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Smale's mean value constant (1981) equals 4

A mix of human-written and AI-generated textHuman understanding: some partsmath.CV — Complex Variables

Contributed by Dave Goel

Version 2 / Nov 11, 2026 / CC BY-SA 4.0

Abstract

For a complex polynomial p of degree d ≥ 2 with p(0) = 0 and p′(0) ≠ 0, let S(p) be the minimum of |p(c)/(c p′(0))| over the critical points c of p, and let K(d) be the supremum of S(p) over all such p of degree d. In 1981 Smale proved that K(d) ≤ 4 for every d, and asked whether 4 can be replaced by 1. We prove that Smale's mean value constant sup_d K(d) equals 4 and that K(d) → 4 as d → ∞, combining a reduction due to Jatar and Ng with an explicit construction. Each K(d) is known to be below 4, so the supremum is not attained. The main result has also been formalized in the Lean 4 proof assistant.

Author comments: The expanded version of this paper is Hexagon 2611.00283, at https://hexagonmath.org/2611.00283 and on Zenodo at https://doi.org/10.5281/zenodo.24069916

Provenance statement

This work was done overwhelmingly by AI models, steered by the author in a continuous critical dialogue, alternating between Anthropic's Claude and OpenAI's ChatGPT. The author takes full responsibility for the paper.

References

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Version history

  1. v1 · 11 pagesSubmitted by Dave GoelAn expanded version of this paper, with further results, is posted on Zenodo: https://doi.org/10.5281/zenodo.24069916SupersededNov 11, 2026
  2. v2 · 11 pagesSubmitted by Dave GoelThe expanded version of this paper is Hexagon 2611.00283, at https://hexagonmath.org/2611.00283 and on Zenodo at https://doi.org/10.5281/zenodo.24069916CurrentNov 11, 2026