complex matrix polynomials

We study the empirical spectral distribution (ESD) for complex [Formula: see text] matrix polynomials of degree [Formula: see text] under relatively mild assumptions on the underlying distributions, thus highlighting universality phenomena. In particular, we assume that the entries of each matrix coefficient of the matrix polynomial have mean zero and finite variance, potentially allowing for distinct distributions for entries of distinct coefficients. We derive the almost sure limit of the ESD in two distinct scenarios: (1) [Formula: see text] with [Formula: see text] constant and (2) [Formula: see text] with [Formula: see text] bounded by [Formula: see text] for some [Formula: see text]; the second result additionally requires that the underlying distributions are continuous and uniformly bounded. Our results are universal in the sense that they depend on the choice of the variances and possibly on [Formula: see text] (if it is kept constant), but not on the underlying distributions. The results can be specialized to specific models by fixing the variances, thus obtaining matrix polynomial analogues of results known for special classes of scalar polynomials, such as Kac, Weyl, elliptic and hyperbolic polynomials.

Hodge-Index Type Inequalities, Hyperbolic Polynomials, and Complex Hessian Equations

It is noted that using complex Hessian equations and the concavity inequalities for elementary symmetric polynomials implies a generalized form of Hodge index inequality. Inspired by this result, using Gårding’s theory for hyperbolic polynomials, we obtain a mixed Hodge-index type theorem for classes of type $(1,1)$. The new feature is that this Hodge-index type theorem holds with respect to mixed polarizations in which some satisfy particular positivity condition but could be degenerate and even negative along some directions.

A continuity result on quadratic matings with respect to parameters of odd denominator rationals

In this paper we prove a continuity result on matings of quadratic lamination maps sp depending on odd denominator rationals p ∈(0,1). One of the two mating components is fixed in the result. Note that our result has its implication on continuity of matings of quadratic hyperbolic polynomials fc(z)=z2 + c, c ∈ M the Mandelbrot set with respect to the usual parameters c. This is because every quadratic hyperbolic polynomial in M is contained in a bounded hyperbolic component. Its center is Thurston equivalent to some quadratic lamination map sp, and there are bounds on sizes of limbs of M and on sizes of limbs of the mating components on the quadratic parameter slice Perm′(0).

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.