Essential minimum of the Zhang-Zagier height

Description of constant

Let $\overline{\mathbb{Q}}$ be the set of all algebraic numbers. The naïve height $h : \overline{\mathbb{Q}} \rightarrow \mathbb{R}$ is defined as follows. Let $\alpha \in \overline{\mathbb{Q}}$ and let $P(x)$ be an irreducible primitive polynomial with integers coefficients such that $P(\alpha)=0$. Let $n$ be the degree and $a$ be the leading coefficient of $P(x)$. Then,

\[h(\alpha):=\frac{1}{n}\left( \log|a|+\sum_{P(\beta)=0} \log \max (1,|\beta|)\right).\]

The Zhang-Zagier height is the function

\[h_Z : \overline{\mathbb{Q}} \rightarrow \mathbb{R}, \quad h_Z(\alpha)=h(\alpha)+h(1-\alpha).\]

The essential minimum of the Zhang-Zagier height is the smallest number $C_{82}$ such that there exists a finite set $S\subseteq \overline{\mathbb{Q}}$ with the property that the inequality $h_Z(\alpha)\geq C_{82}$ holds for all $\alpha \in \overline{\mathbb{Q}}\setminus S$.

Known upper bounds

Bound Reference Comments
0.39679 [D03], Section 7  
0.25594 [Doc01a] $h_Z=\log \mathfrak{h}$ in his notation
0.25444 [Doc01b] Best known upper bound

Known lower bounds

Bound Reference Comments
0 trivial  
0 [Zha95] Strict inequality
$\frac{1}{2}\log \big(\frac{1+\sqrt{5}}{2}\big)=0.2406059…$ [Zag93]  
0.24824 [Doc01a] $h_Z=\log \mathfrak{h}$ in his notation
0.24874 [F18] $h_Z=\log \zeta$ in her notation, best known lower bound

Additional comments

References