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Level B · Reproducible Analysis P-rudin-problem-for-polynomials

Rudin problem for polynomials

Let d ≥ 2 and D ≥ 1. For p ∈ 4,∞, let C^p(d,D) be the maximum of the ratio frac‖u‖_L^p(S^d)‖u‖_L^2(S^d) where u ranges over (real) spherical harmonics of degree D on the d-dimensional sphere S^d, which we normalize to have unit measure.

From the catalogue. Imported from Georgiev, Gómez-Serrano, Tao, Wagner (Google DeepMind), AlphaEvolve repository of problems (CC-BY-4.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite 
@misc{cairn-rudin-problem-for-polynomials,
  title        = {Rudin problem for polynomials},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/rudin-problem-for-polynomials}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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The problem

The problem

Let and . For , let be the maximum of the ratio where ranges over (real) spherical harmonics of degree on the -dimensional sphere , which we normalize to have unit measure. Establish upper and lower bounds on that are as strong as possible.

What counts as progress

  • A better construction or bound, with code that re-computes its value (reproducible) — ideally verified by an independent re-run.
  • A proof that a known construction is optimal, or a better bound on the other side.
  • Literature: earlier or newer records (literature claims).

Source and licence

Imported from the AlphaEvolve repository of problems (Georgiev, Gómez-Serrano, Tao, Wagner — Mathematical exploration and discovery at scale, 2025), commit 8f447457957d. Text under CC BY 4.0, code under Apache 2.0; reformatted for this page.