On Harmonic Polynomials Invariant under Unitary Transformations

Unitary transformations and canonical representatives of a family of real-valued harmonic fourth-degree polynomials in three complex variables are studied. The subject relates to the study of Moser normal equations for real hypersurfaces of four-dimensional complex spaces and isotropy groups (holomorphic stabilizers) of such surfaces. The dimension of the stabilizer for a particular strictly pseudo-convex hypersurface is estimated from above by the dimension of a unitary subgroup preserving the fourth-degree polynomial from its normal equation.