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

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

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

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.