de Bruin-Sharma Problem
For n ≥ 4, let Ω(n) be the set of pairs (α,β) ∈ ℝ_+^2 such that, whenever P is a degree n polynomial whose roots z_1,…,z_n sum to zero, and ξ_1,…,ξ_n-1 are the critical points (roots of P'), that |ξ_1|^4 + … + |ξ_n-1|^4 ≤ α (|z_1|^4 + … + |z_n|^4) + β (|z_1|^2 + … + |z_n|^2)^2. What is Ω(n)?
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.
Cite
@misc{cairn-de-bruin-sharma-problem,
title = {de Bruin-Sharma Problem},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/de-bruin-sharma-problem}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
} Also: CITATION.cff · Atom feed of results
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The problem
The problem
For , let be the set of pairs such that, whenever is a degree polynomial whose roots sum to zero, and are the critical points (roots of ), that What is ?
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.