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PreprintCC BY 4.0arXiv

A new lower bound for the Schur-Siegel-Smyth trace problem

  • David Niedbala Giraudin
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Abstract

We prove that lambda_SSS >= 1.80220 for the smallest limiting trace-to-degree ratio of totally positive algebraic integers, improving the bound 1.80203 obtained by Orloski, Talebizadeh Sardari and Smith. The proof exhibits an explicit probability measure on [0,8] together with eighteen integer polynomials, and verifies their dual inequality on the whole of [0,infinity) by interval bisection. The certificate assumes nothing about the data it is built from: an error in that data can only weaken the bound, never invalidate it.

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Cite

BibTeXRIS
BibTeX
@misc{giraudin2026lower,
  title = {{A new lower bound for the Schur-Siegel-Smyth trace problem}},
  author = {David Niedbala Giraudin},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29298},
  eprint = {2609.29298},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29298},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A new lower bound for the Schur-Siegel-Smyth trace problem
AU  - David Niedbala Giraudin
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29298
UR  - https://arxiv.org/abs/2609.29298
AB  - We prove that lambda_SSS >= 1.80220 for the smallest limiting trace-to-degree ratio of totally positive algebraic integers, improving the bound 1.80203 obtained by Orloski, Talebizadeh Sardari and Smith. The proof exhibits an explicit probability measure on [0,8] together with eighteen integer polynomials, and verifies their dual inequality on the whole of [0,infinity) by interval bisection. The certificate assumes nothing about the data it is built from: an error in that data can only weaken the bound, never invalidate it.
ER  -