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\title[Essential minimum of the Zhang--Zagier height]{A certified lower bound for the essential
minimum of the Zhang--Zagier height}
\author{David Niedbala Giraudin}
\address{ORCID 0009-0009-1526-1178}
\date{3 November 2026}

\begin{document}

\begin{abstract}
Let $h$ be the absolute logarithmic Weil height. We prove that the essential minimum of the
Zhang--Zagier height $h(\alpha)+h(1-\alpha)$ is at least $0.2497815$: for every $\delta>0$, only
finitely many algebraic numbers $\alpha$ satisfy $h(\alpha)+h(1-\alpha)<0.2497815-\delta$. The previous best lower bound was $0.24874$
(Flammang, 2018). The proof combines Smyth's auxiliary-function method, with the polynomials of
Doche and of Flammang, and the condition that the discriminant of the minimal polynomial is a
nonzero integer, in the potential-theoretic form used by Smith and by Orloski, Talebizadeh Sardari
and Smith for the trace problem. The infimum of the resulting auxiliary function is bounded below
rigorously by monotone interval bounds and adaptive bisection, from a stored certificate. The
statement concerns the essential minimum and is weaker in nature than the pointwise bounds of
Doche and Flammang.
\end{abstract}

\maketitle

\section{Introduction}

For an algebraic number $\alpha$ of degree $n$, with primitive minimal polynomial
$P=a_n\prod_{i=1}^n(x-\alpha_i)\in\Z[x]$, let
\[
h(\alpha)=\frac1n\Bigl(\log|a_n|+\sum_{i=1}^n\log\max(1,|\alpha_i|)\Bigr),\qquad
h_Z(\alpha)=h(\alpha)+h(1-\alpha).
\]
Zhang \cite{Zh} and Zagier \cite{Za} showed that $h_Z$ is bounded below by a positive constant
outside $\{0,1,(1\pm i\sqrt3)/2\}$; Zagier proved $h_Z(\alpha)\ge\frac12\log\frac{1+\sqrt5}2$
outside the roots of $x(x-1)(x^2-x+1)$, with equality at the primitive tenth roots of unity. The
\emph{essential minimum} $\ess(h_Z)$ is the supremum of the real numbers $H$ such that
$\{\alpha\in\Qbar:h_Z(\alpha)\le H\}$ is finite. It is entry 82 of the list of optimization
constants of Davis, Ivanisvili and Tao \cite{OC}. The known bounds are
\[
0.24874\ \text{(Flammang \cite{Fl})}\ \le\ \ess(h_Z)\ \le\ 0.2536331\ \text{(Grinsztajn
\cite{Gr})},
\]
after the lower bound $0.248247$ and the upper bound $0.254437$ of Doche \cite{Do1,Do2}, which
Burgos Gil, Menares, Qu and Sombra \cite{BMQS} record as the best known; lower upper bounds have
been announced in self-archived computations \cite[entry 82]{OC}. The lower bounds of Doche and
Flammang are \emph{pointwise}: they hold for every $\alpha$ outside an explicit finite set
(Flammang states hers for algebraic integers; since the essential minimum is attained along
sequences of algebraic integers \cite{BMQS}, her bound also bounds $\ess(h_Z)$).

\begin{theorem}\label{thm:main}
$\ess(h_Z)\ge 0.2497815$. Equivalently, for every $\delta>0$ only finitely many algebraic numbers
satisfy $h(\alpha)+h(1-\alpha)<0.2497815-\delta$, and in multiplicative form
$\liminf\bigl(M(\alpha)M(1-\alpha)\bigr)^{1/\deg\alpha}\ge e^{0.2497815}=1.2837448\ldots$ along any
sequence of distinct algebraic numbers.
\end{theorem}

More precisely, the proof gives, for $\alpha$ of degree $n$ outside a finite explicit set,
\begin{equation}\label{eq:finite}
h_Z(\alpha)\ \ge\ 0.2497815-\lambda_0\Bigl(\frac{\log n}{n}+\frac{599}{n^2}\Bigr),
\qquad \lambda_0=0.0862985\ldots,
\end{equation}
which is not a pointwise bound of the type of \cite{Do1,Fl}; the loss is $8.0\cdot10^{-5}$ at
$n=10^4$.

The method is that of Smyth \cite{Sm}, as used by Doche and Flammang, to which one condition is
added: the discriminant of $P$ is a nonzero integer. In the limit this becomes a condition on
the logarithmic energy of the limiting distribution of conjugates, which is the condition used by
Smith \cite{Smi} and by Orloski, Talebizadeh Sardari and Smith \cite{OTS} for the
Schur--Siegel--Smyth trace problem. In the framework of \cite{BMQS}, where the essential minimum
is characterised by strong duality between Smyth's method and measures satisfying
$\int\log|Q|\,d\mu\ge0$ for all $Q\in\Z[x]\setminus\{0\}$, this energy condition is implied by
the full family of these linear conditions; its use here is computational: a single inequality
replaces a large part of a search over polynomials of high degree. The contribution of this note
is the numerical value and its certification.

\section{The certificate inequality}\label{sec:cert}

Put $g(z)=\log^+|z|+\log^+|1-z|$. For $\alpha$ as above put $L=\frac1n\log|a_n|\ge0$ and
$\mu_\alpha=\frac1n\sum_i\delta_{\alpha_i}$. Since $1-\alpha$ has minimal polynomial $\pm P(1-x)$,
with leading coefficient $\pm a_n$,
\begin{equation}\label{eq:hZ}
h_Z(\alpha)=2L+\int g\,d\mu_\alpha .
\end{equation}

\emph{Resultant conditions.} If $Q\in\Z[x]$ does not vanish at $\alpha$, then
$\operatorname{Res}(P,Q)=a_n^{\deg Q}\prod_iQ(\alpha_i)$ is a nonzero integer, so
\begin{equation}\label{eq:res}
\deg(Q)\,L+\int\log|Q|\,d\mu_\alpha\ \ge\ 0 .
\end{equation}

\emph{Discriminant condition.} $\operatorname{disc}(P)=a_n^{2n-2}\prod_{i<j}(\alpha_i-\alpha_j)^2$ is
a nonzero integer, so
\begin{equation}\label{eq:disc}
2\Bigl(1-\frac1n\Bigr)L+\frac1{n^2}\sum_{i\ne j}\log|\alpha_i-\alpha_j|\ \ge\ 0 .
\end{equation}

For a finite positive measure $\eta$ with compact support put
$U_\eta(z)=\int\log|z-w|\,d\eta(w)$ and $I(\eta)=\iint\log|z-w|\,d\eta\,d\eta$.

\begin{lemma}\label{lem:energy}
Let $\eta$ be a probability measure with compact support and a density bounded by $D$. Then for
every $\varepsilon>0$ and every $\alpha$ of degree $n$,
\[
\frac1{n^2}\sum_{i\ne j}\log|\alpha_i-\alpha_j|
\ \le\ \frac2n\sum_iU_\eta(\alpha_i)-I(\eta)+\frac1n\log\frac1\varepsilon+D\pi\varepsilon^2 .
\]
\end{lemma}

\begin{proof}
Let $\sigma$ be the average of the uniform probability measures on the circles of radius
$\varepsilon$ centred at the $\alpha_i$. The potential of such a circle, centred at $c$, is
$\log\max(|z-c|,\varepsilon)\ge\log|z-c|$, and its energy is $\log\varepsilon$; hence, with no
assumption on the position of the circles,
$I(\sigma)\ge\frac1{n^2}\bigl(\sum_{i\ne j}\log|\alpha_i-\alpha_j|+n\log\varepsilon\bigr)$.
Next,
$\int U_\eta\,d\sigma=\frac1n\sum_i\int\log\max(|\alpha_i-w|,\varepsilon)\,d\eta(w)$, and
$\int\log\max(|a-w|,\varepsilon)\,d\eta(w)-U_\eta(a)=\int_{|w-a|<\varepsilon}
\log\frac{\varepsilon}{|w-a|}\,d\eta(w)\le D\int_{|u|<\varepsilon}\log\frac{\varepsilon}{|u|}\,dA(u)
=\frac{D\pi\varepsilon^2}2$. Finally $\sigma-\eta$ is a compactly supported signed measure of
total mass $0$ and finite energy, so $I(\sigma-\eta)\le0$, that is
$I(\sigma)\le2\int U_\eta\,d\sigma-I(\eta)$. Combining the three estimates gives the lemma.
\end{proof}

\begin{proposition}\label{prop:cert}
Let $Q_1,\dots,Q_m\in\Z[x]\setminus\{0\}$, $c_1,\dots,c_m\ge0$, $\lambda_0\ge0$ with
\begin{equation}\label{eq:budget}
\sum_kc_k\deg Q_k+2\lambda_0\le2,
\end{equation}
and let $\eta$ be as in Lemma~\ref{lem:energy}. Put
\[
A(z)=g(z)-\sum_kc_k\log|Q_k(z)|-2\lambda_0U_\eta(z)+\lambda_0I(\eta).
\]
Then every algebraic $\alpha$ of degree $n$ which is not a root of any $Q_k$ satisfies
$h_Z(\alpha)\ge\inf_{\C}A-\lambda_0\bigl(\frac1n\log n+\frac{D\pi}{n^2}\bigr)$. Consequently
$\ess(h_Z)\ge\inf_{\C}A$.
\end{proposition}

\begin{proof}
Subtract from \eqref{eq:hZ} the left-hand sides of \eqref{eq:res} multiplied by $c_k$ and the
left-hand side of \eqref{eq:disc} multiplied by $\lambda_0$, and bound the double sum by
Lemma~\ref{lem:energy} with $\varepsilon=1/n$. The coefficient of $L$ becomes
$2-\sum_kc_k\deg Q_k-2\lambda_0(1-\frac1n)\ge0$, and $L\ge0$, so this term may be dropped; what
remains is $\int A\,d\mu_\alpha$ minus the stated defect, and $\int A\,d\mu_\alpha\ge\inf A$. For
the last assertion: given $\delta>0$, the defect is below $\delta$ for $n\ge n_0(\delta)$, and by
Northcott's theorem only finitely many $\alpha$ of degree $<n_0$ have $h_Z(\alpha)\le\inf A$; the
roots of the $Q_k$ are finitely many.
\end{proof}

\begin{lemma}[support]\label{lem:circles}
Under \eqref{eq:budget} with strict inequality, $\inf_{\C}A$ is attained on
$\{|z|=1\}\cup\{|z-1|=1\}$.
\end{lemma}

\begin{proof}
The two circles cut $\C$ into four open regions, on each of which $g$ is harmonic ($0$,
$\log|z|$, $\log|1-z|$ or $\log|z(1-z)|$). The functions $-\log|Q_k|$ and $-U_\eta$ are
superharmonic on $\C$, so $A$ is superharmonic on each region, and lower semicontinuous with
values in $(-\infty,+\infty]$. As $|z|\to\infty$,
$A(z)=(2-\sum_kc_k\deg Q_k-2\lambda_0)\log|z|+O(1)\to+\infty$. By the minimum principle, the
infimum of $A$ over the closure of each region is attained on its boundary, which lies in the
union of the two circles.
\end{proof}

This support property is also stated by Flammang \cite{Fl}.

\section{The certificate}\label{sec:data}

The certificate, in the ancillary file \texttt{CERTIFICATE\_C82\_ZhangZagier\_FINAL.json}, consists of:
\begin{itemize}
\item thirteen polynomials $Q_k\in\Z[z]$ with multipliers $c_k>0$: five built from the
exceptional set of Doche \cite{Do1}, namely $z$, $z(z-1)$, $z^2-z+1$, $\Phi_{10}(z)$ and
$\Phi_{10}(z)\Phi_{10}(1-z)$, where $\Phi_{10}(z)=z^4-z^3+z^2-z+1$; and eight polynomials in
$w=z(1-z)$, of degrees $6$, $16$, $16$, $20$, $22$, $24$, $32$, $32$ in $z$, selected by the
optimisation from a list containing the polynomials of Flammang \cite[Table 1]{Fl};
\item $\lambda_0=0.08629858849600838$;
\item a measure $\eta$ which is a sum of uniform measures on $2157$ pairwise disjoint closed discs
centred on the two circles, of total mass $1$ and density at most $D=190.582\ldots\le190.59$. (The stored masses, which are
binary floating-point numbers, sum to $1+7\cdot10^{-17}$; we take for $\eta$ the stored measure
divided by this total, which changes $A$ by less than $10^{-15}$, an amount absorbed by the margin
of Section~\ref{sec:proof}.)
\end{itemize}
The budget \eqref{eq:budget} equals $1.0204024<2$. All quantities involving $\eta$ have closed
forms: the potential of the uniform probability measure on the disc of centre $c$ and radius $r$
is $\log|z-c|$ for $|z-c|\ge r$ and $\log r+\frac12(|z-c|^2/r^2-1)$ otherwise; the mutual energy
of two disjoint discs is $\log|c-c'|$, and the self-energy of a disc is $\log r-\frac14$. This
gives $I(\eta)=4.0048\cdot10^{-5}$. Since $D\pi<599$, Proposition~\ref{prop:cert} gives
\eqref{eq:finite} once $\inf A\ge0.2497815$ is established.

\section{Rigorous bound for the infimum}\label{sec:proof}

By Lemma~\ref{lem:circles} it suffices to bound $A$ from below on the circles
$z=s+e^{i\theta}$, $s\in\{0,1\}$. On an interval $|\theta-t|\le H$ we have
$|z(\theta)-z(t)|\le H$, and each term of $A$ is bounded in the safe direction without any
derivative of $A$:
\begin{enumerate}
\item $g$ is increasing in $|z|$ and $|1-z|$, so
$g\ge\log^+\max(0,|z(t)|-H)+\log^+\max(0,|1-z(t)|-H)$;
\item the disc potential is increasing and continuous in the distance $\rho$ to the centre, so
$U_\eta(z(\theta))\le\sum_jm_ju_j(|z(t)-c_j|+H)$;
\item for each $Q_k$, by Taylor's formula with integral remainder on a segment,
$|Q(\zeta)|\le|Q(x)|+|Q'(x)|d+\frac{d^2}2\max_{|\xi|\le|x|+d}|Q''(\xi)|$ for $|\zeta-x|\le d$,
where $Q(x)$ and $Q'(x)$ are evaluated by Horner's rule, with the rounding error bound added, and
$|Q''|$ is bounded by the sum of the absolute values of its coefficients. For the polynomials of
large degree this is applied to the polynomial $\tilde Q_k$ of half degree with $Q_k(z)=\tilde
Q_k(z-z^2)$ (an exact identity over $\Z$, checked by re-expansion), with $x=w(t)=z(t)-z(t)^2$ and
$d=(|1-2z(t)|+2H)H$, since $w(\theta)-w(t)=(z(\theta)-z(t))(1-z(\theta)-z(t))$;
\item $I(\eta)$ is computed in extended precision and decreased by $10^{-12}$.
\end{enumerate}
Floating-point errors are accounted for explicitly: half-widths are enlarged by $10^{-13}$, which
covers the errors on $z(t)$ and on all distances by the monotonicity of the bounds; the Horner
error bound $\gamma_{4d}\sum_k|a_k||x|^k$, with $\gamma_m=mu/(1-mu)$ and $u=2^{-53}$, covers complex
evaluation; and $10^{-9}$ is subtracted from every lower bound, several hundred times the total
remaining rounding error. Adaptive bisection, starting from $2000$ intervals per circle, then
certifies $A\ge0.2497815$ on both circles. Before the bisection the program checks the hypotheses
of Proposition~\ref{prop:cert} from the certificate file (signs, budget, total mass, disjointness of
the discs, recomputed $I(\eta)$), and checks on random intervals and points that its lower bound
never exceeds the directly evaluated value of $A$.

The program \texttt{prove\_C82.py} (Python with NumPy) does this from the certificate file alone;
its output is in \texttt{prove\_C82\_output.log}. An independent direct evaluation of $A$ at
$4\cdot10^5$ points per circle (\texttt{scan\_C82.py}, output in \texttt{scan\_C82\_output.log}) gives the minimum
value $0.24978197$, so the
margin of the certified bound is about $5\cdot10^{-7}$. This proves Theorem~\ref{thm:main}.


\subsection*{Acknowledgement}
Technical assistance: Claude (Anthropic).

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\end{document}