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Perfect Forms and Torsion Bounds in Algebraic K-Theory over Totally Real and CM Fields

  • Alejandro de la Torre Durán
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Abstract

We extend the method of Van Woerden \cite{woerden} to totally real and CM number fields to obtain an upper bound for the number of GLo−equivalence classes of perfect forms. Moreover, by considering appropriate barycentric subdivisions of the Voronoi tessellation, we extend these estimates to the number of GLo-orbits of k-dimensional faces of the Voronoi tessellation. As a consequence, we improve the asymptotic order of the bounds for the torsion part of Km(rint) obtained by Soulé \cite{soule_hermitian_bounds} and Houriet \cite{Juliet_PHD}. We emphasise the improvement of the general bound for F=ℚ, unchanged since Soulé's work, and the independence of the discriminant exponent from the field degree for totally real and CM fields.

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Cite

BibTeXRIS
BibTeX
@misc{duran2026perfect,
  title = {{Perfect Forms and Torsion Bounds in Algebraic $K$-Theory over Totally Real and CM Fields}},
  author = {Alejandro de la Torre Durán},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06300},
  eprint = {2611.06300},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06300},
}
RIS
TY  - UNPB
TI  - Perfect Forms and Torsion Bounds in Algebraic K-Theory over Totally Real and CM Fields
AU  - Alejandro de la Torre Durán
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06300
UR  - https://arxiv.org/abs/2611.06300
AB  - We extend the method of Van Woerden \cite{woerden} to totally real and CM number fields to obtain an upper bound for the number of GLo−equivalence classes of perfect forms. Moreover, by considering appropriate barycentric subdivisions of the Voronoi tessellation, we extend these estimates to the number of GLo-orbits of k-dimensional faces of the Voronoi tessellation. As a consequence, we improve the asymptotic order of the bounds for the torsion part of K_m(rint) obtained by Soulé \cite{soule_hermitian_bounds} and Houriet \cite{Juliet_PHD}. We emphasise the improvement of the general bound for F=ℚ, unchanged since Soulé's work, and the independence of the discriminant exponent from the field degree for totally real and CM fields.
ER  -