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

We extend the method of Van Woerden {woerden} to totally real and CM number fields to obtain an upper bound for the number of -equivalence classes of perfect forms. Moreover, by considering appropriate barycentric subdivisions of the Voronoi tessellation, we extend these estimates to the number of -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() obtained by Soulé {soule_hermitian_bounds} and Houriet {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.