In a recent publication Roland Bacher showed that the number p d of non-similar perfect d-dimensional quadratic forms satisfies e Ω ( d ) < p d < e O ( d 3 log ( d ) ) . We improve the upper bound to e O ( d 2 log ( d ) ) by a volumetric argument based on Voronoi's first reduction theory.
Perfect quadratic form, Perfect lattice
Advances in Mathematics

van Woerden, W.P.J. (2020). An upper bound on the number of perfect quadratic forms. Advances in Mathematics, 365. doi:10.1016/j.aim.2020.107031