Submit progress

on Rudin problem for polynomials (AlphaEvolve problem 57)

The statement

Let $d \geq 2$ and $D \geq 1$ . For $p \in \{4,\infty\}$ , let $C^p(d,D)$ be the maximum of the ratio

\frac{\|u\|_{L^p({\mathbb S}^d)}}{\|u\|_{L^2({\mathbb S}^d)}}

where $u$ ranges over (real) spherical harmonics of degree $D$ on the $d$ -dimensional sphere $\mathbb S^d$ , which we normalize to have unit measure. Establish upper and lower bounds on $C^p(d,D)$ that are as strong as possible.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.