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The Schur-Siegel-Smyth trace problem: the lower bound 1.8022246

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 11, 2026 / CC BY 4.0

Abstract

We prove lambda_SSS >= 9011123/5000000 = 1.8022246 for the smallest limiting trace-to-degree ratio of totally positive algebraic integers. The previous best bound is 1.8022225 (Tomar). The certificate consists of fourteen integer polynomials with rational multipliers and a continuous piecewise-linear density with 8678 rational hat weights; its dual inequality is verified on (0,infinity) in ball arithmetic. For the same family of polynomials the value of the underlying optimisation problem is 1.8022247 to within 10^-9 (numerical), so the bound is within 1.1*10^-7 of what this family can give.

Author comments: 3 pages. Ancillary files: the exact certificate (cert_SSS_1802224600.json), a rigorous checker in ball arithmetic (check_sss.py, python-flint), its run log, an independent numerical cross-check (pointcheck_mp.py) and SHA-256 sums.

Provenance statement

The mathematics, the computations and the writing were produced with the assistance of an AI model; the author checked the statements and the references. The result is computer-assisted: it rests on the certificate and the verification programs in the ancillary files, which anyone can rerun.

Tools used

Anthropic
Claude Opus

References

  1. D. Niedbala Giraudin, A new lower bound for the Schur--Siegel--Smyth trace problem , arXiv:2609.29298 [math.NT] (2026).arXiv
  2. B. J. Orloski, N. Talebizadeh Sardari and A. Smith, New lower bounds for the Schur--Siegel--Smyth trace problem , Math. Comp. 94 (2025), no. 354, 2005--2040, doi:10.1090/mcom/4004; arXiv:2401.03252.arXivDOI
  3. A. Smith, Algebraic integers with conjugates in a prescribed distribution , Ann. of Math. (2) 200 (2024), no. 1, 71--122, doi:10.4007/annals.2024.200.1.2.DOI
  4. T. S. Tomar, Upper and lower bounds for the Schur--Siegel--Smyth trace problem , version 4, Zenodo, doi:10.5281/zenodo.23563968 (2026).DOI
  5. R. Pinho, A trace upper-bound certificate with classical auxiliaries on an adaptive density , Hexagon 2610.00695v3 (2026).

Version history

  1. v1 · 3 pagesSubmitted by David Niedbala Giraudin3 pages. Ancillary files: the exact certificate (cert_SSS_1802224600.json), a rigorous checker in ball arithmetic (check_sss.py, python-flint), its run log, an independent numerical cross-check (pointcheck_mp.py) and SHA-256 sums.CurrentNov 10, 2026