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 supremum $C_{82}$ over all real numbers $H$ such that the set ${\alpha \in \overline{\mathbb{Q}} | h_Z(\alpha)\le H}$ is finite.
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] | Previous best known upper bound |
| 0.2536331090204145 | [Gri26] | Computation based on Doche’s polynomials |
| 0.2535744749068227* | [Epi26] | Doche’s limit-point construction with unequal denominator multiplicities and two additional factors (Ed5, En6); reported in the Épi results repository (September 2026). An independent Arb re-evaluation of the same construction in [Pin26] encloses it at $0.25356893\ldots$. Asterisked: a self-archived computer-assisted result. |
| 0.2534449* | [Pin26] | Doche’s limit-point construction [Doc01b] on the factor library of [Gri26] and [Epi26], adding $P_7$ to the denominator and 25 new numerator factors found by LP-dual-weighted lattice reduction (LLL), with all multiplicities and exponents re-optimised by a linear program ($\deg Q = 999999992$). Certified upper bound $0.2534446873\ldots$ by two independently written Arb checkers; the repository includes a self-contained proof that the construction yields infinitely many distinct algebraic numbers with $h_Z \le B$. Certificate SHA-256 b33626f367b6f52922746c0fee7bb008449cd0bb18efcb983ab20ff073350fce, repository commit 69d8964. Asterisked: a self-archived computer-assisted certificate, not yet independently replayed. |
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
- All the lower bounds other than [Zha95] are obtained by an algorithm pionered by C.J. Smyth [S80]. The upper bounds are obtained by an algorithm based in the determination the asymptotic distribution of Galois conjugates of small algebraic integers. It is shown (non effectively) in [BMQS] that both algorithms converge to $C_{82}$ if they were to run indefinitely.
- A similar problem is the determination of the essential minimum $\mu^{ess}(h_F)$ of the stable Faltings height on elliptic curves, for which it is known $−0.748629 \leq \mu^{ess}(h_F) \leq −0.748622$ [BMRL]
References
- [BMQS] Burgos Gil, J., Menares, R., Qu, B., Sombra, M. Closing the gap around the essential minimum of height functions with linear programming. arXiv:2601.18978, preprint 2026
- [BMRL] Burgos Gil, J., Menares, R., Rivera-Letelier, J. On the essential minimum of Faltings’ height. Math. Comp. 87 (2018), no. 313, 2425–2459.
- [Doc01a] Doche, Ch. On the spectrum of the Zhang-Zagier height. Math. Comp. 70 (2001), no. 233, 419–430.
- [Doc01b] Doche, Ch. Zhang-Zagier heights of perturbed polynomials. 21st Journées Arithmétiques (Rome, 2001) J. Théor. Nombres Bordeaux 13 (2001), no. 1, 103–110.
- [D03] Dresden, Gregory P. Sums of heights of algebraic numbers. Math. Comp. 72 (2003), no. 243, 1487–1499
- [F18] Flammang, V. On the Zhang-Zagier measure. Int. J. Number Theory 14 (2018), no. 10, 2663–2671
- [Epi26] Épi (BakeLab). Zhang–Zagier C82 record, Épi results repository, September 2026. https://github.com/BakeLab/Epi-results/tree/main/records/zhang-zagier/zhang_zagier_c82
- [Gri26] Max Grinsztajn. Code and certification data for an improved C82 Zhang-Zagier bound. GitHub repository, 2026. https://github.com/maaxgrin/zhang-zagier-c82-bound
- [Pin26] Ricardo Pinho. Certificate and checkers for C82 < 0.2534449. GitHub repository, 2026. https://github.com/rpinho/c82-height-certificate
- [S80] Smyth, C. J. On the measure of totally real algebraic integers. J. Austral. Math. Soc. Ser. A 30 (1980/81), no. 2, 137–149.
- [Zag93] Zagier, D. Algebraic numbers close to both 0 and 1. Math. Comp. 61 (1993), no. 203, 485–491.
- [Zha95] Zhang, S. Positive line bundles on arithmetic varieties. J. Amer. Math. Soc. 8 (1995), no. 1, 187–221.
Contribution notes
Prepared with assistance from GPT-5.5 Pro.
The [Pin26] upper-bound row (October 2026): the factor search, the certificate, the checkers and the row text were prepared with AI assistance (OpenAI Codex/GPT-6 Astra and Claude) under the direction of the submitter, R. Pinho, who reviewed the argument and the references.