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A trace upper-bound certificate with complex-root auxiliaries

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Ricardo Pinho ↗

Version 1 / Oct 31, 2026 / CC BY 4.0

Abstract

A rational piecewise-linear measure with 23 auxiliary polynomials gives C86≤1.80697883C_{86}\le1.80697883 for the Schur--Siegel--Smyth trace constant. The proof uses Smith's arithmetic-measure criterion and converse theorem. Two degree-eight auxiliaries have nonreal roots; both polynomials already occur in Flammang (2009) and McKee (2011). Their use in this upper-bound certificate, rather than their discovery, is the change described here. This note is unrefereed, with no independent human review or formalization.

Provenance statement

Analysis and proof drafting by AI agents under my direction; I reviewed the results at summary level. AI agents also prepared certificates, checking software, and publication materials. No line-by-line human verification is claimed. AI is not an author.

References

  1. A. Smith, Algebraic integers with conjugates in a prescribed distribution, Annals of Mathematics 200 (2024), Theorem 1.5 and Proposition 5.15; https://arxiv.org/html/2111.12660v2.link
  2. B. Orloski and N. Talebizadeh Sardari, A quantitative converse of Fekete's theorem; https://arxiv.org/abs/2304.10021.link
  3. D. Niedbala Giraudin, trace upper-bound certificate, PR 270; https://github.com/teorth/optimizationproblems/pull/270.link
  4. T. Tomar, Schur--Siegel--Smyth trace bounds, unrefereed preprint (21 October 2026), certificate repository; https://github.com/tejasvi/schur-siegel-smyth-trace-bounds.link
  5. B. Orloski, N. Talebizadeh Sardari and A. Smith, trace-bound construction, Mathematics of Computation 94 (2025), 2005--2040; https://arxiv.org/abs/2401.03252.link
  6. V. Flammang, Trace of totally positive algebraic integers and integer transfinite diameter, Mathematics of Computation 78 (2009), 1119--1125, Table 1, rows 22--23; https://doi.org/10.1090/S0025-5718-08-02120-0.DOI
  7. J. McKee, Computing totally positive algebraic integers of small trace, Mathematics of Computation 80 (2011), 1041--1052, Table 1, rows 21 and 23; https://doi.org/10.1090/S0025-5718-2010-02424-X.DOI
  8. R. Pinho, C86 trace certificate , directory complex-auxiliaries , revision placeholder COMMIT_C86 ; https://github.com/rpinho/c86-trace-certificate/tree/e4d9c640ebaf922b8869a81f275c798c09ab89b1/complex-auxiliaries.link

Version history

  1. v1Submitted by Ricardo PinhoInitial depositCurrentOct 30, 2026