Schmeisser's Conjecture

For each n ≥ 2, let C(n) be the smallest constant such that for any complex polynomial f of degree n ≥ 2 with zeros z_1, …, z_n in the unit disk and critical points w_1, …, w_n-1, and for any nonnegative weights l_1, …, l_n ≥ 0 satisfying Σ_k=1^n l_k = 1, we have min_1 ≤ j ≤ n-1 | Σ_k=1^n l_k z_k -…

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

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

For each $n \geq 2$ , let $C (n)$ be the smallest constant such that for any complex polynomial $f$ of degree $n \geq 2$ with zeros $z_{1}, \dots, z_{n}$ in the unit disk and critical points $w_{1}, \dots, w_{n - 1}$ , and for any nonnegative weights $l_{1}, \dots, l_{n} \geq 0$ satisfying $\sum_{k = 1}^{n} l_{k} = 1$ , we have $1 \leq j \leq n - 1 min k = 1 \sum n l_{k} z_{k} - w_{j} \leq C (n) .$ It was conjectured by Schmeisser that $C (n) = 1$ .

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.

Current state

The problem

The problem

What counts as progress

Source and licence