Martinet's constant for totally real number fields ($C_{87}$)

• The nearest visible improvement, and why it is out of reach. The record above cuts by $\tau_{\mathfrak{p}_{13}}^2$, because [HMR2019, Proposition 1.3(iii)] gives only that the generator of tame inertia $\tau_{\mathfrak{p}_{13}}$ has depth at least two in $G_S$. [HMR2019, §3.3.4] observes that if that depth were greater than two, one could cut by $\tau_{\mathfrak{p}_{13}}$ itself; the Golod–Shafarevich polynomial would then be at most $1 - 9t + 20t^2 + t^3$, which still has a root in $(0,1)$. The same degree-$12$ field $\mathrm{K}$ would then have an infinite $2$-Hilbert class field tower — no ramification at $13$ needed at all — and the totally real record would fall from $857.567$ to $\mathrm{rd}_{\mathrm{K}} < 770.644$, a $10\%$ improvement with no new field. The authors write that they “do not see how to check the depth of $\tau_{\mathfrak{p}_{13}}$ in $G_S$”, and the question opening [HMR2019, §5] raises the general obstruction: given $S = \{\mathfrak{p}\}$ with $\mathfrak{p}$ coprime to $p$ and $G_S$ infinite, how deep can the generator $\tau_{\mathfrak{p}}$ of tame inertia be? This is a hope rather than a bound, so nothing in the table above changes; but it is the cheapest known route to the next record.
• 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 (from the repository page)

• [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).

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 60.8395; best known upper bound 857.567. Values marked with an asterisk have minimal verification.

Known upper bounds

Known lower bounds