Distribution results on polynomials with bounded roots

TL;DR

This paper investigates the measure of polynomial root distributions within the Schur-Cohn region, proving conjectures about ratios of measures for polynomials with specific root configurations using advanced integral and determinant techniques.

Contribution

It proves two conjectures regarding the ratios of measures of subsets of polynomials with bounded roots and specific root types, providing explicit formulas for these ratios.

Findings

The ratio v_{2s}^{(s)}/v_{2s}^{(0)} equals 2^{2s(s-1)} times the binomial coefficient C(2s,s).

The ratio v_d^{(s)}/v_d^{(0)} is an integer for all d and s ≤ d/2.

Explicit formulas for v_d^{(s)}/v_d^{(0)} are derived for all d and s ≤ d/2.

Abstract

For $d \in \mathbb{N}$ the well-known Schur-Cohn region $\mathcal{E}_d$ consists of all $d$-dimensional vectors $(a_1,\ldots,a_d)\in\mathbb{R}^d$ corresponding to monic polynomials $X^d+a_1X^{d-1}+\cdots+a_{d-1}X+a_d$ whose roots all lie in the open unit disk. This region has been extensively studied over decades. Recently, Akiyama and Peth\H{o} considered the subsets $\mathcal{E}_d^{(s)}$ of the Schur-Cohn region that correspond to polynomials of degree $d$ with exactly $s$ pairs of nonreal roots. They were especially interested in the $d$-dimensional Lebesgue measures $v_d^{(s)}:=\lambda_d(\mathcal{E}_d^{(s)})$ of these sets and their arithmetic properties, and gave some fundamental results. Moreover, they posed two conjectures that we prove in the present paper. Namely, we show that in the totally complex case $d=2s$ the formula \[ \frac{v_{2s}^{(s)}}{v_{2s}^{(0)}} = 2^{2s(s-1)}\binom {2s}s \] holds for all $s\in\mathbb{N}$ and in the general case the quotient $v_d^{(s)}/v_d^{(0)}$ is an integer for all choices $d\in \mathbb{N}$ and $s\le d/2$. We even go beyond that and prove explicit formul\ae{} for $v_d^{(s)} / v_d^{(0)}$ for arbitrary $d\in \mathbb{N}$, $s\le d/2$. The ingredients of our proofs comprise Selberg type integrals, determinants like the Cauchy double alternant, and partial Hilbert matrices.