A number theoretic problem on the distribution of polynomials with bounded roots

TL;DR

This paper proves a conjecture relating the measure of polynomials with a specific root structure to Legendre polynomials, revealing an unexpected simple formula for the ratio of measures when exactly one pair of roots is complex.

Contribution

It confirms a conjecture by Akiyama and Pethő for the case of one pair of complex conjugate roots and derives a simple explicit formula involving Legendre polynomials.

Findings

The ratio of measures is an integer for all relevant degrees.

The explicit ratio is given by a formula involving Legendre polynomials.

The result confirms a specific case of a broader conjecture.

Abstract

Let $\mathcal{E}_d^{(s)}$ denote the set of coefficient vectors $(a_1,\dots,a_d)\in \mathbb{R}^d$ of contractive polynomials $x^d+a_1x^{d-1}+\dots+a_d\in \mathbb{R}[x]$ that have exactly $s$ pairs of complex conjugate roots and let $v_d^{(s)}=\lambda_d(\mathcal{E}_d^{(s)})$ be its ($d$-dimensional) Lebesgue measure. We settle the instance $s=1$ of a conjecture by Akiyama and Peth\H{o}, stating that the ratio $v_d^{(s)}/v_d^{(0)}$ is an integer for all $d\ge 2s.$ Moreover we establish the surprisingly simple formula $v_d^{(1)}/v_d^{(0)} = (P_d(3)-2d-1)/4,$ where $P_d(x)$ are the Legendre polynomials.