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\title[A counterexample of degree 29\,440 to Smale's mean value problem]{An explicit counterexample of degree 29\,440\\ to Smale's mean value problem with constant one}
\author{David Niedbala Giraudin}
\address{Meaux, France}
\email{mrjack77100@gmail.com}
\thanks{ORCID 0009-0009-1526-1178.}
\date{November 7, 2026}
\subjclass[2020]{30C15, 30C10, 65G20}
\keywords{Smale's mean value problem, critical points of polynomials, interval arithmetic, Krawczyk test}

\begin{document}
\sloppy

\begin{abstract}
For a complex polynomial $p$ with $p(0)=0$ and $p'(0)=1$, let $S(p,0)=\min_{p'(c)=0}|p(c)/c|$.
Adamczewski and Colbrook recently gave an explicit polynomial of degree $29\,984$ 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
$p\in\mathbb{Q}(i)[z]$ of degree $29\,440$ 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
$29\,439$ critical points, and, independently, by the Adamczewski--Colbrook verifier, which
confirms $29\,418$ of them and is inconclusive (insufficient working precision) on the other $21$.
The exact coefficients and both verification programs are provided as ancillary files.
\end{abstract}

\maketitle

\section{Introduction}

Let $p$ be a complex polynomial of degree $n\ge 2$ and $a\in\mathbb{C}$ with $p'(a)\neq 0$. Smale
\cite{Smale} proved that
\[
S(p,a)=\min_{p'(c)=0}\left|\frac{p(c)-p(a)}{(c-a)\,p'(a)}\right|\le 4
\]
and asked whether the constant $4$ can be replaced by $1$, or even by $(n-1)/n$.
The constant-one version was disproved by GPT-6 Astra, with a formal Lean~4 proof \cite{Ad}.
Adamczewski and Colbrook \cite{AC} explained the underlying mechanism and gave an explicit
polynomial $p\in\mathbb{Q}(i)[z]$ of degree $29\,984$ with $S(p,0)>1.00000049595801$, certified by
interval arithmetic \cite{ACcode}. Independently, Jatar and Ng \cite{JN} reduced the determination of
Smale's mean value constant $\mathcal M=\sup S(p,0)$, the supremum over normalized polynomials, to an
extremal problem for admissible pairs of a univalent function and a finite Blaschke product, and showed
that $\mathcal M>1.1018$; their polynomials come from an approximation argument, and no explicit degree
is given. Thus the value $1.00000093\ldots$ below is not a bound on $\mathcal M$; what is new here is only
the degree of an explicit, certified example.

This note records a smaller explicit counterexample. It uses \emph{the same construction} as
\cite[Appendix~A]{AC}: the same lemniscate coordinate, model anchor, Blaschke parameter,
correction centres and correction structure. Only the truncation level $N$, the size of the
high-degree tail and the coefficient tables are changed. The reduction in degree is modest
(about $1.8\%$); no new mechanism is claimed.

\section{The polynomial}

We keep the notation of \cite[Appendix~A]{AC}: $m=16$, $A=1061603/250000$, $S=A+1/A$,
$q(z)=(z^{16}+iA)/S$, the model anchor $a$, the Blaschke parameter $r$ and the truncated model
$h_L$ of \cite[(A.5)]{AC}. We take
\[
N=1840,\qquad L=N-2=1838,\qquad \text{tail blocks } t=48,
\]
and set
\begin{align*}
P(z)&=(z-a)\frac{h_L(z)}{h_L(a)}+\Delta_0(z)+X(z)+H(z),\\
\Delta_0&=(z-a)^2\Bigl(K R_0+\sum_{j=0}^{3}\phi_j(z-\mu)^j\Bigr),\qquad K=(z-\xi_1)^2(z-\xi_2)^2,\quad \mu=\tfrac{\xi_1+\xi_2}{2},\\
X&=(z-a)^2\sum_{\ell=0}^{127}\sum_{k=0}^{15}x_{\ell k}\Bigl(\frac{5z}{6}\Bigr)^{k}q(z)^{\ell},\\
H&=\omega\Bigl(\gamma\, q(z)^{N}+q(z)^{N-t}\sum_{\ell=0}^{t-1}\sum_{k=0}^{15}e_{\ell k}\Bigl(\frac{5z}{6}\Bigr)^{k}q(z)^{\ell}\Bigr),
\end{align*}
with $R_0$ the correction polynomial of \cite{AC} (its $1024$ coefficients re-chosen), rational
$\omega$ with $|\omega|=1$ exactly, rational $\gamma>0$, and a rational base point $\hat a$. The
normalized polynomial is
\[
p(x)=\frac{P(\hat a+x)-P(\hat a)}{P'(\hat a)} .
\]
All data are exact elements of $\mathbb{Q}$ or $\mathbb{Q}(i)$ and are listed in the ancillary
file \texttt{polynomial\_N1840.json}, which follows the format of \cite{ACcode} (field
\texttt{defining\_formula}). In particular $p\in\mathbb{Q}(i)[z]$.

Only $\gamma q^N$ reaches degree $16N=29\,440$; all other terms have degree at most $29\,439$, and
the leading coefficient $\omega\gamma/S^{N}$ is nonzero. Hence $\deg p=29\,440$, and
$p(0)=0$, $p'(0)=1$ by construction.

\begin{theorem}\label{thm:main}
The polynomial $p\in\mathbb{Q}(i)[z]$ above has degree $29\,440$, its $29\,439$ critical points are
simple, and
\[
1.00000093069254310 < S(p,0) < 1.00000093069254311 .
\]
In particular $p$ is a counterexample to Smale's mean value conjecture with constant one.
\end{theorem}

\begin{remark}
Adamczewski and Colbrook note \cite[\S1.1]{AC} that perturbations $\varepsilon z^n$ of an explicit
counterexample, together with Rouch\'e's theorem, give counterexamples in every larger degree.
Applied to $p$, the same argument would give counterexamples in every degree $n\ge 29\,440$; we do
not repeat it here, and Theorem~\ref{thm:main} only concerns degree $29\,440$.
\end{remark}

\section{Verification}

Since $S(p,0)$ is unchanged by the affine normalization, it suffices to work with $P$ and $\hat a$:
$S(p,0)=\min_{P'(c)=0}|Q(c)|$ with $Q(c)=\dfrac{P(c)-P(\hat a)}{(c-\hat a)P'(\hat a)}$.

\subsection*{Certificate 1 (ball arithmetic)}
Arithmetic is done with complex balls (Arb, via \texttt{python-flint}) at $1280$ bits, starting from
the exact rationals of the specification. For each of $29\,439$ approximate critical points:
\begin{enumerate}
\item three Newton steps for $P'$ give a centre $z$;
\item on the square box $D$ of half-side $\rho$ about $z$, an enclosure of $P''(D)$ is obtained from
$P''(z)$ and a Cauchy bound $\sup|P'''|\le 6B/(R-\rho\sqrt2)^3$, where $B$ is a rigorous majorant of
$|P|$ on the disc of radius $R$ about $z$ (term-by-term triangle inequality);
\item the Krawczyk operator $K(D)=z-P'(z)/P''(z)+\bigl(1-P''(D)/P''(z)\bigr)(D-z)$ is shown to lie
in the interior of $D$, which proves that $D$ contains exactly one zero of $P'$, and that it is simple;
\item $|Q|$ is bounded below on all of $D$ by a second-order Taylor enclosure of $P$ with the same
remainder bound, and a point counts as certified only if this lower bound exceeds $1$.
\end{enumerate}
All $29\,439$ points pass. The boxes have half-side at most $7.4\cdot10^{-40}$, the refined centres
differ from the supplied ones by at most $7.7\cdot10^{-12}$, and any two supplied centres are at
distance at least $6.05\cdot10^{-5}$; hence the boxes are pairwise disjoint. They thus contain
$29\,439=\deg P'$ distinct zeros of $P'$, i.e.\ all critical points, each simple. The smallest
certified value is
\[
\min_{P'(c)=0}|Q(c)|\ \ge\ 1.0000009306925431017438\ldots,
\]
attained in a single box, which also gives the upper bound of Theorem~\ref{thm:main}.

\subsection*{Certificate 2 (Adamczewski--Colbrook verifier)}
The program \texttt{verify.cpp} of \cite{ACcode} (outward-rounded interval bounds with MPFR-directed
endpoints, Rouch\'e test on critical disks) was recompiled with \texttt{N\_PARAM=1840},
\texttt{TAIL\_BLOCKS\_PARAM=48}, the expected parameters in \texttt{make\_parameters.py} being read
from the specification; the specification passes all its exact-rational consistency checks. Run disk
by disk on the $29\,439$ critical disks, it certifies $|Q|>1$ on $29\,418$ of them, with smallest
lower bound $1.000000930692128\ldots$, on the same critical point as Certificate~1. On the remaining
$21$ disks (indices, in the order of \texttt{critical\_disks\_N1840.csv}, listed in the ancillary summary) its fixed working precision does not suffice to
conclude; these are covered by Certificate~1 only.

\begin{table}[h]
\centering
\begin{tabular}{lcc}
\toprule
 & \cite{AC} & this note\\
\midrule
$N$ & $1874$ & $1840$\\
degree & $29\,984$ & $29\,440$\\
tail blocks & $8$ & $48$\\
certified lower bound for $S(p,0)$ & $1.00000049595801$ & $1.00000093069254310$\\
\bottomrule
\end{tabular}
\end{table}

\section{Ancillary files}
\begin{itemize}
\item \texttt{polynomial\_N1840.json}: exact specification ($\mathbb{Q}$ and $\mathbb{Q}(i)$ data).
\item \texttt{centres\_N1840.csv}: the $29\,439$ supplied critical-point centres, sorted by increasing $|Q|$.
\item \texttt{verify\_ball\_N1840.py}: Certificate~1 for any range of critical points
(Python with \texttt{python-flint} and \texttt{numpy}).
\item \texttt{certificate\_N1840.log}: summary of both certificates.
\item \texttt{critical\_disks\_N1840.csv}, \texttt{make\_parameters\_N1840.py}: input disks and the
modified parameter script used to run the verifier of \cite{ACcode} (Apache License 2.0).
\end{itemize}

\section*{Status and acknowledgements}
Theorem~\ref{thm:main} is a computer-assisted result: its proof is the pair of certificates above,
which anyone can rerun from the ancillary files. The construction is entirely that of
Adamczewski and Colbrook \cite{AC}, building on the counterexample found by GPT-6 Astra \cite{Ad};
only parameters were changed. The computations were carried out with the assistance of Claude
(Anthropic).

\begin{thebibliography}{9}
\bibitem{Ad} T.~Adamczewski, \emph{Mean-value-problem: Lean 4 disproof of Smale's mean value
conjecture ($K=1$)}, GitHub repository, 2026.
\bibitem{AC} T.~Adamczewski and M.~J.~Colbrook, \emph{From boundary geometry to counterexamples in
Smale's mean value problem}, arXiv:2610.36158, 2026.
\bibitem{ACcode} M.~J.~Colbrook, \emph{Interval\_arithmetic\_counterexample\_Smale\_mean\_value},
GitHub repository, 2026.
\bibitem{JN} A.~Jatar and T.~W.~Ng, \emph{Smale's mean value constant and admissible univalent Blaschke
pairs}, arXiv:2610.21472, 2026.
\bibitem{Smale} S.~Smale, The fundamental theorem of algebra and complexity theory,
\emph{Bull. Amer. Math. Soc. (N.S.)} \textbf{4} (1981), 1--36.
\end{thebibliography}

\end{document}