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OP-12097OpenNumber theory

Essential minimum of the Zhang-Zagier height (C82C_{82})

Essential minimum of the Zhang-Zagier height: determine C82C_{82} or improve its bounds; the repository records 0.24874 (lower) and 0.2536331090204145 (upper).

MSC 11G50 · HeightsMSC 11R06 · PV-numbers and generalizations; other special algebraic numbers; Mahler measure

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

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

h(α):=1n(log⁡∣a∣+∑P(β)=0log⁡max⁡(1,∣β∣)).h(\alpha):=\frac{1}{n}\left( \log|a|+\sum_{P(\beta)=0} \log \max (1,|\beta|)\right).

The Zhang-Zagier height is the function

hZ:Q‾→R,hZ(α)=h(α)+h(1−α).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 C82C_{82} over all real numbers HH such that the set {α∈Q‾∣hZ(α)≤H}\{\alpha \in \overline{\mathbb{Q}} | h_Z(\alpha)\le H\} is finite.

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Background

  • 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 C82C_{82} if they were to run indefinitely.
  • A similar problem is the determination of the essential minimum μess(hF)\mu^{ess}(h_F) of the stable Faltings height on elliptic curves, for which it is known −0.748629≤μess(hF)≤−0.748622−0.748629 \leq \mu^{ess}(h_F) \leq −0.748622 [BMRL]

References (from the repository page)

  • [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
  • [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
  • [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.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 0.24874; best known upper bound 0.2536331090204145. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
0.39679 [D03], Section 7
0.25594 [Doc01a] hZ=log⁡hh_Z=\log \mathfrak{h} in his notation
0.25444 [Doc01b] Previous best known upper bound
0.2536331090204145 [Gri26] Computation based on Doche's polynomials

Known lower bounds

Bound Reference Comments
0 trivial
0 [Zha95] Strict inequality
12log⁡(1+52)=0.2406059...\frac{1}{2}\log \big(\frac{1+\sqrt{5}}{2}\big)=0.2406059... [Zag93]
0.24824 [Doc01a] hZ=log⁡hh_Z=\log \mathfrak{h} in his notation
0.24874 [F18] hZ=log⁡ζh_Z=\log \zeta in her notation, best known lower bound
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Essential minimum of the Zhang-Zagier height ($C_{82}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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