An upper bound on the number of perfect quadratic forms

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.

Additional Metadata
Keywords	Perfect quadratic form, Perfect lattice
Persistent URL	doi.org/10.1016/j.aim.2020.107031
Journal	Advances in Mathematics
Organisation	Cryptology
Citation APA Style AAA Style APA Style Cell Style Chicago Style Harvard Style IEEE Style MLA Style Nature Style Vancouver Style American-Institute-of-Physics Style Council-of-Science-Editors Style BibTex Format Endnote Format RIS Format CSL Format DOIs only Format	van Woerden, W. (2020). An upper bound on the number of perfect quadratic forms. Advances in Mathematics, 365. doi:10.1016/j.aim.2020.107031

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