Schur–Siegel–Smyth trace constant
Description of constant
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
\[1.80203 \;\le\; C_{86} \;\le\; 1.8084990 .\]The upper bound $1.8084990$ is a computer-assisted certificate that is currently asterisked (see the table below); the best upper bound without the asterisk is $1.8216$ [SO24].
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. |
| $1.8117251$* | [Pin26] | The construction of [SO24] with $A$ enlarged from their four polynomials to eleven. The polynomials $Q \in A$, the first four being the [SO24] set, with their weights $b_Q$ (these decimals are the exact rational weights): $x$ ($b_Q = 0.038759530583$), $1-x$ ($b_Q = 0.079868388122$), $x^2-3x+1$ ($b_Q = 0.04631766723$), $x^3-5x^2+6x-1$ ($b_Q = 0.053369420893$), $x^4-7x^3+14x^2-8x+1$ ($b_Q = 0.019753937664$), $x^4-7x^3+13x^2-7x+1$ ($b_Q = 0.036545507753$), $x^5-9x^4+28x^3-35x^2+15x-1$ ($b_Q = 0.016049367094$), $x^5-9x^4+26x^3-29x^2+11x-1$ ($b_Q = 0.009731525491$), $x^5-9x^4+27x^3-32x^2+13x-1$ ($b_Q = 0.007106935573$), $x^3-6x^2+9x-1$ ($b_Q = 0.018100957401$), $x^5-9x^4+27x^3-31x^2+12x-1$ ($b_Q = 0.002893658474$). The measure $\mu$ is a step density on the cells of width $1/200$ in $[0, 6]$, with $629$ nonzero cells, support in $[1/20,\ 1037/200]$ and mass $m = 39999999999997/40000000000000$; the bound is $\int x\, d\mu / m = 144938007612277893/79999999999994000 = 1.81172509515\ldots$. Checked in exact rational and arb ball arithmetic (certificate and checker at commit 116a58f): each $Q \in A$ is irreducible with only real roots; $\sum_Q b_Q \deg Q \approx 0.82978\, m < m$; $\int \log\lvert Q \rvert\, d\mu \approx 10^{-4} > 0$ for each $Q \in A$; and $U_\mu(x) - \sum_Q b_Q \log\lvert Q(x) \rvert \ge 0$ for every $x$ in the support (rigorous bisection), hence on all of $\mathbb{C}$ by the minimum principle. These are the hypotheses of Smith’s sufficiency criterion [Smi24], whose theorem then gives $C_{86} \le \int x\, d\mu / m$. Asterisked: a self-archived computer-assisted certificate, not yet independently replayed. |
| $1.8084990$* | [Kos2026] | The construction of [SO24] as in [Pin26], with the auxiliary set $A$ chosen by solving the discretised [SO24] primal problem over all $883$ irreducible monic integer polynomials of degree $\le 7$ with all roots in $[0, 5.3]$ and trace at most twice the degree. $19$ receive positive weight: the eleven of [Pin26]; $x-2$, $x^2-4x+1$, $x^2-4x+2$ and $x^3-6x^2+9x-3$, of trace-to-degree ratio $2$; and $x^6-11x^5+43x^4-73x^3+53x^2-15x+1$, $x^6-11x^5+43x^4-72x^3+51x^2-14x+1$, $x^7-13x^6+64x^5-150x^4+172x^3-89x^2+18x-1$, $x^7-13x^6+63x^5-143x^4+157x^3-78x^2+16x-1$. Auxiliary polynomials whose trace-to-degree ratio exceeds the bound are allowed: [Smi24, Prop. 5.15] and [SO24, Cor. 1.6] place no condition on their traces. The measure $\mu$ is a step density on cells of width $1/200$, with $633$ nonzero cells, support in $[11/200,\ 519/100]$ and mass $m = 199999999999999703/200000000000000000$; the bound is $\int x\, d\mu / m = 144679917840593158601/79999999999999881200 = 1.80849897300\ldots$. Checked in exact rational and arb ball arithmetic (certificate and checkers at commit 90c57242b2f43aa942257154be6731c29898b537): each $Q \in A$ is irreducible with only real roots; $\sum_Q b_Q \deg Q \approx 0.83542\, m < m$; $\int \log\lvert Q \rvert\, d\mu \ge 1.99 \cdot 10^{-6}$ for each $Q \in A$; and $U_\mu(x) - \sum_Q b_Q \log\lvert Q(x) \rvert \ge 10^{-7}$ for every $x$ in the support (rigorous bisection), hence $\ge 0$ on all of $\mathbb{C}$ by the minimum principle. The certificate passes the unmodified checker of [Pin26], and a second checker written from the certificate format and [Smi24], [SO24] only; both certify the margin $10^{-7}$ (the checker of [Pin26] with its positivity test raised to $10^{-7}$). These are the hypotheses of Smith’s sufficiency criterion [Smi24], whose theorem then gives $C_{86} \le \int x\, d\mu / m$. Asterisked: a self-archived computer-assisted certificate, not yet independently replayed. |
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. |
Additional comments and links
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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”.)
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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.
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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 $[1.80203,\, 1.8084990]$ is the gap between the best certificate and the best construction currently known.
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Intermediate announcement. [OS23] announced the lower bound $1.7982$ as forthcoming work; the published form of that work is [OSS25], with the improved value $1.80203$.
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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.
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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).
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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
- [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.
- [Pin26] Pinho, R. A certificate for $C_{86} \le 1.8117251$. GitHub repository rpinho/c86-trace-certificate, commit
116a58ff99681226ce6299d4a02eeff1e7d3c5c7(MIT), 2026: the exact certificate (cert_n1200_A11.json), an independent checker in exact rational and arb ball arithmetic, and the derivation of the bound from [Smi24]. Not refereed. - [Kos2026] Koshta, Dennis, with Claude (Anthropic, Claude Opus 5.5). A certificate for $C_{86} \le 1.8084990$. GitHub repository DennisKoshta/c86-sss-trace-certificate, commit
90c57242b2f43aa942257154be6731c29898b537(MIT), 23 Oct 2026: the exact certificate (c86_A19d7_robust_1.8084990.json), an independent checker in exact rational and arb ball arithmetic, a margin patch for the checker of [Pin26], and the derivation of the bound from [Smi24]. Not refereed.
Contribution notes
Prepared with assistance from Claude Opus 5, which read the arXiv abstracts and HTML full texts of [OSS25], [SO24], [OS23] and [Fla24], the abstract of [Smi24], and the full text of [Fla19] and the first page of [FGR99] as supplied by the submitter; bibliographic data for the remaining historical references was retrieved via Crossref and Semantic Scholar. The historical lower-bound table follows the table in the introduction of [OSS25], with several values sharpened against the introduction of [Fla19]. The pre-2021 entries have not been checked against the primary sources themselves, and all references should be independently verified before citation.
The [Pin26] upper-bound row (October 2026): the polynomial search, the certificate, the checker and the row text were prepared with AI assistance (Claude) under the direction of the submitter, R. Pinho, who reviewed the argument and the references.
The [Kos2026] upper-bound row was found and prepared with Claude (Anthropic, Claude Opus 5.5), directed by Dennis Koshta, who takes responsibility for the submission: the linear-programming search over auxiliary polynomial sets, the certificate, and the text. The certificate was checked by the unmodified checker of [Pin26] and by a second Claude agent with its own checker, written from the certificate format and [Smi24], [SO24] only, without access to the construction code; no human mathematician has checked the argument.