Several observations on spherical harmonics and their nodal sets are presented: a construction for harmonics with prescribed zeros; a natural representation for harmonics on S²; upper and lower bounds for the nodal length and the inner radius (the upper bounds are sharp); the sharp upper bound for the number of common zeros of two spherical harmonics on S²; the mean Hausdorff measure of the intersection of k nodal sets for harmonics of different degrees on Sᵐ, where k≤ m (in particular, the mean number of common zeros of m harmonics).
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V. M. Gichev (2009) studied this question.
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