On the number of nodal domains of spherical harmonics (Q1913468)

It is well known that the \(n\)-th eigenfunction to one-dimensional Sturm-Liouville eigenvalue problems has exactly \(n - 1\) nodes, i.e. non-degenerate zeros. For higher dimensions, it is much more complicated to obtain general statements on the zeros of eigenfunctions. The author states a new conjecture on the number of nodal domains of spherical harmonics, i.e. of connected components of \(S^2 \backslash N(u)\) with the nodal set \(N(u) = \{x \in S^2 : u(x) = 0\}\) of the eigenfunction \(u\), and proves it for the first six eigenvalues. It is a sharp upper bound, thus improving known bounds as the Courant nodal domain theorem, see \textit{S. Y. Cheng}, Comment. Math. Helv. 51, 43-55 (1976; Zbl 0334.35022). The proof uses facts on real projective plane algebraic curves (see \textit{D. A. Gudkov}, Usp. Mat. Nauk 29(4), 3-79, Russian Math. Surveys 29(4), 1-79 (1979; Zbl 0316.14018)), because they are the zero sets of homogeneous polynomials, and the spherical harmonics are the restrictions of spherical harmonic homogeneous polynomials in the space to the plane.