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Level B · Reproducible Analysis P-smale-s-problem

Smale's Problem

For n ≥ 2, let C(n) be the least constant such that for any polynomial f of degree n, and any z ∈ ℂ with f'(z) ≠ 0, there exists a critical point f'(ξ)=0 such that |f(z)-f(ξ)/z-ξ| ≤ C(n) |f'(z)|. Establish upper and lower bounds for C(n) that are as strong as possible.

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-smale-s-problem,
  title        = {Smale's Problem},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/smale-s-problem}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The problem

For , let be the least constant such that for any polynomial of degree , and any with , there exists a critical point such that Establish upper and lower bounds for 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.