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Smale's mean value constant (1981) equals 4, and other miscellanea

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

Contributed by Dave Goel

Version 1 / 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. We give three further proofs: by a variant of the main construction in which part of the boundary of an auxiliary domain is a level curve of an entire function; by a Riemann map onto a domain with thin hairs; and by a Riemann surface whose conformal radius at its base point is near 1. We also include a complete proof of the known fact that K(d) does not decrease.

Author comments: This is the expanded version, with further results, of the paper "Smale's mean value constant (1981) equals 4": https://doi.org/10.5281/zenodo.24069884

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.

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Version history

  1. v1 · 39 pagesSubmitted by Dave GoelThis is the expanded version, with further results, of the paper "Smale's mean value constant (1981) equals 4": https://doi.org/10.5281/zenodo.24069884SupersededNov 11, 2026
  2. v2 · 39 pagesSubmitted by Dave GoelThis is the expanded version of the paper Hexagon 2611.00282, "Smale's mean value constant (1981) equals 4", at https://hexagonmath.org/2611.00282 and on Zenodo at https://doi.org/10.5281/zenodo.24069884CurrentNov 11, 2026