Converse characterization by equioscillation or stationarity

Determine whether a harmonic cubic whose spherical critical values are only \(\pm M\) and \(0\) must be a Hsiang cubic, and whether a stationary harmonic cubic with an infinite extremal set must be a Hsiang cubic.

Background

Hsiang cubics satisfy a strong equioscillation property: their critical values on the sphere lie in {−M,0,M}\{-M,0,M\}, and their extremal set is the normalized idempotent variety. They are also stationary for the Chebyshev-ratio minimization whenever an appropriate positive idempotent measure exists.

The paper does not establish whether these analytic properties characterize Hsiang cubics. The first question asks whether the critical-value pattern alone forces the Hsiang equation; the second asks whether stationarity together with an infinite extremal set forces the same algebraic structure.

References

Whether, conversely, a harmonic cubic whose critical values on the sphere are $\pm M$ and $0$ is necessarily a Hsiang cubic is open (\Cref{prob:converse}).

— Hsiang cubics as spherical Chebyshev polynomials: equioscillation and idempotent quadrature  (2610.13532 - Tkachev, 9 Oct 2026) in Section 5, Remark following Theorem \ref{the:equi}; Section 7, Problem \ref{prob:converse}