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

Peter Kirschenhofer; Mario Weitzer

paper

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

arXiv:1405.1530

Abstract

Let $E_{d}^{(s)}$ denote the set of coefficient vectors $(a_{1}, \dots, a_{d}) \in R^{d}$ of contractive polynomials $x^{d} + a_{1} x^{d - 1} + \dots + a_{d} \in R [x]$ that have exactly $s$ pairs of complex conjugate roots and let $v_{d}^{(s)} = λ_{d} (E_{d}^{(s)})$ be its ( $d$ -dimensional) Lebesgue measure. We settle the instance $s = 1$ of a conjecture by Akiyama and Pethő, stating that the ratio $v_{d}^{(s)} / v_{d}^{(0)}$ is an integer for all $d \geq 2 s .$ Moreover we establish the surprisingly simple formula $v_{d}^{(1)} / v_{d}^{(0)} = (P_{d} (3) - 2 d - 1) /4,$ where $P_{d} (x)$ are the Legendre polynomials.