Submit progress

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

The 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 .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.