Simplicity of all zeros of D’Arcais polynomials

Prove or disprove that all zeros of every D’Arcais polynomial \(P_n^\sigma(X)\) are simple.

Background

The paper cites a result showing that the root of Pnσ(X)P_n^\sigma(X) with largest absolute value is real and simple. It then reports numerical evidence suggesting that this simplicity property may hold for every zero, but explicitly states that the general assertion remains unresolved. The open problem is therefore to determine whether any D’Arcais polynomial has a multiple zero.

References

Numerical evidence suggests that all zeros are simple but this remains an open question.

— A Positive Proportion of the Reduced D'Arcais Polynomials is not Hurwitz  (2608.18842 - Charlton et al., 19 Aug 2026) in Section 5, “Open challenges”