OP-12102OpenNumber theory

Schur–Siegel–Smyth trace constant ( $C_{86}$ )

Schur–Siegel–Smyth trace constant: determine $C_{86}$ or improve its bounds; the repository records 1.80203 (lower) and 1.8216 (upper).

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

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Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓

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Problem statement

An algebraic integer $\alpha$ of degree $d$ , with conjugates $\alpha_1,\dots,\alpha_d$ , is totally positive if all of its conjugates are real and strictly positive. Its absolute trace (or trace-to-degree ratio) is

\overline{\mathrm{tr}}(\alpha) \;:=\; \frac{\mathrm{tr}(\alpha)}{\deg(\alpha)} \;=\; \frac{1}{d}\sum_{i=1}^{d} \alpha_i .

Let $\mathcal{A}$ denote the set of totally positive algebraic integers. The Schur–Siegel–Smyth trace constant is the smallest limiting trace-to-degree ratio

C_{86} \;:=\; \liminf_{\alpha \in \mathcal{A}} \overline{\mathrm{tr}}(\alpha),

also written $\lambda^{\mathrm{SSS}}$ in the recent literature [OSS25]. Equivalently, $C_{86}$ is the supremum of the $\rho \ge 0$ for which all but finitely many totally positive algebraic integers $\alpha$ satisfy $\mathrm{tr}(\alpha) \ge \rho \deg(\alpha)$ ; so it is an essential rather than an absolute minimum, and finitely many exceptional $\alpha$ are always permitted.

The Schur–Siegel–Smyth trace problem — so named by Borwein [Bor02] — asks to fix $\rho < 2$ and show that all but finitely many totally positive algebraic integers $\alpha$ satisfy $\overline{\mathrm{tr}}(\alpha) > \rho$ ; equivalently, to show that $C_{86} = 2$ , i.e. that $2$ is the smallest limit point of the set of absolute traces [Fla19, §1]. The upper bound $C_{86} \le 2$ is classical, and the problem was to match it from below. This has now been resolved in the negative: Smith [Smi24] showed $C_{86} < 2$ , and the best bounds currently known are

1.80203 \;\le\; C_{86} \;\le\; 1.8216 .

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Background

Serre's obstruction. In an appendix to [AP08] (based on 1998 letters), Serre showed that the auxiliary function method described above can never prove a constant larger than $c_S$ , where $c_S \in (1.898302, 1.898303)$ [SO24, §1]. In the words of [Fla19], "Serre showed that this method does not give such an inequality for any $\rho$ larger than $1.898302\dots$ ". So the classical method could not have settled the conjecture $C_{86}=2$ , whatever the choice of polynomials. Serre's bound was a limitation of the method; it became a genuine upper bound on $C_{86}$ only after [Smi24]. (Note that [Fla19] locates this appendix in [AP07] rather than [AP08]; it is [AP08] that is published "with an appendix by Jean-Pierre Serre".)
A conjugate-dependent refinement. A parallel line of work bounds $\overline{\mathrm{tr}}(\alpha)$ in terms of the least conjugate $\alpha_1$ : [FRS97] obtained $\overline{\mathrm{tr}}(\alpha) \ge 1.6 + \alpha_1$ , improved to $1.66 + \alpha_1$ by [ABP06] and to $1.68 + \alpha_1$ by [Fla19], in each case with a finite explicit list of exceptions. These do not bound $C_{86}$ directly, since $\alpha_1$ can be arbitrarily small.
Primal and dual. The problem has a natural linear-programming formulation over probability measures $\mu$ on $[0,\infty)$ : minimize $\int x\, d\mu$ subject to energy conditions $\int \log\lvert Q(x)\rvert\, d\mu \ge 0$ indexed by integer polynomials $Q$ . The lower bounds above are dual certificates (auxiliary functions); the upper bounds are explicit feasible measures, which by the converse-Fekete theorem of [Smi24] are realized as limiting distributions of conjugates of actual algebraic integers. The remaining gap $[1.80203,\, 1.8216]$ is the gap between the best certificate and the best construction currently known.
Intermediate announcement. [OS23] announced the lower bound $1.7982$ as forthcoming work; the published form of that work is [OSS25], with the improved value $1.80203$ .
Related constants in this repository. $C_{82}$ , the essential minimum of the Zhang–Zagier height, has the same essential-minimum structure and is attacked with the same Smyth auxiliary-function machinery. $C_{40a}$ (Lehmer's Mahler measure constant) and $C_{40b}$ (asymptotic Dobrowolski constant) are the corresponding extremal problems for the Mahler measure of an algebraic integer rather than its trace.
Variants. Analogues of the trace problem have been studied for other symmetric functions of the conjugates and for restricted classes of algebraic integers, including totally positive reciprocal integers, Rhin's measure [Fla24], and a matrix analogue (arXiv:2206.12871).
Caution on the literature. A paper titled "A generalization of the Schur–Siegel–Smyth trace problem" (K. Pratt, G. Shakan, A. Zaharescu, J. Math. Anal. Appl. 436 (2016), 489–500, arXiv:1511.08837) was retracted in 2023 (J. Math. Anal. Appl. 518 (2023), 126784, DOI: 10.1016/j.jmaa.2022.126784). It still appears prominently in searches on this topic.

References (from the repository page)

[Sch18] Schur, I. Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Mathematische Zeitschrift 1 (1918), 377–402. DOI: 10.1007/BF01465096.
[Sie45] Siegel, C. L. The trace of totally positive and real algebraic integers. Annals of Mathematics 46 (1945), 302–312. DOI: 10.2307/1969025.
[Bor02] Borwein, P. Computational Excursions in Analysis and Number Theory. CMS Books in Mathematics 10, Springer-Verlag, New York, 2002. The source of the name "Schur–Siegel–Smyth trace problem".
[Smy84] Smyth, C. J. Totally positive algebraic integers of small trace. Annales de l'Institut Fourier (Grenoble) 34 (1984), no. 3, 1–28. DOI: 10.5802/aif.975. This is the paper cited for the $1.7719$ bound by [Fla19]; [OSS25] cite instead Smyth, C. J. The mean values of totally real algebraic integers. Mathematics of Computation 42 (1984), no. 166, 663–681, DOI: 10.1090/S0025-5718-1984-0736460-2.
[FGR99] Flammang, V.; Grandcolas, M.; Rhin, G. Small Salem numbers. In: Number Theory in Progress, Vol. 1 (Zakopane-Kościelisko, 1997), de Gruyter, Berlin, 1999, pp. 165–168. Note that the stated subject of this note is the computation of all Salem numbers $< 1.3$ of degree $\le 40$ ; the trace bound $1.7735$ is attributed to it by [Fla19] and [OSS25] and is presumably obtained there as an auxiliary result.
[FRS97] Flammang, V.; Rhin, G.; Smyth, C. J. The integer transfinite diameter of intervals and totally real algebraic integers. Journal de Théorie des Nombres de Bordeaux 9 (1997), no. 1, 137–168.
[Wu03] Wu, Q. On the linear independence measure of logarithms of rational numbers. Mathematics of Computation 72 (2003), 901–911.
[MS04] McKee, J.; Smyth, C. J. Salem numbers of trace $-2$ and traces of totally positive algebraic integers. In: Algorithmic Number Theory (ANTS-VI), Lecture Notes in Computer Science 3076, Springer, 2004, pp. 327–337. DOI: 10.1007/978-3-540-24847-7_25.
[ABP06] Aguirre, J.; Bilbao, M.; Peral, J. C. The trace of totally positive algebraic integers. Mathematics of Computation 75 (2006), no. 253, 385–393. DOI: 10.1090/S0025-5718-05-01776-X.
[AP07] Aguirre, J.; Peral, J. C. The integer Chebyshev constant of Farey intervals. Publicacions Matemàtiques 51 (2007), Proceedings of the Primeras Jornadas de Teoría de Números, 11–27. DOI: 10.5565/publmat_pjtn05_01.
[AP08] Aguirre, J.; Peral, J. C. The trace problem for totally positive algebraic integers. With an appendix by Jean-Pierre Serre. In: Number Theory and Polynomials, London Mathematical Society Lecture Note Series 352, Cambridge University Press, 2008, pp. 1–19. DOI: 10.1017/CBO9780511721274.003.
[Fla09] Flammang, V. Trace of totally positive algebraic integers and integer transfinite diameter. Mathematics of Computation 78 (2009), no. 266, 1119–1125. DOI: 10.1090/S0025-5718-08-02120-0.
[McK11] McKee, J. Computing totally positive algebraic integers of small trace. Mathematics of Computation 80 (2011), no. 274, 1041–1052. DOI: 10.1090/S0025-5718-2010-02424-X.
[LW11] Liang, Y.; Wu, Q. The trace problem for totally positive algebraic integers. Journal of the Australian Mathematical Society 90 (2011), no. 3, 341–354. DOI: 10.1017/S1446788711001030.
[Fla16] Flammang, V. Une nouvelle minoration pour la trace absolue des entiers algébriques totalement positifs. Preprint (2016), HAL: hal-01346165. The same title and result were posted as arXiv:1907.09407 (2019); [OSS25] cite the arXiv posting under the key [Fla19] while dating the result to 2016.
[Fla19] Flammang, V. The absolute trace of totally positive algebraic integers. International Journal of Number Theory 15 (2019), no. 1, 173–181. DOI: 10.1142/S1793042119500064. A different paper from [Fla16]: it proves the conjugate-dependent bound $\overline{\mathrm{tr}}(\alpha) \ge 1.68 + \alpha_1$ , and its introduction is the source of several of the sharper historical values quoted in the table above.
[WWW21] Wang, C.; Wu, J.; Wu, Q. Totally positive algebraic integers with small trace. Mathematics of Computation 90 (2021), no. 331, 2317–2332. DOI: 10.1090/mcom/3636.
[OS23] Orloski, B. J.; Talebizadeh Sardari, N. Limiting distributions of conjugate algebraic integers. Preprint (2023). arXiv:2302.02872.
[SO24] Talebizadeh Sardari, N.; Orloski, B. J. A quantitative converse of Fekete's theorem. Preprint (2023; v2, 25 March 2024). arXiv:2304.10021.
[Smi24] Smith, A. Algebraic integers with conjugates in a prescribed distribution. Annals of Mathematics 200 (2024), no. 1, 71–122. DOI: 10.4007/annals.2024.200.1.2. arXiv:2111.12660.
[Fla24] Flammang, V. To answer a question of Professor Georges Rhin. Preprint (2024). arXiv:2401.12951.
[OSS25] Orloski, B. J.; Talebizadeh Sardari, N.; Smith, A. New lower bounds for the Schur–Siegel–Smyth trace problem. Mathematics of Computation 94 (2025), no. 354, 2005–2040. DOI: 10.1090/mcom/4004. arXiv:2401.03252.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 1.80203; best known upper bound 1.8216. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound	Reference	Comments
$2$	Classical	For $n \ge 3$ the totally positive algebraic integer $\alpha_n = 4\cos^2(\pi/n) = 2 + 2\cos(2\pi/n)$ has degree $\varphi(n)/2$ and $\overline{\mathrm{tr}}(\alpha_n) = 2 + 2\mu(n)/\varphi(n)$ , which is $< 2$ for $n$ prime and tends to $2$ . Corresponds to the trivial case $\Lambda_\emptyset = 2$ in the framework of [SO24].
$1.89831$	[Smi24]	"There are infinitely many totally positive algebraic integers $\alpha$ with $\mathrm{tr}(\alpha) < 1.89831 \cdot \deg(\alpha)$ ." Obtained by combining Serre's measure $\nu_S$ (appendix to [AP08]) with Smith's converse to Fekete's theorem, which shows that the necessary conditions on a limiting distribution of conjugates are also sufficient. This is the result that disproves the conjecture $C_{86} = 2$ . Stated as $C_{86} \le 1.898304$ in [SO24].
$1.8216$	[SO24]	Explicit measures satisfying the conditions of [Smi24], optimized by gradient descent. Corollary 1.10 of [SO24] gives $\lvert \Lambda_A - 1.8215998 \rvert \le 10^{-7}$ for $A = \{x,\ 1-x,\ x^2-3x+1,\ x^3-5x^2+6x-1\}$ , and $C_{86} \le \Lambda_A$ . Intermediate values in the same family: $\Lambda_{\{x\}} \approx 1.898302$ (Serre's constant), then $1.84701204$ and $1.8224798$ . Preprint; not listed as journal-published as of this writing.

Known lower bounds

Bound	Reference	Comments
$1$	Trivial	By AM–GM, $\overline{\mathrm{tr}}(\alpha) \ge \lvert N(\alpha)\rvert^{1/d} \ge 1$ , since the norm of a nonzero algebraic integer is a nonzero rational integer.
$\sqrt{e} \approx 1.6487$	[Sch18]	Schur, via the arithmetic–geometric mean inequality applied to the discriminant.
$\approx 1.73361$	[Sie45]	Siegel; the bound is $e(1+\nu^{-1})^{-\nu}$ for a root $\nu$ of a transcendental equation.
$1.7719$	[Smy84]	Smyth's auxiliary function method: if $x - \sum_Q c_Q \log\lvert Q(x)\rvert \ge c$ for all $x > 0$ , with $c_Q > 0$ and $Q$ ranging over a finite set of integer polynomials, then $\overline{\mathrm{tr}}(\alpha) \ge c$ for all but finitely many $\alpha$ . Every subsequent entry in this table refines the choice and optimization of the $Q$ and $c_Q$ .
$1.7735$	[FGR99]	Dated 1997 in [Fla19] and [OSS25] (the conference volume appeared in 1999). See the note on this reference below.
$1.7783786$	[MS04]	Value as reported in [Fla19]; rounded to $1.7783$ in the historical table of [OSS25].
$1.78002$	[ABP06]
$1.7836$	[AP07]
$1.784109$	[AP08]	Value as reported in [Fla19]; rounded to $1.7841$ in [OSS25]. This paper also carries Serre's appendix; see "Additional comments and links" below.
$1.78702$	[Fla09]	Introduces the recursive algorithm: the auxiliary polynomials are generated inductively via LLL rather than found heuristically, building on the link between auxiliary functions and the integer transfinite diameter established in [Wu03].
$1.78839$	[McK11]	First to use auxiliary polynomials with complex roots.
$1.79193$	[LW11]	Value as reported in [Fla19]; rounded to $1.7919$ in [OSS25].
$1.792812$	[Fla16]	A variant in the use of the recursive algorithm of [Fla09]. Holds unless the minimal polynomial of $\alpha$ is one of $x-1$ , $x^2-3x+1$ , $x^3-5x^2+6x-1$ , $x^4-7x^3+13x^2-7x+1$ , $x^4-7x^3+14x^2-8x+1$ . Rounded to $1.7928$ and dated 2016 in the historical table of [OSS25], which cites the later arXiv posting.
$1.793145$	[WWW21]	Rounded to $1.7931$ in the historical table of [OSS25]; the value $1.793145$ is as reported in [Fla24].
$1.80203$	[OSS25]	Adds constraints coming from the logarithmic energy of the limiting measure to Smyth's linear program, sharply reducing the number of variables required. The authors prove existence and uniqueness of an optimal solution, characterize it in terms of polynomials, and solve numerically by gradient descent. Schur's and Siegel's bounds are recovered as the two simplest cases.

References

[1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Schur–Siegel–Smyth trace constant ($C_{86}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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https://teorth.github.io/optimizationproblems/constants/86a.html

Constant page from github.com/teorth/optimizationproblems at commit 2c1968c, Apache-2.0. The maintainers review submissions for plausibility and replicability but do not certify correctness; asterisked bounds have minimal verification.

Source date: Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differ
Imported: 2026-10-10

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Posed by: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers) and contributors
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Source: Curated problem listOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOriginal statement ↗
Citation: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Schur–Siegel–Smyth trace constant ($C_{86}$)
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