The minimum numbers of nodal domains of spherical Grushin-harmonics

In this paper, nodal lines and nodal domains of homogeneous Grushin-harmonic polynomials of degree kk in three variables are studied. It is proved that, for the cases k≠4mk≠4m, m∈Nm∈N, there exist spherical Grushin-harmonics FF of degree kk such that the nodal curves of FF divide the unit gauge sphere SρSρ into two parts; and for the case k=4mk=4m, m∈Nm∈N, there is no a spherical Grushin harmonic FF of degree kk such that the nodal curve of FF divides SρSρ into two parts. This provides a different picture comparing with that of classical spherical harmonics.