An upper bound on the number of perfect quadratic forms

Abstract

In a recent preprint on arXiv Roland Bacher showed that the number $p_d$ of non-similar perfect $d$-dimensional quadratic forms satisfies $e^{\Omega (d)}<p_d<e^{O(d^3\log (d))}$. We improve the upper bound to $e^{O(d^2\log (d))}$ by a volumetric argument based on Voronoi's first reduction theory.