93-08-01.ley
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On the number of nodal domains of spherical harmonics

### Review from Zentralblatt für Mathematik

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 $S^2\setminus
N(u)$ 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 {\it S. Y. Cheng}, Comment. Math. Helv. 51,
43-55 (1976; Zbl 334.35022). The proof uses facts on real projective
plane algebraic curves
(see {\it D. A. Gudkov}, Usp. Mat. Nauk 29(4), 3-79, Russian
Math. Surveys 29(4), 1-79 (1979; Zbl 316.14018)), 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.

**Mathematics Subject Classification:**
33C35 (Spherical functions),
14H05 (Algebraic functions)

**Key Words:**
spherical harmonics, algebraic curves

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Josef.Leydold@statistik.wu-wien.ac.at