Nonnegativity of the semi-axis expressions for roots of the degree-19 polynomial

Determine whether the quantities R_2(r) and T_2(r), defined in terms of the parameters a, s, and a real root r of the polynomial v(t)=0 in Theorem 4.1, are always non-negative for every a>0 and s>0.

Background

Section 4 characterizes when the numerical range of the five-dimensional tridiagonal matrix A(5,s,a,r) is a circular or non-degenerate elliptical disk centered at the origin. The characterization involves a degree-19 polynomial v(t), together with the quantities R_2(r) and T_2(r), which determine the relevant horizontal and vertical semi-axis parameters when beta_5(r)>0.

The theorem establishes the elliptical-disk conclusion when a real root r of v(t) satisfies R_2(r)>=0 and T_2(r)>=0, or when beta_5(r)=0. The subsequent question asks whether the nonnegativity conditions on R_2(r) and T_2(r) are automatic for every positive a and s and every real root of v(t), which would simplify the characterization in Theorem 4.1.