Martinet’s constant for totally real number fields
Description of constant
For a number field $K$, let $\Delta_K$ denote the absolute value of its discriminant and let $[K:\mathbb{Q}]$ denote its degree. The root discriminant of $K$ is \(\mathrm{rd}(K) := \Delta_K^{1/[K:\mathbb{Q}]}.\) Martinet’s constant for totally real number fields (also called the asymptotic root-discriminant infimum in the totally real signature) is \(C_{87} := \liminf_{K}\ \mathrm{rd}(K),\) where the $\liminf$ is over all totally real number fields $K$ ordered by degree. Equivalently, $C_{87}$ is the smallest constant $\alpha$ such that there exist infinite towers $K_1 \subset K_2 \subset \dots$ of totally real number fields with $\mathrm{rd}(K_i) \to \alpha$ from above (or $\le \alpha + o(1)$).
Upper bounds on $C_{87}$ come from explicit constructions of infinite tamely ramified towers of totally real number fields with controlled root discriminant, via the Golod–Shafarevich criterion ([GS1964]) applied to restricted-ramification pro-$p$ Galois groups; lower bounds come from the Odlyzko–Serre analysis of Weil’s explicit formula, either unconditional or under GRH.
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $1058.565$ | [Mar1978] | Martinet’s original construction of infinite $2$-class field towers of totally real number fields. |
| $954.293$ | [HM2002] | Hajir–Maire, refined Golod–Shafarevich with tame ramification. |
| $913.493$ | [Mar2006] | Martin, further refinement of the [HM2002] construction; the degree-$8$ field of [HMR2019, §3.3.1] has discriminant $3^4\cdot 5^4\cdot 7^4\cdot 13^2\cdot 29^4\cdot 53^2\cdot 109^2$ and root discriminant $< 913.4927$. |
| $857.567$ | [HMR2019] | Hajir–Maire–Ramakrishna, “cutting towers” via the refined Golod–Shafarevich criterion. The totally real example [HMR2019, §3.3.3] is a degree-$12$ field $\mathrm{K}$ with $\mathrm{rd}_{\mathrm{K}} < 770.6432$, cut at a single prime above $13$ of norm $13$, giving $\mathrm{rd}_{\mathrm{K}_S^{[1]}} = \mathrm{rd}_{\mathrm{K}}\cdot 13^{\frac{1}{12}(1-\frac{1}{2})} < 857.5662\dots$ — a saving of a factor $13^{1/24}$. Current record. |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $1$ | Trivial | $\Delta_K \ge 1$ for every number field. |
| $60.8395\dots$ | [Odl1990] | Unconditional Odlyzko lower bound on the root discriminant of any totally real number field of large degree. |
| $215.333\dots$ | [Odl1990] | GRH-conditional Odlyzko–Serre lower bound in the totally real signature. |
Additional comments and links
- Sibling: totally complex Martinet constant. There is an entirely analogous constant $C^{-}$ defined by taking the $\liminf$ over totally complex (a.k.a. totally imaginary) number fields. For that variant the corresponding history is: Martinet 1978 gives $92.368$; Hajir–Maire 2002 give $82.1004$; and Hajir–Maire–Ramakrishna 2019 give the current record $78.427$ (via a $9$-th root class-field tower, with $\mathrm{rd} \le 78.427$). The GRH lower bound in this signature is $44.763$ and the unconditional lower bound is $22.382$ ([Odl1990]). If the totally complex variant later becomes the subject of separate literature it could be split off as
87b. - Martinet’s constant has geometric implications far beyond number theory. Tsfasman–Vlăduț [TV1991] used towers with bounded root discriminant to construct explicit lattice sphere packings in high dimension. More recently, the same towers underpin the large-degree number-field constructions used in the OpenAI counterexample to the Erdős unit distance conjecture ([ABGLSSTWW2026]) and in the disproof of the sum-product conjecture over the reals (84b; [BSSZ2026]). In both of these applications one needs $\mathrm{rd} \le O(1)^d$, which is exactly the Martinet regime; the [HMR2019] bound $857.567$ is precisely the “Martinet’s constant” $C_2 \le 857.57$ quoted in [BSSZ2026, §5].
- Conditional vs. unconditional. All the upper bounds above are unconditional: they are explicit constructions of infinite towers. The two [Odl1990] lower bounds have very different status — the unconditional bound $60.8$ can be improved only via genuine analytic progress on zero-density estimates for Dedekind zeta functions, whereas the GRH-conditional $215.3$ bound would immediately follow from a proof of GRH. Closing the gap between $215.3$ and $857.6$ would in particular disprove (a strong form of) GRH.
- The problem is not known to be finite in any effective sense at present: no one has proved a subexponential-in-$d$ upper bound on the smallest root discriminant of a totally real number field of degree $d$, only that the $\liminf$ is at most $857.567$.
References
- [GS1964] Golod, E. S.; Shafarevich, I. R. On the class field tower. Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 261–272.
- [Mar1978] Martinet, Jacques. Tours de corps de classes et estimations de discriminants. Inventiones Mathematicae 44 (1978), no. 1, 65–73. DOI: 10.1007/BF01389902.
- [Odl1990] Odlyzko, Andrew M. Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results. Journal de Théorie des Nombres de Bordeaux 2 (1990), no. 1, 119–141.
- [HM2002] Hajir, Farshid; Maire, Christian. Tamely ramified towers and discriminant bounds for number fields — II. Journal of Symbolic Computation 33 (2002), no. 4, 415–423. DOI: 10.1006/jsco.2001.0514.
- [Mar2006] Martin, John S. Improved root-discriminant bounds via ramification-restricted class-field towers. (2006). Cited as “Martin [25]” in [HMR2019]; specific bibliographic details should be verified.
- [HMR2019] Hajir, Farshid; Maire, Christian; Ramakrishna, Ravi. Cutting towers of number fields. arXiv:1901.04354 (2019).
- [TV1991] Tsfasman, Michael A.; Vlăduţ, Serge G. Asymptotic properties of global fields and generalized Brauer–Siegel theorem. Moscow Mathematical Journal (early 1990s work reviewed in later papers); see e.g. Tsfasman, M. A. Global fields, codes and sphere packings. Astérisque 198–200 (1991), 373–396.
- [ABGLSSTWW2026] Alon, N.; Bloom, T. F.; Gowers, W. T.; Litt, D.; Sawin, W.; Shankar, A.; Tsimerman, J.; Wang, V.; Wood, M. M. Remarks on the disproof of the unit distance conjecture. arXiv:2605.20695 (2026).
- [BSSZ2026] Bloom, T. F.; Sawin, W.; Schildkraut, C.; Zhelezov, D. The sum-product conjecture is false for real numbers. arXiv:2605.28781 (2026).
Contribution notes
Prepared with assistance from Claude Opus 4.7, which read the [HMR2019] preprint (parsed via pdftotext) and drew on already-cached references from earlier work on this repository. Historical citations before [HMR2019] were transcribed from the “new records” table in [HMR2019, §3]; the [Mar2006] entry in particular is cited via [HMR2019] and its journal reference should be verified before publication. The exact numerical values quoted for [Odl1990] are standard textbook figures; their exact form in the original 1990 survey should be checked.