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A trace upper-bound certificate with classical auxiliaries on an adaptive density

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

Contributed by Ricardo Pinho

Version 3 / Oct 31, 2026 / CC BY 4.0

Abstract

An exactly specified rational piecewise-linear density gives C86≤1.80560940C_{86}\le1.80560940 for the Schur--Siegel--Smyth trace constant. It uses 28 classical auxiliary polynomials, ten with nonreal roots, and 5,375 rational nodes. Their recorded list occurrences are: Flammang 2019 (all 28), Liang--Wu 2011 (26), Flammang 2009 (21), McKee 2011 (21), Tomar's public A22 pool (18), Aguirre--Peral 2007 (16), and Flammang 2024 (9). The density refines infrastructure originating in Tomar's adaptive-node certificate. We give the full deduction from finite potential and logarithmic-integral checks through Smith's arithmetic-measure criterion and converse theorem, and describe the rigorous continuous enclosures. The contribution is the explicit feasible measure and its smaller upper bound. 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), 71--122; Theorem 1.5 and Proposition 5.15. https://doi.org/10.4007/annals.2024.200.1.2. Author manuscript: https://arxiv.org/html/2111.12660v2.DOI
  2. B. Orloski and N. Talebizadeh Sardari, A quantitative converse of Fekete's theorem, preprint, version 2 (2024). https://arxiv.org/abs/2304.10021v2.link
  3. B. Orloski, N. Talebizadeh Sardari and A. Smith, New lower bounds for the Schur--Siegel--Smyth trace problem, Mathematics of Computation 94 (2025), 2005--2040. https://doi.org/10.1090/mcom/4004.DOI
  4. T. Ransford, Potential Theory in the Complex Plane , London Mathematical Society Student Texts 28, Cambridge University Press, 1995; harmonic minimum principle and capacity of a real interval.
  5. V. Flammang, Trace of totally positive algebraic integers and integer transfinite diameter, Mathematics of Computation 78 (2009), 1119--1125. https://doi.org/10.1090/S0025-5718-08-02120-0.DOI
  6. J. McKee, Computing totally positive algebraic integers of small trace, Mathematics of Computation 80 (2011), 1041--1052. https://doi.org/10.1090/S0025-5718-2010-02424-X.DOI
  7. V. Flammang, Une nouvelle minoration pour la trace des entiers algébriques totalement positifs, preprint (2019), first polynomial list. https://arxiv.org/abs/1907.09407v1.link
  8. Y. Liang and Q. Wu, The trace problem for totally positive algebraic integers, Journal of the Australian Mathematical Society 90 (2011), 341--354; Table 1. https://doi.org/10.1017/S1446788711001030.DOI
  9. J. Aguirre and J. C. Peral, The integer Chebyshev constant of Farey intervals, Publicacions Matemàtiques (2007), Proceedings of the Primeras Jornadas de Teoría de Números, 11--27; Table 2. https://ddd.uab.cat/pub/pubmat/pubmat_a2007vEXTRA/pubmat_a2007vextrap11.pdf.link
  10. V. Flammang, To answer a question of Professor Georges Rhin, preprint (2024), Rhin-measure and sector lists. https://arxiv.org/abs/2401.12951v1.link
Show all 17 referencesShow fewer references
  1. R. Pinho, C86 step-density certificate, historical account and data description, at the pinned repository commit of adaptive . https://github.com/rpinho/math-certificates/tree/848f899c973666f64b73fb6f37760849a600285f/c86.link
  2. D. Koshta, C86 trace certificate, unrefereed computational result (2026), as recorded in the Tao table, entry 86a. https://teorth.github.io/optimizationproblems/constants/86a.html.link
  3. V. Patankar, Improved bounds for C16, C1a, C1c, C86, C89 (exact certificates), PR 270 (2026). https://github.com/teorth/optimizationproblems/pull/270.link
  4. T. S. Tomar, Upper and lower bounds for the Schur--Siegel--Smyth trace problem, unrefereed preprint, first posted 21 October 2026. https://www.mefyi.com/papers/schur-siegel-smyth-trace-bounds/.link
  5. R. Pinho, C86 classical half-cell certificate, exact data, checkers and receipts (2026), at the pinned commit of adaptive . https://github.com/rpinho/math-certificates/tree/848f899c973666f64b73fb6f37760849a600285f/c86/classical-half-cells.link
  6. R. Pinho, C86 adaptive-density certificate, exact data, proof, checkers, receipts and AI review (2026), commit 848f899c973666f64b73fb6f37760849a600285f . https://github.com/rpinho/math-certificates/tree/848f899c973666f64b73fb6f37760849a600285f/c86/adaptive-cone.link
  7. R. Pinho, C86: upper bound 1.80560940 with classical auxiliaries on an adaptive density (unrefereed), PR 298 (2026). https://github.com/teorth/optimizationproblems/pull/298.link

Version history

  1. v1Submitted by Ricardo PinhoInitial depositSupersededOct 30, 2026
  2. v2Submitted by Ricardo PinhoSubmitted as a new versionSupersededOct 31, 2026
  3. v3Submitted by Ricardo PinhoSubmitted as a new versionCurrentNov 03, 2026