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Take a spherical harmonic $y_d$ of degree $d$ on the sphere $\mathbb{S}^2$ and a spherical disk of radius $\frac{1}{d^2}$ centered at any point (let's say the north pole).

Is there an upper bound, independent of d, to the number of critical points of $y_d$ contained in the spherical disk?

(We may allow $d$ to grow arbitrarily big). This article shows that there always exists a spherical harmonic of degree $d$ with the maximum allowed number of critical points, but the construction it employs cannot put a large number of critical points close together. I was thinking that maybe the geometry can stabilize on a local scale when $d$ gets big.

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  • $\begingroup$ If my memory does not betray me, you have the standard basis of the kind $e^{i m\varphi}P_m(\cos\theta)$ in the spherical coordinates where the polynomials $P_m$ are even and non-zero at $0$ for a certain parity of $m$. Thus, it suffices to construct a trigonometric polynomial on the circle (the equator) in frequencies of given parity that has a cluster of many critical points near, say, the "East pole", which is certainly possible. $\endgroup$
    – fedja
    Commented Jul 20, 2018 at 12:49
  • $\begingroup$ Pardon me if I have not understood correctly, but have you considered that the spherical disk shrinks as the degree of the spherical harmonic grows? The question is if you can cluster the critical points faster than the disk can shrink. $\endgroup$
    – un umile appassionato
    Commented Jul 23, 2018 at 8:12
  • $\begingroup$ You can cluster the critical points of the trigonometric polynomials at any speed you wish. cannot you? $\endgroup$
    – fedja
    Commented Jul 23, 2018 at 13:43

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