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An explicit counterexample of degree 29440 to Smale's mean value problem with constant one

Primarily AI-generated textHuman understanding: no partsmath.CV — Complex Variablesmath.NA — Numerical Analysis

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

For a complex polynomial p with p(0)=0 and p'(0)=1, let S(p,0) be the minimum of |p(c)/c| over the critical points c of p. Adamczewski and Colbrook recently gave an explicit polynomial of degree 29984 with S(p,0)>1, certified by interval arithmetic, built on the counterexample found by GPT-6 Astra. Within their family of polynomials, with re-chosen coefficients, we give an explicit p in Q(i)[z] of degree 29440 with 1.00000093069254310 < S(p,0) < 1.00000093069254311. Jatar and Ng have shown that Smale's constant sup_p S(p,0) exceeds 1.1018, through polynomials of unspecified degree; the point here is an explicit polynomial with a certified bound. The bound is certified twice: by a ball-arithmetic Krawczyk certificate covering all 29439 critical points, and, independently, by the Adamczewski-Colbrook verifier, which confirms 29418 of them and is inconclusive (insufficient working precision) on the other 21. The exact coefficients and both verification programs are provided as ancillary files.

Author comments: 4 pages. Ancillary files: exact specification (polynomial_N1840.json), critical-point centres and disks, ball-arithmetic verifier (verify_ball_N1840.py), certificate summary, and the parameter patch used to run the Adamczewski-Colbrook verifier.

Provenance statement

The mathematics, the computations and the writing were produced with the assistance of an AI model; the author checked the statements and the references. The construction is that of Adamczewski and Colbrook (arXiv:2610.36158); only its parameters were changed.

Tools used

Anthropic
Claude Opus

References

  1. T. Adamczewski, Mean-value-problem: Lean 4 disproof of Smale's mean value conjecture (K=1K=1) , GitHub repository, 2026.
  2. T. Adamczewski and M. J. Colbrook, From boundary geometry to counterexamples in Smale's mean value problem , arXiv:2610.36158, 2026.arXiv
  3. M. J. Colbrook, Interval_arithmetic_counterexample_Smale_mean_value , GitHub repository, 2026.
  4. A. Jatar and T. W. Ng, Smale's mean value constant and admissible univalent Blaschke pairs , arXiv:2610.21472, 2026.arXiv
  5. S. Smale, The fundamental theorem of algebra and complexity theory, Bull. Amer. Math. Soc. (N.S.) 4 (1981), 1--36.

Version history

  1. v1 · 4 pagesSubmitted by David Niedbala Giraudin4 pages. Ancillary files: exact specification (polynomial_N1840.json), critical-point centres and disks, ball-arithmetic verifier (verify_ball_N1840.py), certificate summary, and the parameter patch used to run the Adamczewski-Colbrook verifier.CurrentNov 07, 2026