Do rational component masses of the equilibrium measure characterize polynomial pre-images beyond intervals?

Determine whether, for any compact set E = ∪_{j=1}^ℓ E_j consisting of ℓ pairwise disjoint simply connected compact subsets of the complex plane, the condition that the equilibrium measure μ_E assigns rational masses to each component, i.e., μ_E(E_j) ∈ Q for j = 1, …, ℓ, implies that E is a polynomial pre-image of a simply connected compact set Ω; equivalently, ascertain whether there exist a nonconstant polynomial P and a simply connected compact set Ω ⊂ C such that E = P^{-1}(Ω).

Background

Walsh’s conformal map associates to a compact set E = ∪_{j=1}^ℓ E_j a lemniscatic domain with exponents m_j summing to 1. This paper proves that these exponents are given by the equilibrium measure masses: m_j = μ_E(E_j).

For polynomial pre-images E = P^{-1}(Ω) of simply connected compact sets Ω, it follows that μ_E(E_j) are rational numbers. In the special case where E is a finite union of real intervals, an equivalence is known: E is a polynomial pre-image of [-1,1] if and only if the component masses μ_E(E_j) are rational.

The authors observe that while the implication “polynomial pre-image ⇒ rational component masses” holds for more general compact sets, it is unknown whether the converse implication also holds in this generality. Establishing this would extend the intervals equivalence to arbitrary unions of simply connected compact components.