Distribution results on polynomials with bounded roots

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 Pethő 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 v2s(s)v2s(0)=22s(s-1)2ss\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{v_{2s}^{(s)}}{v_{2s}^{(0)}} = 2^{2s(s-1)}\\left( {\\begin{array}{c}2s\\\\ s\\end{array}}\\right) \\end{aligned}$$\\end{document}holds for all sin mathbb {N} and in the general case the quotient v_d^{(s)}/v_d^{(0)} is an integer for all choices din mathbb {N} and sle d/2. We even go beyond that and prove explicit formulæ for v_d^{(s)} / v_d^{(0)} for arbitrary din mathbb {N}, sle d/2. The ingredients of our proofs comprise Selberg type integrals, determinants like the Cauchy double alternant, and partial Hilbert matrices.